# Why Energy Is Quantised: From Standing Waves to Coloured Molecules > Beginning with a string fixed at both ends, this lecture shows how boundary conditions select discrete standing waves and then applies the same reasoning to a particle confined in a box. It derives the allowed energies, demonstrates how a wider box compresses their spacing toward the classical limit, interprets the first probability densities and their nodes, and concludes by using the model to explain absorption trends and colour in conjugated dye molecules. - Canonical watch page: [Why Energy Is Quantised: From Standing Waves to Coloured Molecules](https://academa.ai/lectures/the-particle-in-a-box) - Publisher: [Academa, Inc.](https://academa.ai) - Subject: Physics - Published: 2026-08-28T22:51:52.000Z - Updated: 2026-08-28T22:51:52.000Z - Duration: PT802S (13 minutes 22 seconds) - Chapters: 5 - Views: 1 - Language: en-US - Access: Free - Video stream: [HLS content](https://academa.ai/media/l/01M14TZZSX238T89EBZHGVYW3N/0/dark/master.m3u8) - Audiovisual record: [Semantic JSON](https://academa.ai/media/l/01M14TZZSX238T89EBZHGVYW3N/0/semantic.json) - Thumbnail: [Image](https://academa.ai/media/l/01M14TZZSX238T89EBZHGVYW3N/0/dark/poster.jpg) ## Description Why confined waves admit discrete energies, how box size controls spacing, and how the same model explains trends in conjugated dye colours. ## Chapters - [00:00–02:5.434 · Only Certain Waves Fit](https://academa.ai/lectures/the-particle-in-a-box?t=0) - [02:5.434–04:44.938 · A Particle as a Confined Wave](https://academa.ai/lectures/the-particle-in-a-box?t=125.43370833333334) - [04:44.938–07:12.269 · How the Energy Gaps Shrink](https://academa.ai/lectures/the-particle-in-a-box?t=284.93758333333335) - [07:12.269–09:33.096 · Probability and the Places Never Found](https://academa.ai/lectures/the-particle-in-a-box?t=432.26925000000006) - [09:33.096–13:22 · Why Conjugated Molecules Have Colour](https://academa.ai/lectures/the-particle-in-a-box?t=573.0964166666668) ## Transcript ### [00:00 · Only Certain Waves Fit](https://academa.ai/lectures/the-particle-in-a-box?t=0) Why should energy ever come in separate amounts? I do not want to begin by declaring that it does. Let us begin with a familiar object instead: a stretched string, fixed at both ends, and the waves that can actually live on it. The string runs from zero to L. Each end is clamped, so neither endpoint can move, however strongly the rest of the string vibrates. That physical fact becomes a boundary condition. The displacement y must be zero at the left wall and zero again at the right wall. Try an arbitrary sinusoidal candidate. It begins correctly at zero, but look at its right end. The curve reaches the wall above the fixed point, so this shape cannot be a standing wave of this string. The simplest accepted shape places one half-wave between the walls. Both endpoints are nodes, and the entire string rises and falls as one arch. The next shape fits two half-waves. It still vanishes at both walls, and it gains another node exactly in the middle. A third shape fits three half-waves. Two interior nodes divide the string into three vibrating sections, with neighboring sections moving in opposite directions. We can continue forever, but not continuously. The length L must contain a whole number n of half-wavelengths. Solving for wavelength gives lambda n equals two L over n. The allowed wavelengths form a list: two L, L, two L over three, and so on. The same statement can be written using wave number k. Only n pi over L is allowed, with n equal to one, two, three, and onward. The important word is allowed. The material string could have many amplitudes, but its spatial patterns are selected by the two boundaries. Continuous guesses went in; a discrete family came out. ### [02:5.434 · A Particle as a Confined Wave](https://academa.ai/lectures/the-particle-in-a-box?t=125.43370833333334) Now replace the string by a particle trapped between two perfectly impenetrable walls. Quantum mechanics describes its state with a wavefunction, psi. What survives from the string argument? Here is the region available to the particle. Outside the two walls the particle cannot exist, so the wavefunction is zero there. At an ideal infinite wall, the wavefunction must meet that outside value continuously. Therefore psi is zero at x equals zero and again at x equals L. The first allowed state is one smooth arch. This is not a little particle following the curve. The curve is the wavefunction that will determine probabilities. The second state has two lobes and one interior zero. The third has three lobes and two interior zeros. They are the same spatial fitting patterns we met on the string. Write a sinusoidal candidate A sine k x. The left boundary is already satisfied because sine zero is zero. At the right wall, x becomes L. The amplitude there is A sine k L, and the wall requires that expression to vanish. Sine vanishes at whole multiples of pi. So k L must equal n pi, not an arbitrary number. Dividing by L gives the allowed wave numbers. Once again the boundaries have turned a continuous range of guesses into a numbered list. A fitting rule selects wave numbers. To find energies, we now use the stationary Schrödinger equation inside the box. Its left side measures the curvature of the wavefunction. The coefficient h squared over eight pi squared m is the familiar h bar squared over two m, written using Planck's constant h. The right side is energy times the same wavefunction. Differentiate sine twice and the original wave returns with a factor of minus k squared. Substitute that curvature into the equation. The common wavefunction cancels, leaving E equal to h squared k squared over eight pi squared m. Finally substitute the allowed k values. The energy of state n is n squared h squared over eight m L squared. That is where quantised energy enters. The equation relates energy to wavelength, while the walls permit only certain wavelengths. The discrete energies are the consequence of both facts together. ### [04:44.938 · How the Energy Gaps Shrink](https://academa.ai/lectures/the-particle-in-a-box?t=284.93758333333335) The energy formula contains a powerful prediction. Keep the particle and its mass unchanged, but make the confining region wider. Beside the box are its first three energies. They are separate horizontal lines, and even the lowest state sits above zero. That nonzero ground-state energy is unavoidable. A wave trapped in a finite region cannot be perfectly flat, so it cannot have zero curvature and zero kinetic energy. The difference between neighboring levels is delta E n. Start with E n plus one minus E n. Insert the box energies. The numerator contains n plus one squared minus n squared, while every level carries the same factor one over L squared. The difference of the squares is two n plus one. So the gap is also proportional to one over L squared. Now widen the box. Watch both walls move apart while the three energy levels descend and crowd together. Nothing has changed about the integer labels. The allowed states are still numbered one, two, three, and onward. What changed is the energy scale attached to that list. A very wide box therefore has many levels inside any modest energy interval. The spectrum is discrete in principle, but the gaps may become too small for an experiment to distinguish. This is one route back toward classical physics. First, increasing L lowers the entire spectrum and shrinks every gap as one over L squared. There is a second effect at high quantum number. Divide a gap by the energy itself, and most constants cancel. The relative gap is two n plus one over n squared, which approaches two over n. At large n, neighboring states differ by a tiny fraction of their total energy. Finally, every real measurement has limited resolution. If thousands of allowed energies fall inside one unresolved band, changing energy appears continuous even though the microscopic list remains discrete. The classical world does not require the quantum rules to switch off. Large dimensions, high quantum numbers, and limited resolution make the steps too fine to notice. ### [07:12.269 · Probability and the Places Never Found](https://academa.ai/lectures/the-particle-in-a-box?t=432.26925000000006) A wavefunction is not itself a probability. It can be positive, negative, or even complex. What predicts where the particle may be detected is its absolute square. Here is the second state. Its left lobe is positive and its right lobe is negative, but a probability density cannot be negative. Square the amplitude at every position. Both lobes rise above the axis, producing two regions where detection is likely. The total shaded area is one after normalization. That does not say the particle has a known position. It says some detection somewhere in the box has total probability one. At the middle, the wavefunction is exactly zero. Squaring zero still gives zero, so the particle is never detected at this node. Now compare the first three stationary states. The first density has one broad peak and no internal node. The probability still vanishes at both walls. Those zeros come from the confinement condition shared by every state. The second density has two peaks. Between them is one internal node at L over two, a position with exactly zero probability. The third density has three peaks and two internal nodes, at L over three and two L over three. A node is stronger than a low-probability region. At a node the wavefunction vanishes exactly, so an ideal position measurement never returns that point while the particle remains in that state. The pattern is systematic. State n has n lobes in its density and n minus one internal nodes. Higher states oscillate more rapidly because their allowed wavelengths are shorter. More oscillations produce more exact cancellations and therefore more nodes. These densities also warn us against imagining a classical bead bouncing between the walls. A classical bead can pass through every interior position. A stationary quantum state can forbid particular positions completely. The allowed energy tells us which stationary pattern we have. The square of that pattern tells us where detections can occur, and its nodes mark where they cannot. ### [09:33.096 · Why Conjugated Molecules Have Colour](https://academa.ai/lectures/the-particle-in-a-box?t=573.0964166666668) A toy model earns its keep only if it explains something real. Consider a conjugated dye molecule, built from a chain of neighboring atoms with overlapping p orbitals. The green skeleton marks the chain, while the blue region represents the shared pi-electron system. These electrons are not assigned to one particular bond; they are delocalised along much of the chain. That finite delocalised length acts, approximately, like a one-dimensional box. The detailed molecular potential is not perfectly square, but the familiar expression makes the dependence clear: every energy contains one over L squared. So the molecular orbitals form a numbered set of wave-like states. Their exact shapes are more complicated than sines, but confinement still produces separated energies. Electrons now fill the allowed levels from the bottom upward. Each spatial level can hold two electrons with opposite spin. If the conjugated system has N pi electrons and N is even, the highest filled level has index n equal to N over two. The next level is empty. Absorbing light can lift an electron across this frontier gap from level n to level n plus one. Begin with the difference E n plus one minus E n. Insert the particle-in-a-box energies. The two levels differ only in their squared quantum numbers. Simplifying gives a gap proportional to two n plus one divided by L squared. A photon is absorbed when its energy matches that gap. Delta E equals h f, or h c over wavelength. Solving for wavelength makes the trend explicit. A smaller energy gap corresponds to a longer absorbed wavelength. Compare two related conjugated systems. The shorter chain confines its pi electrons more tightly, so the frontier levels stand farther apart. The longer chain spreads the wavefunctions across a greater distance. Its relevant levels are closer together, so the transition needs a lower-energy photon. Lower photon energy means longer wavelength. As conjugation is extended through a family of dyes, the absorption band commonly shifts toward longer visible wavelengths. A dye looks coloured because it absorbs some visible wavelengths more strongly than others. The colour we observe comes from the light left over or transmitted, so it is not simply the colour of the absorbed photon. Now the complete chain of reasoning fits together. First, finite boundaries select standing-wave states. Second, electrons fill those states, leaving a particular gap between the highest filled and lowest empty levels. Third, light is absorbed when a photon matches that gap. Extending the conjugated region generally reduces the gap and moves the absorption toward longer wavelength. Finally, this remains a model. Real molecules have non-square potentials, electron interactions, vibrations, and environmental effects. Detailed spectra require more complete quantum chemistry. But the model has earned its keep. Fixed ends selected wavelengths; confined wavefunctions selected energies; molecular length controlled an optical gap. Energy is quantised because a bounded wave cannot fit in arbitrary ways. ## About Academa, Inc. Academa makes technical knowledge easier to understand through visual lectures and lets learners request new lecture videos on the topics they need. ## Complete audiovisual record Immutable source: [semantic.json](https://academa.ai/media/l/01M14TZZSX238T89EBZHGVYW3N/0/semantic.json) Record version: 1. Render attempt: 0. ### How to read this timeline Each scene owns its object identifiers. A beat's board is the complete board when listed, empty when marked empty, and unchanged from the nearest earlier listed board in the same scene when marked unchanged. Action times are absolute positions in the published video. ### Scene 1: [Only Certain Waves Fit](https://academa.ai/lectures/the-particle-in-a-box?t=0) Span: 00:00–02:5.434 (0s–125.43370833333334s). #### Objects - allowed: a Math \[text\] that says "$n=1,2,3,dots$" - boundary: a Math \[text\] that says "$y(0)=0, quad y(L)=0$" - card: a Title that says "Introductory Quantum Physics — Why Energy Is Quantised: From Standing Waves to Coloured Molecules" - fit\_work: a Derivation \[text\] that says "$L &= n frac(lambda, 2) \\ lambda\_n &= frac(2L, n) \\ k\_n &= frac(n pi, L)$" - heading: a Heading that says "A String With Two Fixed Ends" - left\_end: a Point \[yellow\] drawn in string - left\_wall: a Line \[gray\] drawn in string (start=(0.0, -1.2), end=(0.0, 1.2)) - middle\_node: a Point \[yellow\] labelled "upright("node")" drawn in string (location=(0.5, 0.0)) - miss: a ParametricCurve \[gray\] labelled "upright("rejected")" drawn in string (function=\) - missed\_end: a Point \[red\] labelled "y(L) eq.not 0" drawn in string (location=(1.0, 1.0)) - mode\_1: a ParametricCurve \[blue\] labelled "n=1" drawn in string (function=\) - mode\_2: a ParametricCurve \[green\] labelled "n=2" drawn in string (function=\) - mode\_3: a ParametricCurve \[red\] labelled "n=3" drawn in string (function=\) - rest\_line: a Line \[gray\] drawn in string (dashed=True) - right\_end: a Point \[yellow\] drawn in string (location=(1.0, 0.0)) - right\_wall: a Line \[gray\] drawn in string (start=(1.0, -1.2), end=(1.0, 1.2)) - string: a Figure (x\_range=(-0.08, 1.08), y\_range=(-1.3, 1.3)) - third\_node\_1: a Point \[yellow\] drawn in string (location=(0.3333333333333333, 0.0)) - third\_node\_2: a Point \[yellow\] drawn in string (location=(0.6666666666666666, 0.0)) #### Beats ##### [00:00](https://academa.ai/lectures/the-particle-in-a-box?t=0) Narration: Why should energy ever come in separate amounts? I do not want to begin by declaring that it does. Let us begin with a familiar object instead: a stretched string, fixed at both ends, and the waves that can actually live on it. Board: Empty. Actions: - [00:00](https://academa.ai/lectures/the-particle-in-a-box?t=0): card is shown on the screen, written out. - [00:1.5](https://academa.ai/lectures/the-particle-in-a-box?t=1.5): card: enter:write-left-to-right. - [00:14.466](https://academa.ai/lectures/the-particle-in-a-box?t=14.4665): card is hidden from the screen — left the board. ##### [00:15.666](https://academa.ai/lectures/the-particle-in-a-box?t=15.6665) Narration: The string runs from zero to L. Each end is clamped, so neither endpoint can move, however strongly the rest of the string vibrates. Board: Unchanged from the preceding beat in this scene. Actions: - [00:15.666](https://academa.ai/lectures/the-particle-in-a-box?t=15.6665): heading is shown on the screen, written out. - [00:16.212](https://academa.ai/lectures/the-particle-in-a-box?t=16.212): string is shown on the screen, written out. - [00:16.212](https://academa.ai/lectures/the-particle-in-a-box?t=16.212): rest\_line is shown on the screen, written out. - [00:19.66](https://academa.ai/lectures/the-particle-in-a-box?t=19.66): left\_wall is shown on the screen, written out. - [00:19.66](https://academa.ai/lectures/the-particle-in-a-box?t=19.66): right\_wall is shown on the screen, written out. - [00:20.891](https://academa.ai/lectures/the-particle-in-a-box?t=20.891000000000002): left\_end is shown on the screen, written out. - [00:20.891](https://academa.ai/lectures/the-particle-in-a-box?t=20.891000000000002): right\_end is shown on the screen, written out. ##### [00:25.682](https://academa.ai/lectures/the-particle-in-a-box?t=25.682) Narration: That physical fact becomes a boundary condition. The displacement y must be zero at the left wall and zero again at the right wall. Board: string — a Figure (x\_range=(-0.08, 1.08), y\_range=(-1.3, 1.3)); heading — a Heading that says "A String With Two Fixed Ends"; left\_wall — a Line \[gray\] drawn in string (start=(0.0, -1.2), end=(0.0, 1.2)); right\_wall — a Line \[gray\] drawn in string (start=(1.0, -1.2), end=(1.0, 1.2)); rest\_line — a Line \[gray\] drawn in string (dashed=True); left\_end — a Point \[yellow\] drawn in string; right\_end — a Point \[yellow\] drawn in string (location=(1.0, 0.0)) Actions: - [00:27.691](https://academa.ai/lectures/the-particle-in-a-box?t=27.690999999999995): string moves to a new place on the board. - [00:27.691](https://academa.ai/lectures/the-particle-in-a-box?t=27.690999999999995): boundary is shown on the screen, written out. - [00:31.882](https://academa.ai/lectures/the-particle-in-a-box?t=31.881999999999998): boundary (the "y(0)=0" part) is indicated — a transient flash. - [00:33.971](https://academa.ai/lectures/the-particle-in-a-box?t=33.971): boundary (the "y(L)=0" part) is indicated — a transient flash. ##### [00:35.558](https://academa.ai/lectures/the-particle-in-a-box?t=35.558) Narration: Try an arbitrary sinusoidal candidate. It begins correctly at zero, but look at its right end. The curve reaches the wall above the fixed point, so this shape cannot be a standing wave of this string. Board: boundary — a Math \[text\] that says "$y(0)=0, quad y(L)=0$"; string — a Figure (x\_range=(-0.08, 1.08), y\_range=(-1.3, 1.3)); heading — a Heading that says "A String With Two Fixed Ends"; left\_wall — a Line \[gray\] drawn in string (start=(0.0, -1.2), end=(0.0, 1.2)); right\_wall — a Line \[gray\] drawn in string (start=(1.0, -1.2), end=(1.0, 1.2)); rest\_line — a Line \[gray\] drawn in string (dashed=True); left\_end — a Point \[yellow\] drawn in string; right\_end — a Point \[yellow\] drawn in string (location=(1.0, 0.0)) Actions: - [00:37.799](https://academa.ai/lectures/the-particle-in-a-box?t=37.799): miss is shown on the screen, drawn. - [00:41.781](https://academa.ai/lectures/the-particle-in-a-box?t=41.781): missed\_end is shown on the screen, written out. - [00:44.567](https://academa.ai/lectures/the-particle-in-a-box?t=44.567): missed\_end is indicated — a transient flash. - [00:49.211](https://academa.ai/lectures/the-particle-in-a-box?t=49.211): missed\_end is hidden from the screen. ##### [00:49.811](https://academa.ai/lectures/the-particle-in-a-box?t=49.811) Narration: The simplest accepted shape places one half-wave between the walls. Both endpoints are nodes, and the entire string rises and falls as one arch. Board: boundary — a Math \[text\] that says "$y(0)=0, quad y(L)=0$"; string — a Figure (x\_range=(-0.08, 1.08), y\_range=(-1.3, 1.3)); heading — a Heading that says "A String With Two Fixed Ends"; left\_wall — a Line \[gray\] drawn in string (start=(0.0, -1.2), end=(0.0, 1.2)); right\_wall — a Line \[gray\] drawn in string (start=(1.0, -1.2), end=(1.0, 1.2)); rest\_line — a Line \[gray\] drawn in string (dashed=True); left\_end — a Point \[yellow\] drawn in string; right\_end — a Point \[yellow\] drawn in string (location=(1.0, 0.0)); miss — a ParametricCurve \[gray\] labelled "upright("rejected")" drawn in string (function=\) Actions: - [00:49.811](https://academa.ai/lectures/the-particle-in-a-box?t=49.811): miss is hidden from the screen. - [00:52.551](https://academa.ai/lectures/the-particle-in-a-box?t=52.551): mode\_1 is shown on the screen, drawn. - [00:56.336](https://academa.ai/lectures/the-particle-in-a-box?t=56.336): left\_end is indicated — a transient flash. - [00:56.336](https://academa.ai/lectures/the-particle-in-a-box?t=56.336): right\_end is indicated — a transient flash. ##### [01:0.988](https://academa.ai/lectures/the-particle-in-a-box?t=60.988) Narration: The next shape fits two half-waves. It still vanishes at both walls, and it gains another node exactly in the middle. Board: boundary — a Math \[text\] that says "$y(0)=0, quad y(L)=0$"; string — a Figure (x\_range=(-0.08, 1.08), y\_range=(-1.3, 1.3)); heading — a Heading that says "A String With Two Fixed Ends"; left\_wall — a Line \[gray\] drawn in string (start=(0.0, -1.2), end=(0.0, 1.2)); right\_wall — a Line \[gray\] drawn in string (start=(1.0, -1.2), end=(1.0, 1.2)); rest\_line — a Line \[gray\] drawn in string (dashed=True); left\_end — a Point \[yellow\] drawn in string; right\_end — a Point \[yellow\] drawn in string (location=(1.0, 0.0)); mode\_1 — a ParametricCurve \[blue\] labelled "n=1" drawn in string (function=\) Actions: - [01:0.988](https://academa.ai/lectures/the-particle-in-a-box?t=60.988): mode\_1 is hidden from the screen. - [01:2.358](https://academa.ai/lectures/the-particle-in-a-box?t=62.358000000000004): mode\_2 is shown on the screen, drawn. - [01:6.991](https://academa.ai/lectures/the-particle-in-a-box?t=66.99100000000001): middle\_node is shown on the screen, written out. ##### [01:9.404](https://academa.ai/lectures/the-particle-in-a-box?t=69.4045) Narration: A third shape fits three half-waves. Two interior nodes divide the string into three vibrating sections, with neighboring sections moving in opposite directions. Board: boundary — a Math \[text\] that says "$y(0)=0, quad y(L)=0$"; string — a Figure (x\_range=(-0.08, 1.08), y\_range=(-1.3, 1.3)); heading — a Heading that says "A String With Two Fixed Ends"; left\_wall — a Line \[gray\] drawn in string (start=(0.0, -1.2), end=(0.0, 1.2)); right\_wall — a Line \[gray\] drawn in string (start=(1.0, -1.2), end=(1.0, 1.2)); rest\_line — a Line \[gray\] drawn in string (dashed=True); left\_end — a Point \[yellow\] drawn in string; right\_end — a Point \[yellow\] drawn in string (location=(1.0, 0.0)); mode\_2 — a ParametricCurve \[green\] labelled "n=2" drawn in string (function=\); middle\_node — a Point \[yellow\] labelled "upright("node")" drawn in string (location=(0.5, 0.0)) Actions: - [01:9.404](https://academa.ai/lectures/the-particle-in-a-box?t=69.4045): mode\_2 is hidden from the screen. - [01:9.404](https://academa.ai/lectures/the-particle-in-a-box?t=69.4045): middle\_node is hidden from the screen. - [01:10.786](https://academa.ai/lectures/the-particle-in-a-box?t=70.78600000000002): mode\_3 is shown on the screen, drawn. - [01:12.574](https://academa.ai/lectures/the-particle-in-a-box?t=72.57400000000001): third\_node\_1 is shown on the screen, written out. - [01:12.574](https://academa.ai/lectures/the-particle-in-a-box?t=72.57400000000001): third\_node\_2 is shown on the screen, written out. - [01:17.764](https://academa.ai/lectures/the-particle-in-a-box?t=77.76400000000001): mode\_3 is indicated — a transient flash. ##### [01:19.85](https://academa.ai/lectures/the-particle-in-a-box?t=79.8495) Narration: We can continue forever, but not continuously. The length L must contain a whole number n of half-wavelengths. Board: boundary — a Math \[text\] that says "$y(0)=0, quad y(L)=0$"; string — a Figure (x\_range=(-0.08, 1.08), y\_range=(-1.3, 1.3)); heading — a Heading that says "A String With Two Fixed Ends"; left\_wall — a Line \[gray\] drawn in string (start=(0.0, -1.2), end=(0.0, 1.2)); right\_wall — a Line \[gray\] drawn in string (start=(1.0, -1.2), end=(1.0, 1.2)); rest\_line — a Line \[gray\] drawn in string (dashed=True); left\_end — a Point \[yellow\] drawn in string; right\_end — a Point \[yellow\] drawn in string (location=(1.0, 0.0)); mode\_3 — a ParametricCurve \[red\] labelled "n=3" drawn in string (function=\); third\_node\_1 — a Point \[yellow\] drawn in string (location=(0.3333333333333333, 0.0)); third\_node\_2 — a Point \[yellow\] drawn in string (location=(0.6666666666666666, 0.0)) Actions: - [01:25.33](https://academa.ai/lectures/the-particle-in-a-box?t=85.33): fit\_work is shown on the screen, written out. - [01:25.887](https://academa.ai/lectures/the-particle-in-a-box?t=85.887): fit\_work (the "n" part) is indicated — a transient flash. - [01:26.305](https://academa.ai/lectures/the-particle-in-a-box?t=86.305): fit\_work (the "frac(lambda, 2)" part) is indicated — a transient flash. ##### [01:28.345](https://academa.ai/lectures/the-particle-in-a-box?t=88.345) Narration: Solving for wavelength gives lambda n equals two L over n. The allowed wavelengths form a list: two L, L, two L over three, and so on. Board: Unchanged from the preceding beat in this scene. Actions: - [01:28.693](https://academa.ai/lectures/the-particle-in-a-box?t=88.693): fit\_work is shown on the screen, written out. - [01:31.259](https://academa.ai/lectures/the-particle-in-a-box?t=91.259): fit\_work (the "frac(2L, n)" part) is indicated — a transient flash. ##### [01:40.114](https://academa.ai/lectures/the-particle-in-a-box?t=100.1135) Narration: The same statement can be written using wave number k. Only n pi over L is allowed, with n equal to one, two, three, and onward. Board: Unchanged from the preceding beat in this scene. Actions: - [01:42.354](https://academa.ai/lectures/the-particle-in-a-box?t=102.35400000000001): fit\_work is shown on the screen, written out. - [01:44.386](https://academa.ai/lectures/the-particle-in-a-box?t=104.38600000000001): fit\_work (the "frac(n pi, L)" part) is indicated — a transient flash. - [01:47.625](https://academa.ai/lectures/the-particle-in-a-box?t=107.62500000000001): allowed is shown on the screen, written out. ##### [01:51.07](https://academa.ai/lectures/the-particle-in-a-box?t=111.0695) Narration: The important word is allowed. The material string could have many amplitudes, but its spatial patterns are selected by the two boundaries. Continuous guesses went in; a discrete family came out. Board: boundary — a Math \[text\] that says "$y(0)=0, quad y(L)=0$"; allowed — a Math \[text\] that says "$n=1,2,3,dots$"; string — a Figure (x\_range=(-0.08, 1.08), y\_range=(-1.3, 1.3)); heading — a Heading that says "A String With Two Fixed Ends"; left\_wall — a Line \[gray\] drawn in string (start=(0.0, -1.2), end=(0.0, 1.2)); right\_wall — a Line \[gray\] drawn in string (start=(1.0, -1.2), end=(1.0, 1.2)); rest\_line — a Line \[gray\] drawn in string (dashed=True); left\_end — a Point \[yellow\] drawn in string; right\_end — a Point \[yellow\] drawn in string (location=(1.0, 0.0)); mode\_3 — a ParametricCurve \[red\] labelled "n=3" drawn in string (function=\); third\_node\_1 — a Point \[yellow\] drawn in string (location=(0.3333333333333333, 0.0)); third\_node\_2 — a Point \[yellow\] drawn in string (location=(0.6666666666666666, 0.0)) Actions: - [02:2.563](https://academa.ai/lectures/the-particle-in-a-box?t=122.56300000000003): A box is drawn around allowed. - [02:4.392](https://academa.ai/lectures/the-particle-in-a-box?t=124.39204166666667): allowed is hidden from the screen — left the board. - [02:4.392](https://academa.ai/lectures/the-particle-in-a-box?t=124.39204166666667): boundary is hidden from the screen — left the board. - [02:4.392](https://academa.ai/lectures/the-particle-in-a-box?t=124.39204166666667): fit\_work is hidden from the screen — left the board. - [02:4.392](https://academa.ai/lectures/the-particle-in-a-box?t=124.39204166666667): heading is hidden from the screen — left the board. - [02:4.392](https://academa.ai/lectures/the-particle-in-a-box?t=124.39204166666667): string is hidden from the screen — left the board. - [02:4.392](https://academa.ai/lectures/the-particle-in-a-box?t=124.39204166666667): left\_wall is hidden from the screen — string left the board. - [02:4.392](https://academa.ai/lectures/the-particle-in-a-box?t=124.39204166666667): right\_wall is hidden from the screen — string left the board. - [02:4.392](https://academa.ai/lectures/the-particle-in-a-box?t=124.39204166666667): rest\_line is hidden from the screen — string left the board. - [02:4.392](https://academa.ai/lectures/the-particle-in-a-box?t=124.39204166666667): left\_end is hidden from the screen — string left the board. - [02:4.392](https://academa.ai/lectures/the-particle-in-a-box?t=124.39204166666667): right\_end is hidden from the screen — string left the board. - [02:4.392](https://academa.ai/lectures/the-particle-in-a-box?t=124.39204166666667): mode\_3 is hidden from the screen — string left the board. - [02:4.392](https://academa.ai/lectures/the-particle-in-a-box?t=124.39204166666667): third\_node\_1 is hidden from the screen — string left the board. - [02:4.392](https://academa.ai/lectures/the-particle-in-a-box?t=124.39204166666667): third\_node\_2 is hidden from the screen — string left the board. ### Scene 2: [A Particle as a Confined Wave](https://academa.ai/lectures/the-particle-in-a-box?t=125.43370833333334) Span: 02:5.434–04:44.938 (125.43370833333334s–284.93758333333335s). #### Objects - baseline: a Line \[gray\] drawn in box (dashed=True) - boundary: a Math \[text\] that says "$psi(0)=0, quad psi(L)=0$" - box: an Axes (x\_range=(-0.08, 1.08), y\_range=(-1.25, 1.25), include\_ticks=False) - energy\_result: a Math \[text\] that says "$E\_1, E\_2, E\_3, dots$" - energy\_work: a Derivation \[text\] that says "$-frac(h^2, 8 pi^2 m) psi''(x) &= E psi(x) \\ psi''(x) &= -k^2 psi(x) \\ E &= frac(h^2 k^2, 8 pi^2 m) \\ E\_n &= frac(n^2 h^2, 8 m L^2)$" - f3: a Math \[text\] that says "$k\_n &= frac(n pi, L)$" - fit\_work: a Derivation \[text\] that says "$psi(x) &= A sin(k x) \\ psi(L) &= A sin(k L)=0 \\ k L &= n pi \\ k\_n &= frac(n pi, L)$" - heading\_energy: a Heading that says "From Wavelength to Energy" - heading\_fit: a Heading that says "The Same Fitting Condition" - left\_wall: a Line \[gray\] drawn in box (start=(0.0, -1.15), end=(0.0, 1.15)) - question: a Panel that says "What changes when the confined object is a particle described by a wavefunction?" - right\_wall: a Line \[gray\] drawn in box (start=(1.0, -1.15), end=(1.0, 1.15)) - wave\_1: a FunctionPlot \[blue\] labelled "psi\_1" drawn in box (function=\, x\_range=(0.0, 1.0)) - wave\_2: a FunctionPlot \[green\] labelled "psi\_2" drawn in box (function=\, x\_range=(0.0, 1.0)) - wave\_3: a FunctionPlot \[red\] labelled "psi\_3" drawn in box (function=\, x\_range=(0.0, 1.0)) #### Beats ##### [02:5.434](https://academa.ai/lectures/the-particle-in-a-box?t=125.43370833333334) Narration: Now replace the string by a particle trapped between two perfectly impenetrable walls. Quantum mechanics describes its state with a wavefunction, psi. What survives from the string argument? Board: Empty. Actions: - [02:5.434](https://academa.ai/lectures/the-particle-in-a-box?t=125.43370833333334): question is shown on the screen, written out. - [02:17.566](https://academa.ai/lectures/the-particle-in-a-box?t=137.56620833333335): question is hidden from the screen — left the board. ##### [02:18.166](https://academa.ai/lectures/the-particle-in-a-box?t=138.16620833333334) Narration: Here is the region available to the particle. Outside the two walls the particle cannot exist, so the wavefunction is zero there. Board: Unchanged from the preceding beat in this scene. Actions: - [02:18.166](https://academa.ai/lectures/the-particle-in-a-box?t=138.16620833333334): heading\_fit is shown on the screen, written out. - [02:19.118](https://academa.ai/lectures/the-particle-in-a-box?t=139.11770833333335): box is shown on the screen, written out. - [02:22.184](https://academa.ai/lectures/the-particle-in-a-box?t=142.18370833333336): left\_wall is shown on the screen, written out. - [02:22.184](https://academa.ai/lectures/the-particle-in-a-box?t=142.18370833333336): right\_wall is shown on the screen, written out. - [02:25.295](https://academa.ai/lectures/the-particle-in-a-box?t=145.29470833333335): baseline is shown on the screen, written out. ##### [02:26.928](https://academa.ai/lectures/the-particle-in-a-box?t=146.92820833333334) Narration: At an ideal infinite wall, the wavefunction must meet that outside value continuously. Therefore psi is zero at x equals zero and again at x equals L. Board: box — an Axes (x\_range=(-0.08, 1.08), y\_range=(-1.25, 1.25), include\_ticks=False); heading\_fit — a Heading that says "The Same Fitting Condition"; left\_wall — a Line \[gray\] drawn in box (start=(0.0, -1.15), end=(0.0, 1.15)); right\_wall — a Line \[gray\] drawn in box (start=(1.0, -1.15), end=(1.0, 1.15)); baseline — a Line \[gray\] drawn in box (dashed=True) Actions: - [02:33.279](https://academa.ai/lectures/the-particle-in-a-box?t=153.27870833333336): box moves to a new place on the board. - [02:33.279](https://academa.ai/lectures/the-particle-in-a-box?t=153.27870833333336): boundary is shown on the screen, written out. - [02:35.102](https://academa.ai/lectures/the-particle-in-a-box?t=155.10170833333333): boundary (the "psi(0)=0" part) is indicated — a transient flash. - [02:37.226](https://academa.ai/lectures/the-particle-in-a-box?t=157.22570833333336): boundary (the "psi(L)=0" part) is indicated — a transient flash. ##### [02:39.858](https://academa.ai/lectures/the-particle-in-a-box?t=159.85770833333333) Narration: The first allowed state is one smooth arch. This is not a little particle following the curve. The curve is the wavefunction that will determine probabilities. Board: boundary — a Math \[text\] that says "$psi(0)=0, quad psi(L)=0$"; box — an Axes (x\_range=(-0.08, 1.08), y\_range=(-1.25, 1.25), include\_ticks=False); heading\_fit — a Heading that says "The Same Fitting Condition"; left\_wall — a Line \[gray\] drawn in box (start=(0.0, -1.15), end=(0.0, 1.15)); right\_wall — a Line \[gray\] drawn in box (start=(1.0, -1.15), end=(1.0, 1.15)); baseline — a Line \[gray\] drawn in box (dashed=True) Actions: - [02:40.415](https://academa.ai/lectures/the-particle-in-a-box?t=160.41470833333335): wave\_1 is shown on the screen, drawn. - [02:47.172](https://academa.ai/lectures/the-particle-in-a-box?t=167.17170833333336): wave\_1 is indicated — a transient flash. ##### [02:50.349](https://academa.ai/lectures/the-particle-in-a-box?t=170.34920833333334) Narration: The second state has two lobes and one interior zero. The third has three lobes and two interior zeros. They are the same spatial fitting patterns we met on the string. Board: boundary — a Math \[text\] that says "$psi(0)=0, quad psi(L)=0$"; box — an Axes (x\_range=(-0.08, 1.08), y\_range=(-1.25, 1.25), include\_ticks=False); heading\_fit — a Heading that says "The Same Fitting Condition"; left\_wall — a Line \[gray\] drawn in box (start=(0.0, -1.15), end=(0.0, 1.15)); right\_wall — a Line \[gray\] drawn in box (start=(1.0, -1.15), end=(1.0, 1.15)); baseline — a Line \[gray\] drawn in box (dashed=True); wave\_1 — a FunctionPlot \[blue\] labelled "psi\_1" drawn in box (function=\, x\_range=(0.0, 1.0)) Actions: - [02:50.349](https://academa.ai/lectures/the-particle-in-a-box?t=170.34920833333334): wave\_1 is hidden from the screen. - [02:50.768](https://academa.ai/lectures/the-particle-in-a-box?t=170.76770833333336): wave\_2 is shown on the screen, drawn. - [02:54.378](https://academa.ai/lectures/the-particle-in-a-box?t=174.37770833333335): wave\_2 is hidden from the screen. - [02:54.378](https://academa.ai/lectures/the-particle-in-a-box?t=174.37770833333335): wave\_3 is shown on the screen, drawn. ##### [03:2.315](https://academa.ai/lectures/the-particle-in-a-box?t=182.31520833333335) Narration: Write a sinusoidal candidate A sine k x. The left boundary is already satisfied because sine zero is zero. Board: boundary — a Math \[text\] that says "$psi(0)=0, quad psi(L)=0$"; box — an Axes (x\_range=(-0.08, 1.08), y\_range=(-1.25, 1.25), include\_ticks=False); heading\_fit — a Heading that says "The Same Fitting Condition"; left\_wall — a Line \[gray\] drawn in box (start=(0.0, -1.15), end=(0.0, 1.15)); right\_wall — a Line \[gray\] drawn in box (start=(1.0, -1.15), end=(1.0, 1.15)); baseline — a Line \[gray\] drawn in box (dashed=True); wave\_3 — a FunctionPlot \[red\] labelled "psi\_3" drawn in box (function=\, x\_range=(0.0, 1.0)) Actions: - [03:3.906](https://academa.ai/lectures/the-particle-in-a-box?t=183.90570833333334): fit\_work is shown on the screen, written out. - [03:4.893](https://academa.ai/lectures/the-particle-in-a-box?t=184.89270833333336): fit\_work (the "sin(k x)" part) is indicated — a transient flash. ##### [03:10.985](https://academa.ai/lectures/the-particle-in-a-box?t=190.98470833333334) Narration: At the right wall, x becomes L. The amplitude there is A sine k L, and the wall requires that expression to vanish. Board: Unchanged from the preceding beat in this scene. Actions: - [03:11.589](https://academa.ai/lectures/the-particle-in-a-box?t=191.58870833333333): fit\_work is shown on the screen, written out. - [03:15.699](https://academa.ai/lectures/the-particle-in-a-box?t=195.69870833333334): fit\_work (the "sin(k L)" part) is indicated — a transient flash. ##### [03:20.559](https://academa.ai/lectures/the-particle-in-a-box?t=200.55920833333334) Narration: Sine vanishes at whole multiples of pi. So k L must equal n pi, not an arbitrary number. Board: Unchanged from the preceding beat in this scene. Actions: - [03:22.034](https://academa.ai/lectures/the-particle-in-a-box?t=202.03370833333335): fit\_work is shown on the screen, written out. - [03:25.703](https://academa.ai/lectures/the-particle-in-a-box?t=205.70270833333336): fit\_work (the "n pi" part) is indicated — a transient flash. ##### [03:28.648](https://academa.ai/lectures/the-particle-in-a-box?t=208.64770833333336) Narration: Dividing by L gives the allowed wave numbers. Once again the boundaries have turned a continuous range of guesses into a numbered list. Board: Unchanged from the preceding beat in this scene. Actions: - [03:28.996](https://academa.ai/lectures/the-particle-in-a-box?t=208.99570833333334): fit\_work is shown on the screen, written out. - [03:35.95](https://academa.ai/lectures/the-particle-in-a-box?t=215.94970833333335): A box is drawn around fit\_work. - [03:37.111](https://academa.ai/lectures/the-particle-in-a-box?t=217.11070833333332): boundary is hidden from the screen — left the board. - [03:37.111](https://academa.ai/lectures/the-particle-in-a-box?t=217.11070833333332): fit\_work is hidden from the screen — left the board. - [03:37.111](https://academa.ai/lectures/the-particle-in-a-box?t=217.11070833333332): heading\_fit is hidden from the screen — left the board. ##### [03:37.711](https://academa.ai/lectures/the-particle-in-a-box?t=217.71070833333334) Narration: A fitting rule selects wave numbers. To find energies, we now use the stationary Schrödinger equation inside the box. Board: box — an Axes (x\_range=(-0.08, 1.08), y\_range=(-1.25, 1.25), include\_ticks=False); left\_wall — a Line \[gray\] drawn in box (start=(0.0, -1.15), end=(0.0, 1.15)); right\_wall — a Line \[gray\] drawn in box (start=(1.0, -1.15), end=(1.0, 1.15)); baseline — a Line \[gray\] drawn in box (dashed=True); wave\_3 — a FunctionPlot \[red\] labelled "psi\_3" drawn in box (function=\, x\_range=(0.0, 1.0)) Actions: - [03:37.711](https://academa.ai/lectures/the-particle-in-a-box?t=217.71070833333334): heading\_energy is shown on the screen, written out. - [03:43.667](https://academa.ai/lectures/the-particle-in-a-box?t=223.66670833333336): energy\_work is shown on the screen, written out. ##### [03:46.589](https://academa.ai/lectures/the-particle-in-a-box?t=226.58920833333335) Narration: Its left side measures the curvature of the wavefunction. The coefficient h squared over eight pi squared m is the familiar h bar squared over two m, written using Planck's constant h. The right side is energy times the same wavefunction. Board: box — an Axes (x\_range=(-0.08, 1.08), y\_range=(-1.25, 1.25), include\_ticks=False); left\_wall — a Line \[gray\] drawn in box (start=(0.0, -1.15), end=(0.0, 1.15)); right\_wall — a Line \[gray\] drawn in box (start=(1.0, -1.15), end=(1.0, 1.15)); baseline — a Line \[gray\] drawn in box (dashed=True); wave\_3 — a FunctionPlot \[red\] labelled "psi\_3" drawn in box (function=\, x\_range=(0.0, 1.0)); heading\_energy — a Heading that says "From Wavelength to Energy" Actions: - [03:48.401](https://academa.ai/lectures/the-particle-in-a-box?t=228.40070833333334): energy\_work (the "psi''(x)" part) is indicated — a transient flash. - [04:1.706](https://academa.ai/lectures/the-particle-in-a-box?t=241.70570833333335): energy\_work (the "E" part) is indicated — a transient flash. ##### [04:4.709](https://academa.ai/lectures/the-particle-in-a-box?t=244.70870833333333) Narration: Differentiate sine twice and the original wave returns with a factor of minus k squared. Board: Unchanged from the preceding beat in this scene. Actions: - [04:5.057](https://academa.ai/lectures/the-particle-in-a-box?t=245.05670833333335): energy\_work is shown on the screen, written out. - [04:9.202](https://academa.ai/lectures/the-particle-in-a-box?t=249.20170833333336): energy\_work (the "-k^2" part) is indicated — a transient flash. ##### [04:11.404](https://academa.ai/lectures/the-particle-in-a-box?t=251.40420833333334) Narration: Substitute that curvature into the equation. The common wavefunction cancels, leaving E equal to h squared k squared over eight pi squared m. Board: Unchanged from the preceding beat in this scene. Actions: - [04:17.209](https://academa.ai/lectures/the-particle-in-a-box?t=257.20870833333333): energy\_work is shown on the screen, written out. - [04:19.276](https://academa.ai/lectures/the-particle-in-a-box?t=259.27570833333334): energy\_work (the "k^2" part) is indicated — a transient flash. ##### [04:22.569](https://academa.ai/lectures/the-particle-in-a-box?t=262.56920833333334) Narration: Finally substitute the allowed k values. The energy of state n is n squared h squared over eight m L squared. Board: Unchanged from the preceding beat in this scene. Actions: - [04:22.918](https://academa.ai/lectures/the-particle-in-a-box?t=262.91770833333334): energy\_work is shown on the screen, written out. - [04:27.585](https://academa.ai/lectures/the-particle-in-a-box?t=267.58470833333337): energy\_work (the "n^2" part) is indicated — a transient flash. - [04:29.977](https://academa.ai/lectures/the-particle-in-a-box?t=269.97670833333336): energy\_work (the "L^2" part) is indicated — a transient flash. ##### [04:31.726](https://academa.ai/lectures/the-particle-in-a-box?t=271.7262083333334) Narration: That is where quantised energy enters. The equation relates energy to wavelength, while the walls permit only certain wavelengths. The discrete energies are the consequence of both facts together. Board: Unchanged from the preceding beat in this scene. Actions: - [04:40.55](https://academa.ai/lectures/the-particle-in-a-box?t=280.54970833333334): energy\_result is shown on the screen, written out. - [04:42.86](https://academa.ai/lectures/the-particle-in-a-box?t=282.85970833333334): A box is drawn around energy\_work. - [04:43.896](https://academa.ai/lectures/the-particle-in-a-box?t=283.8959166666667): box is hidden from the screen — left the board. - [04:43.896](https://academa.ai/lectures/the-particle-in-a-box?t=283.8959166666667): left\_wall is hidden from the screen — box left the board. - [04:43.896](https://academa.ai/lectures/the-particle-in-a-box?t=283.8959166666667): right\_wall is hidden from the screen — box left the board. - [04:43.896](https://academa.ai/lectures/the-particle-in-a-box?t=283.8959166666667): baseline is hidden from the screen — box left the board. - [04:43.896](https://academa.ai/lectures/the-particle-in-a-box?t=283.8959166666667): wave\_3 is hidden from the screen — box left the board. - [04:43.896](https://academa.ai/lectures/the-particle-in-a-box?t=283.8959166666667): energy\_result is hidden from the screen — left the board. - [04:43.896](https://academa.ai/lectures/the-particle-in-a-box?t=283.8959166666667): energy\_work is hidden from the screen — left the board. - [04:43.896](https://academa.ai/lectures/the-particle-in-a-box?t=283.8959166666667): heading\_energy is hidden from the screen — left the board. ### Scene 3: [How the Energy Gaps Shrink](https://academa.ai/lectures/the-particle-in-a-box?t=284.93758333333335) Span: 04:44.938–07:12.269 (284.93758333333335s–432.26925000000006s). #### Objects - base\_formula: a Math \[text\] that says "$E\_n=frac(n^2 h^2, 8 m L^2)$" - box\_floor: a Line \[gray\] drawn in picture (start=((7.0 - (width / 2.0)), 5.25), end=((7.0 + (width / 2.0)), 5.25)) - gap\_1: a Line \[yellow\] labelled "Delta E\_1" drawn in picture (start=(4.45, (0.55 + (2.2 / (width \*\* 2.0)))), end=(4.45, (0.55 + (8.8 / (width \*\* 2.0))))) - heading: a Heading that says "Widen the Box" - heading\_limit: a Heading that says "When Discrete Levels Look Continuous" - item\_1: a Text \[text\] that says "Larger $L$ makes every energy proportional to $1/L^2$." - item\_2: a Text \[text\] that says "At large $n$, neighboring levels are close relative to $E\_n$." - item\_3: a Text \[text\] that says "Finite measurements then cannot resolve the individual levels." - left\_wall: a Line \[blue\] drawn in picture (start=((7.0 - (width / 2.0)), 5.25), end=((7.0 - (width / 2.0)), 7.15)) - level\_1: a Line \[green\] labelled "E\_1" drawn in picture (start=(1.0, (0.55 + (2.2 / (width \*\* 2.0)))), end=(4.0, (0.55 + (2.2 / (width \*\* 2.0))))) - level\_2: a Line \[yellow\] labelled "E\_2" drawn in picture (start=(1.0, (0.55 + (8.8 / (width \*\* 2.0)))), end=(4.0, (0.55 + (8.8 / (width \*\* 2.0))))) - level\_3: a Line \[red\] labelled "E\_3" drawn in picture (start=(1.0, (0.55 + (19.8 / (width \*\* 2.0)))), end=(4.0, (0.55 + (19.8 / (width \*\* 2.0))))) - picture: a Figure (x\_range=(0.0, 10.0), y\_range=(0.0, 7.5)) - relative: a Math \[text\] that says "$frac(Delta E\_n, E\_n)=frac(2n+1, n^2) approx frac(2, n)$" - right\_wall: a Line \[blue\] drawn in picture (start=((7.0 + (width / 2.0)), 5.25), end=((7.0 + (width / 2.0)), 7.15)) - spacing\_work: a Derivation \[text\] that says "$Delta E\_n &= E\_(n+1)-E\_n \\ &= frac(((n+1)^2-n^2) h^2, 8 m L^2) \\ &= frac((2n+1) h^2, 8 m L^2)$" - width: a VariableNumber (initial\_value=2.0) - width\_brace: a Brace \[yellow\] labelled "L" drawn in picture (x\_start=(7.0 - (width / 2.0)), x\_end=(7.0 + (width / 2.0)), y=5.25) - zero\_line: a Line \[gray\] labelled "E=0" drawn in picture (start=(0.8, 0.55), end=(4.2, 0.55), dashed=True) #### Beats ##### [04:44.938](https://academa.ai/lectures/the-particle-in-a-box?t=284.93758333333335) Narration: The energy formula contains a powerful prediction. Keep the particle and its mass unchanged, but make the confining region wider. Board: Empty. Actions: - [04:44.938](https://academa.ai/lectures/the-particle-in-a-box?t=284.93758333333335): heading is shown on the screen, written out. - [04:45.542](https://academa.ai/lectures/the-particle-in-a-box?t=285.54158333333334): base\_formula is shown on the screen, written out. - [04:52.136](https://academa.ai/lectures/the-particle-in-a-box?t=292.13558333333333): left\_wall is shown on the screen, written out. - [04:52.136](https://academa.ai/lectures/the-particle-in-a-box?t=292.13558333333333): right\_wall is shown on the screen, written out. - [04:52.136](https://academa.ai/lectures/the-particle-in-a-box?t=292.13558333333333): box\_floor is shown on the screen, written out. - [04:52.496](https://academa.ai/lectures/the-particle-in-a-box?t=292.49558333333334): picture is shown on the screen, written out. - [04:52.496](https://academa.ai/lectures/the-particle-in-a-box?t=292.49558333333334): width\_brace is shown on the screen, written out. ##### [04:54.036](https://academa.ai/lectures/the-particle-in-a-box?t=294.03608333333335) Narration: Beside the box are its first three energies. They are separate horizontal lines, and even the lowest state sits above zero. Board: base\_formula — a Math \[text\] that says "$E\_n=frac(n^2 h^2, 8 m L^2)$"; picture — a Figure (x\_range=(0.0, 10.0), y\_range=(0.0, 7.5)); heading — a Heading that says "Widen the Box"; left\_wall — a Line \[blue\] drawn in picture (start=((7.0 - (width / 2.0)), 5.25), end=((7.0 - (width / 2.0)), 7.15)); right\_wall — a Line \[blue\] drawn in picture (start=((7.0 + (width / 2.0)), 5.25), end=((7.0 + (width / 2.0)), 7.15)); box\_floor — a Line \[gray\] drawn in picture (start=((7.0 - (width / 2.0)), 5.25), end=((7.0 + (width / 2.0)), 5.25)); width\_brace — a Brace \[yellow\] labelled "L" drawn in picture (x\_start=(7.0 - (width / 2.0)), x\_end=(7.0 + (width / 2.0)), y=5.25) Actions: - [04:55.778](https://academa.ai/lectures/the-particle-in-a-box?t=295.7775833333333): level\_1 is shown on the screen, written out. - [04:56.138](https://academa.ai/lectures/the-particle-in-a-box?t=296.13758333333334): level\_2 is shown on the screen, written out. - [04:56.288](https://academa.ai/lectures/the-particle-in-a-box?t=296.2875833333334): level\_3 is shown on the screen, written out. - [05:1.559](https://academa.ai/lectures/the-particle-in-a-box?t=301.55858333333333): zero\_line is shown on the screen, written out. ##### [05:3.03](https://academa.ai/lectures/the-particle-in-a-box?t=303.02958333333333) Narration: That nonzero ground-state energy is unavoidable. A wave trapped in a finite region cannot be perfectly flat, so it cannot have zero curvature and zero kinetic energy. Board: base\_formula — a Math \[text\] that says "$E\_n=frac(n^2 h^2, 8 m L^2)$"; picture — a Figure (x\_range=(0.0, 10.0), y\_range=(0.0, 7.5)); heading — a Heading that says "Widen the Box"; left\_wall — a Line \[blue\] drawn in picture (start=((7.0 - (width / 2.0)), 5.25), end=((7.0 - (width / 2.0)), 7.15)); right\_wall — a Line \[blue\] drawn in picture (start=((7.0 + (width / 2.0)), 5.25), end=((7.0 + (width / 2.0)), 7.15)); box\_floor — a Line \[gray\] drawn in picture (start=((7.0 - (width / 2.0)), 5.25), end=((7.0 + (width / 2.0)), 5.25)); width\_brace — a Brace \[yellow\] labelled "L" drawn in picture (x\_start=(7.0 - (width / 2.0)), x\_end=(7.0 + (width / 2.0)), y=5.25); zero\_line — a Line \[gray\] labelled "E=0" drawn in picture (start=(0.8, 0.55), end=(4.2, 0.55), dashed=True); level\_1 — a Line \[green\] labelled "E\_1" drawn in picture (start=(1.0, (0.55 + (2.2 / (width \*\* 2.0)))), end=(4.0, (0.55 + (2.2 / (width \*\* 2.0))))); level\_2 — a Line \[yellow\] labelled "E\_2" drawn in picture (start=(1.0, (0.55 + (8.8 / (width \*\* 2.0)))), end=(4.0, (0.55 + (8.8 / (width \*\* 2.0))))); level\_3 — a Line \[red\] labelled "E\_3" drawn in picture (start=(1.0, (0.55 + (19.8 / (width \*\* 2.0)))), end=(4.0, (0.55 + (19.8 / (width \*\* 2.0))))) Actions: - [05:4.238](https://academa.ai/lectures/the-particle-in-a-box?t=304.23758333333336): level\_1 is indicated — a transient flash. - [05:8.034](https://academa.ai/lectures/the-particle-in-a-box?t=308.03358333333335): left\_wall is indicated — a transient flash. - [05:8.034](https://academa.ai/lectures/the-particle-in-a-box?t=308.03358333333335): right\_wall is indicated — a transient flash. ##### [05:15.472](https://academa.ai/lectures/the-particle-in-a-box?t=315.47208333333333) Narration: The difference between neighboring levels is delta E n. Start with E n plus one minus E n. Board: Unchanged from the preceding beat in this scene. Actions: - [05:15.983](https://academa.ai/lectures/the-particle-in-a-box?t=315.98258333333337): gap\_1 is shown on the screen, written out. - [05:21.057](https://academa.ai/lectures/the-particle-in-a-box?t=321.0565833333334): spacing\_work is shown on the screen, written out. ##### [05:23.061](https://academa.ai/lectures/the-particle-in-a-box?t=323.06108333333333) Narration: Insert the box energies. The numerator contains n plus one squared minus n squared, while every level carries the same factor one over L squared. Board: base\_formula — a Math \[text\] that says "$E\_n=frac(n^2 h^2, 8 m L^2)$"; picture — a Figure (x\_range=(0.0, 10.0), y\_range=(0.0, 7.5)); heading — a Heading that says "Widen the Box"; left\_wall — a Line \[blue\] drawn in picture (start=((7.0 - (width / 2.0)), 5.25), end=((7.0 - (width / 2.0)), 7.15)); right\_wall — a Line \[blue\] drawn in picture (start=((7.0 + (width / 2.0)), 5.25), end=((7.0 + (width / 2.0)), 7.15)); box\_floor — a Line \[gray\] drawn in picture (start=((7.0 - (width / 2.0)), 5.25), end=((7.0 + (width / 2.0)), 5.25)); width\_brace — a Brace \[yellow\] labelled "L" drawn in picture (x\_start=(7.0 - (width / 2.0)), x\_end=(7.0 + (width / 2.0)), y=5.25); zero\_line — a Line \[gray\] labelled "E=0" drawn in picture (start=(0.8, 0.55), end=(4.2, 0.55), dashed=True); level\_1 — a Line \[green\] labelled "E\_1" drawn in picture (start=(1.0, (0.55 + (2.2 / (width \*\* 2.0)))), end=(4.0, (0.55 + (2.2 / (width \*\* 2.0))))); level\_2 — a Line \[yellow\] labelled "E\_2" drawn in picture (start=(1.0, (0.55 + (8.8 / (width \*\* 2.0)))), end=(4.0, (0.55 + (8.8 / (width \*\* 2.0))))); level\_3 — a Line \[red\] labelled "E\_3" drawn in picture (start=(1.0, (0.55 + (19.8 / (width \*\* 2.0)))), end=(4.0, (0.55 + (19.8 / (width \*\* 2.0))))); gap\_1 — a Line \[yellow\] labelled "Delta E\_1" drawn in picture (start=(4.45, (0.55 + (2.2 / (width \*\* 2.0)))), end=(4.45, (0.55 + (8.8 / (width \*\* 2.0))))) Actions: - [05:23.282](https://academa.ai/lectures/the-particle-in-a-box?t=323.28158333333334): spacing\_work is shown on the screen, written out. - [05:32.872](https://academa.ai/lectures/the-particle-in-a-box?t=332.8715833333333): spacing\_work (the "L^2" part) is indicated — a transient flash. ##### [05:34.586](https://academa.ai/lectures/the-particle-in-a-box?t=334.58608333333336) Narration: The difference of the squares is two n plus one. So the gap is also proportional to one over L squared. Board: Unchanged from the preceding beat in this scene. Actions: - [05:36.363](https://academa.ai/lectures/the-particle-in-a-box?t=336.36258333333336): spacing\_work is shown on the screen, written out. - [05:36.363](https://academa.ai/lectures/the-particle-in-a-box?t=336.36258333333336): spacing\_work (the "2n+1" part) is indicated — a transient flash. - [05:40.74](https://academa.ai/lectures/the-particle-in-a-box?t=340.73958333333337): spacing\_work (the "L^2" part) is indicated — a transient flash. ##### [05:42.489](https://academa.ai/lectures/the-particle-in-a-box?t=342.48858333333334) Narration: Now widen the box. Watch both walls move apart while the three energy levels descend and crowd together. Board: Unchanged from the preceding beat in this scene. Actions: - [05:43.244](https://academa.ai/lectures/the-particle-in-a-box?t=343.24358333333333): left\_wall is redrawn as the numbers it depends on change. - [05:43.244](https://academa.ai/lectures/the-particle-in-a-box?t=343.24358333333333): right\_wall is redrawn as the numbers it depends on change. - [05:43.244](https://academa.ai/lectures/the-particle-in-a-box?t=343.24358333333333): box\_floor is redrawn as the numbers it depends on change. - [05:43.244](https://academa.ai/lectures/the-particle-in-a-box?t=343.24358333333333): width\_brace is redrawn as the numbers it depends on change. - [05:43.244](https://academa.ai/lectures/the-particle-in-a-box?t=343.24358333333333): level\_1 is redrawn as the numbers it depends on change. - [05:43.244](https://academa.ai/lectures/the-particle-in-a-box?t=343.24358333333333): level\_2 is redrawn as the numbers it depends on change. - [05:43.244](https://academa.ai/lectures/the-particle-in-a-box?t=343.24358333333333): level\_3 is redrawn as the numbers it depends on change. - [05:43.244](https://academa.ai/lectures/the-particle-in-a-box?t=343.24358333333333): gap\_1 is redrawn as the numbers it depends on change. - [05:43.244](https://academa.ai/lectures/the-particle-in-a-box?t=343.24358333333333): width ticks to 4.2. ##### [05:51.123](https://academa.ai/lectures/the-particle-in-a-box?t=351.12258333333335) Narration: Nothing has changed about the integer labels. The allowed states are still numbered one, two, three, and onward. What changed is the energy scale attached to that list. Board: Unchanged from the preceding beat in this scene. Actions: - [05:56.464](https://academa.ai/lectures/the-particle-in-a-box?t=356.46358333333336): level\_1 is indicated — a transient flash. - [05:57.044](https://academa.ai/lectures/the-particle-in-a-box?t=357.04358333333334): level\_2 is indicated — a transient flash. - [05:57.695](https://academa.ai/lectures/the-particle-in-a-box?t=357.69458333333336): level\_3 is indicated — a transient flash. ##### [06:4.111](https://academa.ai/lectures/the-particle-in-a-box?t=364.11058333333335) Narration: A very wide box therefore has many levels inside any modest energy interval. The spectrum is discrete in principle, but the gaps may become too small for an experiment to distinguish. Board: Unchanged from the preceding beat in this scene. Actions: - [06:12.203](https://academa.ai/lectures/the-particle-in-a-box?t=372.20258333333334): gap\_1 is indicated — a transient flash. - [06:15.558](https://academa.ai/lectures/the-particle-in-a-box?t=375.55808333333334): base\_formula is hidden from the screen — left the board. - [06:15.558](https://academa.ai/lectures/the-particle-in-a-box?t=375.55808333333334): heading is hidden from the screen — left the board. - [06:15.558](https://academa.ai/lectures/the-particle-in-a-box?t=375.55808333333334): picture is hidden from the screen — left the board. - [06:15.558](https://academa.ai/lectures/the-particle-in-a-box?t=375.55808333333334): left\_wall is hidden from the screen — picture left the board. - [06:15.558](https://academa.ai/lectures/the-particle-in-a-box?t=375.55808333333334): right\_wall is hidden from the screen — picture left the board. - [06:15.558](https://academa.ai/lectures/the-particle-in-a-box?t=375.55808333333334): box\_floor is hidden from the screen — picture left the board. - [06:15.558](https://academa.ai/lectures/the-particle-in-a-box?t=375.55808333333334): width\_brace is hidden from the screen — picture left the board. - [06:15.558](https://academa.ai/lectures/the-particle-in-a-box?t=375.55808333333334): zero\_line is hidden from the screen — picture left the board. - [06:15.558](https://academa.ai/lectures/the-particle-in-a-box?t=375.55808333333334): level\_1 is hidden from the screen — picture left the board. - [06:15.558](https://academa.ai/lectures/the-particle-in-a-box?t=375.55808333333334): level\_2 is hidden from the screen — picture left the board. - [06:15.558](https://academa.ai/lectures/the-particle-in-a-box?t=375.55808333333334): level\_3 is hidden from the screen — picture left the board. - [06:15.558](https://academa.ai/lectures/the-particle-in-a-box?t=375.55808333333334): gap\_1 is hidden from the screen — picture left the board. - [06:15.558](https://academa.ai/lectures/the-particle-in-a-box?t=375.55808333333334): spacing\_work is hidden from the screen — left the board. ##### [06:16.158](https://academa.ai/lectures/the-particle-in-a-box?t=376.15808333333337) Narration: This is one route back toward classical physics. First, increasing L lowers the entire spectrum and shrinks every gap as one over L squared. Board: Empty. Actions: - [06:16.158](https://academa.ai/lectures/the-particle-in-a-box?t=376.15808333333337): heading\_limit is shown on the screen, written out. - [06:20.501](https://academa.ai/lectures/the-particle-in-a-box?t=380.50058333333334): item\_1 is shown on the screen, written out. - [06:24.924](https://academa.ai/lectures/the-particle-in-a-box?t=384.92358333333334): item\_1 (the "1/L^2" part) is indicated — a transient flash. ##### [06:26.615](https://academa.ai/lectures/the-particle-in-a-box?t=386.61458333333337) Narration: There is a second effect at high quantum number. Divide a gap by the energy itself, and most constants cancel. Board: item\_1 — a Text \[text\] that says "Larger $L$ makes every energy proportional to $1/L^2$."; heading\_limit — a Heading that says "When Discrete Levels Look Continuous" Actions: - [06:29.982](https://academa.ai/lectures/the-particle-in-a-box?t=389.98158333333333): relative is shown on the screen, written out. ##### [06:34.622](https://academa.ai/lectures/the-particle-in-a-box?t=394.62208333333336) Narration: The relative gap is two n plus one over n squared, which approaches two over n. At large n, neighboring states differ by a tiny fraction of their total energy. Board: item\_1 — a Text \[text\] that says "Larger $L$ makes every energy proportional to $1/L^2$."; relative — a Math \[text\] that says "$frac(Delta E\_n, E\_n)=frac(2n+1, n^2) approx frac(2, n)$"; heading\_limit — a Heading that says "When Discrete Levels Look Continuous" Actions: - [06:36.027](https://academa.ai/lectures/the-particle-in-a-box?t=396.02658333333335): relative (the "frac(2n+1, n^2)" part) is indicated — a transient flash. - [06:39.127](https://academa.ai/lectures/the-particle-in-a-box?t=399.1265833333334): relative (the "frac(2, n)" part) is indicated — a transient flash. - [06:41.24](https://academa.ai/lectures/the-particle-in-a-box?t=401.23958333333337): item\_2 is shown on the screen, written out. ##### [06:46.53](https://academa.ai/lectures/the-particle-in-a-box?t=406.5300833333334) Narration: Finally, every real measurement has limited resolution. If thousands of allowed energies fall inside one unresolved band, changing energy appears continuous even though the microscopic list remains discrete. Board: item\_1 — a Text \[text\] that says "Larger $L$ makes every energy proportional to $1/L^2$."; relative — a Math \[text\] that says "$frac(Delta E\_n, E\_n)=frac(2n+1, n^2) approx frac(2, n)$"; item\_2 — a Text \[text\] that says "At large $n$, neighboring levels are close relative to $E\_n$."; heading\_limit — a Heading that says "When Discrete Levels Look Continuous" Actions: - [06:49.119](https://academa.ai/lectures/the-particle-in-a-box?t=409.1185833333334): item\_3 is shown on the screen, written out. - [06:53.624](https://academa.ai/lectures/the-particle-in-a-box?t=413.6235833333334): item\_3 (the "cannot resolve" part) is indicated — a transient flash. ##### [07:0.865](https://academa.ai/lectures/the-particle-in-a-box?t=420.86458333333337) Narration: The classical world does not require the quantum rules to switch off. Large dimensions, high quantum numbers, and limited resolution make the steps too fine to notice. Board: item\_1 — a Text \[text\] that says "Larger $L$ makes every energy proportional to $1/L^2$."; relative — a Math \[text\] that says "$frac(Delta E\_n, E\_n)=frac(2n+1, n^2) approx frac(2, n)$"; item\_2 — a Text \[text\] that says "At large $n$, neighboring levels are close relative to $E\_n$."; item\_3 — a Text \[text\] that says "Finite measurements then cannot resolve the individual levels."; heading\_limit — a Heading that says "When Discrete Levels Look Continuous" Actions: - [07:9.828](https://academa.ai/lectures/the-particle-in-a-box?t=429.82758333333334): A box is drawn around relative. - [07:11.228](https://academa.ai/lectures/the-particle-in-a-box?t=431.22758333333337): heading\_limit is hidden from the screen — left the board. - [07:11.228](https://academa.ai/lectures/the-particle-in-a-box?t=431.22758333333337): item\_1 is hidden from the screen — left the board. - [07:11.228](https://academa.ai/lectures/the-particle-in-a-box?t=431.22758333333337): item\_2 is hidden from the screen — left the board. - [07:11.228](https://academa.ai/lectures/the-particle-in-a-box?t=431.22758333333337): item\_3 is hidden from the screen — left the board. - [07:11.228](https://academa.ai/lectures/the-particle-in-a-box?t=431.22758333333337): relative is hidden from the screen — left the board. ### Scene 4: [Probability and the Places Never Found](https://academa.ai/lectures/the-particle-in-a-box?t=432.26925000000006) Span: 07:12.269–09:33.096 (432.26925000000006s–573.0964166666668s). #### Objects - area\_1: an AreaUnder \[blue\] drawn in case\_1 (x\_range=(0.0, 1.0), target='curve\_1') - area\_2: an AreaUnder \[green\] drawn in case\_2 (x\_range=(0.0, 1.0), target='curve\_2') - area\_3: an AreaUnder \[red\] drawn in case\_3 (x\_range=(0.0, 1.0), target='curve\_3') - born: a Math \[text\] that says "$rho(x)=abs(psi(x))^2$" - caption\_1: a Math \[text\] that says "$n=1$" - caption\_2: a Math \[text\] that says "$n=2$" - caption\_3: a Math \[text\] that says "$n=3$" - case\_1: an Axes (y\_range=(0.0, 2.2), x\_ticks\_every=0.5, x\_label='x/L') - case\_2: an Axes (y\_range=(0.0, 2.2), x\_ticks\_every=0.5, x\_label='x/L') - case\_3: an Axes (y\_range=(0.0, 2.2), x\_ticks\_every=0.5, x\_label='x/L') - curve\_1: a FunctionPlot \[blue\] drawn in case\_1 (function=\, x\_range=(0.0, 1.0)) - curve\_2: a FunctionPlot \[green\] drawn in case\_2 (function=\, x\_range=(0.0, 1.0)) - curve\_3: a FunctionPlot \[red\] drawn in case\_3 (function=\, x\_range=(0.0, 1.0)) - density: a FunctionPlot \[green\] drawn in density\_axes (function=\, x\_range=(0.0, 1.0)) - density\_area: an AreaUnder \[green\] drawn in density\_axes (x\_range=(0.0, 1.0), target='density') - density\_axes: an Axes (y\_range=(0.0, 2.2), x\_ticks\_every=0.5, y\_ticks\_every=1.0) - density\_node: a Point \[yellow\] labelled "upright("zero probability")" drawn in density\_axes (location=(0.5, 0.0)) - heading\_square: a Heading that says "From Amplitude to Probability" - heading\_states: a Heading that says "The First Three Probability Densities" - math: a Math \[text\] that says "$psi\_2(x)$" - math\_2: a Math \[text\] that says "$abs(psi\_2(x))^2$" - node\_2: a Point \[yellow\] labelled "L/2" drawn in case\_2 (location=(0.5, 0.0)) - node\_3a: a Point \[yellow\] labelled "L/3" drawn in case\_3 (location=(0.3333333333333333, 0.0)) - node\_3b: a Point \[yellow\] labelled "2L/3" drawn in case\_3 (location=(0.6666666666666666, 0.0)) - node\_count: a Math \[text\] that says "$N\_(upright("nodes"))=n-1$" - normalization: a Math \[text\] that says "$integral\_0^L abs(psi\_n(x))^2 thin dif x=1$" - wave: a FunctionPlot \[blue\] drawn in wave\_axes (function=\, x\_range=(0.0, 1.0)) - wave\_axes: an Axes (y\_range=(-1.65, 1.65), x\_ticks\_every=0.5, y\_ticks\_every=1.0) - wave\_node: a Point \[yellow\] labelled "x=L/2" drawn in wave\_axes (location=(0.5, 0.0)) #### Beats ##### [07:12.269](https://academa.ai/lectures/the-particle-in-a-box?t=432.26925000000006) Narration: A wavefunction is not itself a probability. It can be positive, negative, or even complex. What predicts where the particle may be detected is its absolute square. Board: Empty. Actions: - [07:12.269](https://academa.ai/lectures/the-particle-in-a-box?t=432.26925000000006): heading\_square is shown on the screen, written out. - [07:12.478](https://academa.ai/lectures/the-particle-in-a-box?t=432.47825000000006): wave\_axes is shown on the screen, written out. - [07:12.478](https://academa.ai/lectures/the-particle-in-a-box?t=432.47825000000006): wave is shown on the screen, drawn. - [07:21.766](https://academa.ai/lectures/the-particle-in-a-box?t=441.76625000000007): wave\_axes moves to a new place on the board. - [07:21.766](https://academa.ai/lectures/the-particle-in-a-box?t=441.76625000000007): born is shown on the screen, written out. ##### [07:23.649](https://academa.ai/lectures/the-particle-in-a-box?t=443.64875000000006) Narration: Here is the second state. Its left lobe is positive and its right lobe is negative, but a probability density cannot be negative. Board: wave\_axes — an Axes (y\_range=(-1.65, 1.65), x\_ticks\_every=0.5, y\_ticks\_every=1.0); born — a Math \[text\] that says "$rho(x)=abs(psi(x))^2$"; heading\_square — a Heading that says "From Amplitude to Probability"; wave — a FunctionPlot \[blue\] drawn in wave\_axes (function=\, x\_range=(0.0, 1.0)) Actions: - [07:25.559](https://academa.ai/lectures/the-particle-in-a-box?t=445.5592500000001): wave is indicated — a transient flash. - [07:27.126](https://academa.ai/lectures/the-particle-in-a-box?t=447.12625): wave is indicated — a transient flash. ##### [07:31.906](https://academa.ai/lectures/the-particle-in-a-box?t=451.90575000000007) Narration: Square the amplitude at every position. Both lobes rise above the axis, producing two regions where detection is likely. Board: Unchanged from the preceding beat in this scene. Actions: - [07:32.312](https://academa.ai/lectures/the-particle-in-a-box?t=452.31225000000006): density\_axes is shown on the screen, written out. - [07:32.312](https://academa.ai/lectures/the-particle-in-a-box?t=452.31225000000006): density is shown on the screen, drawn. - [07:38.047](https://academa.ai/lectures/the-particle-in-a-box?t=458.0472500000001): density\_area is shown on the screen, written out. ##### [07:40.761](https://academa.ai/lectures/the-particle-in-a-box?t=460.76075000000003) Narration: The total shaded area is one after normalization. That does not say the particle has a known position. It says some detection somewhere in the box has total probability one. Board: wave\_axes — an Axes (y\_range=(-1.65, 1.65), x\_ticks\_every=0.5, y\_ticks\_every=1.0); born — a Math \[text\] that says "$rho(x)=abs(psi(x))^2$"; density\_axes — an Axes (y\_range=(0.0, 2.2), x\_ticks\_every=0.5, y\_ticks\_every=1.0); heading\_square — a Heading that says "From Amplitude to Probability"; wave — a FunctionPlot \[blue\] drawn in wave\_axes (function=\, x\_range=(0.0, 1.0)); density — a FunctionPlot \[green\] drawn in density\_axes (function=\, x\_range=(0.0, 1.0)); density\_area — an AreaUnder \[green\] drawn in density\_axes (x\_range=(0.0, 1.0), target='density') Actions: - [07:41.77](https://academa.ai/lectures/the-particle-in-a-box?t=461.77025000000003): density\_area is indicated — a transient flash. - [07:42.711](https://academa.ai/lectures/the-particle-in-a-box?t=462.71125000000006): normalization is shown on the screen, written out. - [07:51.453](https://academa.ai/lectures/the-particle-in-a-box?t=471.45325): normalization (the "1" part) is indicated — a transient flash. ##### [07:52.832](https://academa.ai/lectures/the-particle-in-a-box?t=472.83175000000006) Narration: At the middle, the wavefunction is exactly zero. Squaring zero still gives zero, so the particle is never detected at this node. Board: wave\_axes — an Axes (y\_range=(-1.65, 1.65), x\_ticks\_every=0.5, y\_ticks\_every=1.0); born — a Math \[text\] that says "$rho(x)=abs(psi(x))^2$"; density\_axes — an Axes (y\_range=(0.0, 2.2), x\_ticks\_every=0.5, y\_ticks\_every=1.0); normalization — a Math \[text\] that says "$integral\_0^L abs(psi\_n(x))^2 thin dif x=1$"; heading\_square — a Heading that says "From Amplitude to Probability"; wave — a FunctionPlot \[blue\] drawn in wave\_axes (function=\, x\_range=(0.0, 1.0)); density — a FunctionPlot \[green\] drawn in density\_axes (function=\, x\_range=(0.0, 1.0)); density\_area — an AreaUnder \[green\] drawn in density\_axes (x\_range=(0.0, 1.0), target='density') Actions: - [07:53.551](https://academa.ai/lectures/the-particle-in-a-box?t=473.55125000000004): wave\_node is shown on the screen, written out. - [07:55.513](https://academa.ai/lectures/the-particle-in-a-box?t=475.5132500000001): density\_node is shown on the screen, written out. - [07:59.588](https://academa.ai/lectures/the-particle-in-a-box?t=479.5882500000001): density\_node is indicated — a transient flash. - [08:1.446](https://academa.ai/lectures/the-particle-in-a-box?t=481.4462500000001): born is hidden from the screen — left the board. - [08:1.446](https://academa.ai/lectures/the-particle-in-a-box?t=481.4462500000001): density\_axes is hidden from the screen — left the board. - [08:1.446](https://academa.ai/lectures/the-particle-in-a-box?t=481.4462500000001): density is hidden from the screen — density\_axes left the board. - [08:1.446](https://academa.ai/lectures/the-particle-in-a-box?t=481.4462500000001): density\_area is hidden from the screen — density\_axes left the board. - [08:1.446](https://academa.ai/lectures/the-particle-in-a-box?t=481.4462500000001): density\_node is hidden from the screen — density\_axes left the board. - [08:1.446](https://academa.ai/lectures/the-particle-in-a-box?t=481.4462500000001): heading\_square is hidden from the screen — left the board. - [08:1.446](https://academa.ai/lectures/the-particle-in-a-box?t=481.4462500000001): normalization is hidden from the screen — left the board. - [08:1.446](https://academa.ai/lectures/the-particle-in-a-box?t=481.4462500000001): wave\_axes is hidden from the screen — left the board. - [08:1.446](https://academa.ai/lectures/the-particle-in-a-box?t=481.4462500000001): wave is hidden from the screen — wave\_axes left the board. - [08:1.446](https://academa.ai/lectures/the-particle-in-a-box?t=481.4462500000001): wave\_node is hidden from the screen — wave\_axes left the board. ##### [08:2.046](https://academa.ai/lectures/the-particle-in-a-box?t=482.04625000000004) Narration: Now compare the first three stationary states. The first density has one broad peak and no internal node. Board: Empty. Actions: - [08:2.046](https://academa.ai/lectures/the-particle-in-a-box?t=482.04625000000004): heading\_states is shown on the screen, written out. - [08:3.079](https://academa.ai/lectures/the-particle-in-a-box?t=483.07925000000006): case\_1 is shown on the screen, written out. - [08:3.079](https://academa.ai/lectures/the-particle-in-a-box?t=483.07925000000006): curve\_1 is shown on the screen, written out. - [08:6.76](https://academa.ai/lectures/the-particle-in-a-box?t=486.76025000000004): case\_1 moves to a new place on the board. - [08:6.76](https://academa.ai/lectures/the-particle-in-a-box?t=486.76025000000004): area\_1 is shown on the screen, written out. - [08:6.76](https://academa.ai/lectures/the-particle-in-a-box?t=486.76025000000004): caption\_1 is shown on the screen, written out. ##### [08:9.554](https://academa.ai/lectures/the-particle-in-a-box?t=489.55375000000004) Narration: The probability still vanishes at both walls. Those zeros come from the confinement condition shared by every state. Board: case\_1 — an Axes (y\_range=(0.0, 2.2), x\_ticks\_every=0.5, x\_label='x/L'); caption\_1 — a Math \[text\] that says "$n=1$"; heading\_states — a Heading that says "The First Three Probability Densities"; curve\_1 — a FunctionPlot \[blue\] drawn in case\_1 (function=\, x\_range=(0.0, 1.0)); area\_1 — an AreaUnder \[blue\] drawn in case\_1 (x\_range=(0.0, 1.0), target='curve\_1') Actions: - [08:11.655](https://academa.ai/lectures/the-particle-in-a-box?t=491.6552500000001): The point (0.0, 0.0) in case\_1 is lit up. - [08:11.655](https://academa.ai/lectures/the-particle-in-a-box?t=491.6552500000001): The point (1.0, 0.0) in case\_1 is lit up. - [08:16.398](https://academa.ai/lectures/the-particle-in-a-box?t=496.3982500000001): case\_1: retire a lit point (unemphasize\_point). - [08:16.398](https://academa.ai/lectures/the-particle-in-a-box?t=496.3982500000001): case\_1: retire a lit point (unemphasize\_point). ##### [08:16.998](https://academa.ai/lectures/the-particle-in-a-box?t=496.99825000000004) Narration: The second density has two peaks. Between them is one internal node at L over two, a position with exactly zero probability. Board: Unchanged from the preceding beat in this scene. Actions: - [08:17.283](https://academa.ai/lectures/the-particle-in-a-box?t=497.28325000000007): case\_2 is shown on the screen, written out. - [08:17.283](https://academa.ai/lectures/the-particle-in-a-box?t=497.28325000000007): curve\_2 is shown on the screen, written out. - [08:18.27](https://academa.ai/lectures/the-particle-in-a-box?t=498.27025000000003): area\_2 is shown on the screen, written out. - [08:18.27](https://academa.ai/lectures/the-particle-in-a-box?t=498.27025000000003): caption\_2 is shown on the screen, written out. - [08:20.824](https://academa.ai/lectures/the-particle-in-a-box?t=500.82425000000006): node\_2 is shown on the screen, written out. - [08:23.97](https://academa.ai/lectures/the-particle-in-a-box?t=503.9702500000001): node\_2 is indicated — a transient flash. ##### [08:25.998](https://academa.ai/lectures/the-particle-in-a-box?t=505.99825000000004) Narration: The third density has three peaks and two internal nodes, at L over three and two L over three. Board: case\_1 — an Axes (y\_range=(0.0, 2.2), x\_ticks\_every=0.5, x\_label='x/L'); caption\_1 — a Math \[text\] that says "$n=1$"; case\_2 — an Axes (y\_range=(0.0, 2.2), x\_ticks\_every=0.5, x\_label='x/L'); caption\_2 — a Math \[text\] that says "$n=2$"; heading\_states — a Heading that says "The First Three Probability Densities"; curve\_1 — a FunctionPlot \[blue\] drawn in case\_1 (function=\, x\_range=(0.0, 1.0)); area\_1 — an AreaUnder \[blue\] drawn in case\_1 (x\_range=(0.0, 1.0), target='curve\_1'); curve\_2 — a FunctionPlot \[green\] drawn in case\_2 (function=\, x\_range=(0.0, 1.0)); area\_2 — an AreaUnder \[green\] drawn in case\_2 (x\_range=(0.0, 1.0), target='curve\_2'); node\_2 — a Point \[yellow\] labelled "L/2" drawn in case\_2 (location=(0.5, 0.0)) Actions: - [08:26.532](https://academa.ai/lectures/the-particle-in-a-box?t=506.53225000000003): case\_3 is shown on the screen, written out. - [08:26.532](https://academa.ai/lectures/the-particle-in-a-box?t=506.53225000000003): curve\_3 is shown on the screen, written out. - [08:27.449](https://academa.ai/lectures/the-particle-in-a-box?t=507.44925000000006): area\_3 is shown on the screen, written out. - [08:27.449](https://academa.ai/lectures/the-particle-in-a-box?t=507.44925000000006): caption\_3 is shown on the screen, written out. - [08:30.061](https://academa.ai/lectures/the-particle-in-a-box?t=510.0612500000001): node\_3a is shown on the screen, written out. - [08:31.129](https://academa.ai/lectures/the-particle-in-a-box?t=511.12925000000007): node\_3b is shown on the screen, written out. ##### [08:33.529](https://academa.ai/lectures/the-particle-in-a-box?t=513.52925) Narration: A node is stronger than a low-probability region. At a node the wavefunction vanishes exactly, so an ideal position measurement never returns that point while the particle remains in that state. Board: case\_1 — an Axes (y\_range=(0.0, 2.2), x\_ticks\_every=0.5, x\_label='x/L'); caption\_1 — a Math \[text\] that says "$n=1$"; case\_2 — an Axes (y\_range=(0.0, 2.2), x\_ticks\_every=0.5, x\_label='x/L'); caption\_2 — a Math \[text\] that says "$n=2$"; case\_3 — an Axes (y\_range=(0.0, 2.2), x\_ticks\_every=0.5, x\_label='x/L'); caption\_3 — a Math \[text\] that says "$n=3$"; heading\_states — a Heading that says "The First Three Probability Densities"; curve\_1 — a FunctionPlot \[blue\] drawn in case\_1 (function=\, x\_range=(0.0, 1.0)); area\_1 — an AreaUnder \[blue\] drawn in case\_1 (x\_range=(0.0, 1.0), target='curve\_1'); curve\_2 — a FunctionPlot \[green\] drawn in case\_2 (function=\, x\_range=(0.0, 1.0)); area\_2 — an AreaUnder \[green\] drawn in case\_2 (x\_range=(0.0, 1.0), target='curve\_2'); node\_2 — a Point \[yellow\] labelled "L/2" drawn in case\_2 (location=(0.5, 0.0)); curve\_3 — a FunctionPlot \[red\] drawn in case\_3 (function=\, x\_range=(0.0, 1.0)); area\_3 — an AreaUnder \[red\] drawn in case\_3 (x\_range=(0.0, 1.0), target='curve\_3'); node\_3a — a Point \[yellow\] labelled "L/3" drawn in case\_3 (location=(0.3333333333333333, 0.0)); node\_3b — a Point \[yellow\] labelled "2L/3" drawn in case\_3 (location=(0.6666666666666666, 0.0)) Actions: - [08:34.144](https://academa.ai/lectures/the-particle-in-a-box?t=514.14425): node\_2 is indicated — a transient flash. - [08:34.344](https://academa.ai/lectures/the-particle-in-a-box?t=514.3442500000001): node\_3a is indicated — a transient flash. - [08:34.544](https://academa.ai/lectures/the-particle-in-a-box?t=514.54425): node\_3b is indicated — a transient flash. ##### [08:47.04](https://academa.ai/lectures/the-particle-in-a-box?t=527.03975) Narration: The pattern is systematic. State n has n lobes in its density and n minus one internal nodes. Board: Unchanged from the preceding beat in this scene. Actions: - [08:48.073](https://academa.ai/lectures/the-particle-in-a-box?t=528.07325): node\_count is shown on the screen, written out. - [08:53.042](https://academa.ai/lectures/the-particle-in-a-box?t=533.0422500000001): node\_count (the "n-1" part) is indicated — a transient flash. ##### [08:56.092](https://academa.ai/lectures/the-particle-in-a-box?t=536.09225) Narration: Higher states oscillate more rapidly because their allowed wavelengths are shorter. More oscillations produce more exact cancellations and therefore more nodes. Board: case\_1 — an Axes (y\_range=(0.0, 2.2), x\_ticks\_every=0.5, x\_label='x/L'); caption\_1 — a Math \[text\] that says "$n=1$"; case\_2 — an Axes (y\_range=(0.0, 2.2), x\_ticks\_every=0.5, x\_label='x/L'); caption\_2 — a Math \[text\] that says "$n=2$"; case\_3 — an Axes (y\_range=(0.0, 2.2), x\_ticks\_every=0.5, x\_label='x/L'); caption\_3 — a Math \[text\] that says "$n=3$"; node\_count — a Math \[text\] that says "$N\_(upright("nodes"))=n-1$"; heading\_states — a Heading that says "The First Three Probability Densities"; curve\_1 — a FunctionPlot \[blue\] drawn in case\_1 (function=\, x\_range=(0.0, 1.0)); area\_1 — an AreaUnder \[blue\] drawn in case\_1 (x\_range=(0.0, 1.0), target='curve\_1'); curve\_2 — a FunctionPlot \[green\] drawn in case\_2 (function=\, x\_range=(0.0, 1.0)); area\_2 — an AreaUnder \[green\] drawn in case\_2 (x\_range=(0.0, 1.0), target='curve\_2'); node\_2 — a Point \[yellow\] labelled "L/2" drawn in case\_2 (location=(0.5, 0.0)); curve\_3 — a FunctionPlot \[red\] drawn in case\_3 (function=\, x\_range=(0.0, 1.0)); area\_3 — an AreaUnder \[red\] drawn in case\_3 (x\_range=(0.0, 1.0), target='curve\_3'); node\_3a — a Point \[yellow\] labelled "L/3" drawn in case\_3 (location=(0.3333333333333333, 0.0)); node\_3b — a Point \[yellow\] labelled "2L/3" drawn in case\_3 (location=(0.6666666666666666, 0.0)) Actions: - [08:56.44](https://academa.ai/lectures/the-particle-in-a-box?t=536.4402500000001): curve\_1 is indicated — a transient flash. - [09:5.427](https://academa.ai/lectures/the-particle-in-a-box?t=545.4272500000001): curve\_3 is indicated — a transient flash. ##### [09:7.188](https://academa.ai/lectures/the-particle-in-a-box?t=547.18775) Narration: These densities also warn us against imagining a classical bead bouncing between the walls. A classical bead can pass through every interior position. A stationary quantum state can forbid particular positions completely. Board: Unchanged from the preceding beat in this scene. Actions: - [09:19.216](https://academa.ai/lectures/the-particle-in-a-box?t=559.2162500000001): node\_2 is indicated — a transient flash. - [09:19.416](https://academa.ai/lectures/the-particle-in-a-box?t=559.41625): node\_3a is indicated — a transient flash. - [09:19.616](https://academa.ai/lectures/the-particle-in-a-box?t=559.61625): node\_3b is indicated — a transient flash. ##### [09:22.405](https://academa.ai/lectures/the-particle-in-a-box?t=562.40475) Narration: The allowed energy tells us which stationary pattern we have. The square of that pattern tells us where detections can occur, and its nodes mark where they cannot. Board: Unchanged from the preceding beat in this scene. Actions: - [09:31.031](https://academa.ai/lectures/the-particle-in-a-box?t=571.0312500000001): A box is drawn around node\_count. - [09:32.055](https://academa.ai/lectures/the-particle-in-a-box?t=572.05475): caption\_1 is hidden from the screen — left the board. - [09:32.055](https://academa.ai/lectures/the-particle-in-a-box?t=572.05475): caption\_2 is hidden from the screen — left the board. - [09:32.055](https://academa.ai/lectures/the-particle-in-a-box?t=572.05475): caption\_3 is hidden from the screen — left the board. - [09:32.055](https://academa.ai/lectures/the-particle-in-a-box?t=572.05475): case\_1 is hidden from the screen — left the board. - [09:32.055](https://academa.ai/lectures/the-particle-in-a-box?t=572.05475): curve\_1 is hidden from the screen — case\_1 left the board. - [09:32.055](https://academa.ai/lectures/the-particle-in-a-box?t=572.05475): area\_1 is hidden from the screen — case\_1 left the board. - [09:32.055](https://academa.ai/lectures/the-particle-in-a-box?t=572.05475): case\_2 is hidden from the screen — left the board. - [09:32.055](https://academa.ai/lectures/the-particle-in-a-box?t=572.05475): curve\_2 is hidden from the screen — case\_2 left the board. - [09:32.055](https://academa.ai/lectures/the-particle-in-a-box?t=572.05475): area\_2 is hidden from the screen — case\_2 left the board. - [09:32.055](https://academa.ai/lectures/the-particle-in-a-box?t=572.05475): node\_2 is hidden from the screen — case\_2 left the board. - [09:32.055](https://academa.ai/lectures/the-particle-in-a-box?t=572.05475): case\_3 is hidden from the screen — left the board. - [09:32.055](https://academa.ai/lectures/the-particle-in-a-box?t=572.05475): curve\_3 is hidden from the screen — case\_3 left the board. - [09:32.055](https://academa.ai/lectures/the-particle-in-a-box?t=572.05475): area\_3 is hidden from the screen — case\_3 left the board. - [09:32.055](https://academa.ai/lectures/the-particle-in-a-box?t=572.05475): node\_3a is hidden from the screen — case\_3 left the board. - [09:32.055](https://academa.ai/lectures/the-particle-in-a-box?t=572.05475): node\_3b is hidden from the screen — case\_3 left the board. - [09:32.055](https://academa.ai/lectures/the-particle-in-a-box?t=572.05475): heading\_states is hidden from the screen — left the board. - [09:32.055](https://academa.ai/lectures/the-particle-in-a-box?t=572.05475): node\_count is hidden from the screen — left the board. ### Scene 5: [Why Conjugated Molecules Have Colour](https://academa.ai/lectures/the-particle-in-a-box?t=573.0964166666668) Span: 09:33.096–13:22.498 (573.0964166666668s–802.4982916666668s). #### Objects - bond\_1: a Line \[green\] drawn in molecule (start=(1.0, 2.0), end=(2.5, 2.45)) - bond\_2: a Line \[green\] drawn in molecule (start=(2.5, 2.45), end=(4.0, 2.0)) - bond\_3: a Line \[green\] drawn in molecule (start=(4.0, 2.0), end=(5.5, 2.45)) - bond\_4: a Line \[green\] drawn in molecule (start=(5.5, 2.45), end=(7.0, 2.0)) - bond\_5: a Line \[green\] drawn in molecule (start=(7.0, 2.0), end=(8.5, 2.45)) - box\_energy: a Math \[text\] that says "$E\_n=frac(n^2 h^2, 8 m L^2)$" - carbon\_1: a Point \[yellow\] drawn in molecule (location=(1.0, 2.0)) - carbon\_2: a Point \[yellow\] drawn in molecule (location=(2.5, 2.45)) - carbon\_3: a Point \[yellow\] drawn in molecule (location=(4.0, 2.0)) - carbon\_4: a Point \[yellow\] drawn in molecule (location=(5.5, 2.45)) - carbon\_5: a Point \[yellow\] drawn in molecule (location=(7.0, 2.0)) - carbon\_6: a Point \[yellow\] drawn in molecule (location=(8.5, 2.45)) - cloud: a Polygon \[blue\] labelled "upright("delocalised pi electrons")" drawn in molecule (vertices=((0.6, 1.05), (9.4, 1.05), (9.4, 2.95), (0.6, 2.95)), fill\_opacity=0.14) - concept: a Panel that says "In a conjugated chain, neighboring p orbitals overlap. Their pi electrons are spread over much of the molecular length rather than confined to one bond." - electron\_1a: a Point \[blue\] drawn in levels (location=(1.8, 0.8)) - electron\_1b: a Point \[blue\] drawn in levels (location=(2.2, 0.8)) - electron\_2a: a Point \[blue\] drawn in levels (location=(1.8, 1.55)) - electron\_2b: a Point \[blue\] drawn in levels (location=(2.2, 1.55)) - electron\_3a: a Point \[blue\] drawn in levels (location=(1.8, 2.55)) - electron\_3b: a Point \[blue\] drawn in levels (location=(2.2, 2.55)) - excitation: a Vector \[red\] labelled "Delta E" drawn in levels (start=(3.1, 2.55), end=(3.1, 4.15)) - fill\_1: a Math \[text\] that says "$upright("two electrons per level")$" - fill\_2: a Math \[text\] that says "$n=frac(N,2)$" - gap\_work: a Derivation \[text\] that says "$Delta E &= E\_(n+1)-E\_n \\ &= frac(((n+1)^2-n^2) h^2, 8 m L^2) \\ &= frac((2n+1) h^2, 8 m L^2)$" - heading\_compare: a Heading that says "Longer Conjugation, Smaller Gap" - heading\_levels: a Heading that says "Filling the Allowed Levels" - heading\_molecule: a Heading that says "A Molecular Box" - heading\_summary: a Heading that says "What the Toy Model Earns" - level\_1: a Line \[gray\] labelled "E\_1" drawn in levels (start=(0.7, 0.8), end=(4.3, 0.8)) - level\_2: a Line \[gray\] labelled "E\_2" drawn in levels (start=(0.7, 1.55), end=(4.3, 1.55)) - level\_3: a Line \[green\] labelled "E\_n" drawn in levels (start=(0.7, 2.55), end=(4.3, 2.55)) - level\_4: a Line \[yellow\] labelled "E\_(n+1)" drawn in levels (start=(0.7, 4.15), end=(4.3, 4.15)) - levels: a Figure (x\_range=(0.0, 5.0), y\_range=(0.0, 5.2)) - long: a Figure (x\_range=(0.0, 8.0), y\_range=(0.0, 5.0)) - long\_chain: a Line \[green\] drawn in long (start=(0.7, 3.9), end=(7.3, 3.9)) - long\_cloud: a Polygon \[blue\] drawn in long (vertices=((0.5, 3.4), (7.5, 3.4), (7.5, 4.4), (0.5, 4.4)), fill\_opacity=0.14) - long\_high: a Line \[yellow\] drawn in long (start=(1.1, 2.35), end=(6.9, 2.35)) - long\_jump: a Vector \[red\] labelled "Delta E\_(upright("long"))" drawn in long (start=(4.0, 1.25), end=(4.0, 2.35)) - long\_low: a Line \[green\] drawn in long (start=(1.1, 1.25), end=(6.9, 1.25)) - molecular\_length: a Brace \[yellow\] labelled "L" drawn in molecule (x\_start=1.0, x\_end=8.5, y=1.0) - molecule: a Figure (x\_range=(0.0, 10.0), y\_range=(0.0, 4.0)) - photon\_work: a Derivation \[text\] that says "$Delta E=h f=frac(h c, lambda) \\ lambda=frac(h c, Delta E)$" - short: a Figure (x\_range=(0.0, 8.0), y\_range=(0.0, 5.0)) - short\_chain: a Line \[green\] drawn in short (start=(1.0, 3.9), end=(4.8, 3.9)) - short\_cloud: a Polygon \[blue\] drawn in short (vertices=((0.8, 3.4), (5.0, 3.4), (5.0, 4.4), (0.8, 4.4)), fill\_opacity=0.14) - short\_high: a Line \[yellow\] drawn in short (start=(1.1, 2.8), end=(5.1, 2.8)) - short\_jump: a Vector \[red\] labelled "Delta E\_(upright("short"))" drawn in short (start=(3.1, 0.9), end=(3.1, 2.8)) - short\_low: a Line \[green\] drawn in short (start=(1.1, 0.9), end=(5.1, 0.9)) - summary\_1: a Text \[text\] that says "Boundaries select molecular standing-wave states." - summary\_2: a Text \[text\] that says "Electron filling makes the first empty state a specific energy away." - summary\_3: a Text \[text\] that says "A smaller gap absorbs a longer wavelength of light." - summary\_4: a Text \[text\] that says "The model predicts trends, not exact spectra or perceived colour." - text: a Text \[text\] that says "Shorter conjugated chain" - text\_2: a Text \[text\] that says "Longer conjugated chain" #### Beats ##### [09:33.096](https://academa.ai/lectures/the-particle-in-a-box?t=573.0964166666668) Narration: A toy model earns its keep only if it explains something real. Consider a conjugated dye molecule, built from a chain of neighboring atoms with overlapping p orbitals. Board: Empty. Actions: - [09:33.096](https://academa.ai/lectures/the-particle-in-a-box?t=573.0964166666668): heading\_molecule is shown on the screen, written out. - [09:39.075](https://academa.ai/lectures/the-particle-in-a-box?t=579.0754166666668): molecule is shown on the screen, written out. - [09:39.075](https://academa.ai/lectures/the-particle-in-a-box?t=579.0754166666668): cloud is shown on the screen, written out. - [09:39.075](https://academa.ai/lectures/the-particle-in-a-box?t=579.0754166666668): carbon\_1 is shown on the screen, written out. - [09:39.075](https://academa.ai/lectures/the-particle-in-a-box?t=579.0754166666668): carbon\_2 is shown on the screen, written out. - [09:39.075](https://academa.ai/lectures/the-particle-in-a-box?t=579.0754166666668): carbon\_3 is shown on the screen, written out. - [09:39.075](https://academa.ai/lectures/the-particle-in-a-box?t=579.0754166666668): carbon\_4 is shown on the screen, written out. - [09:39.075](https://academa.ai/lectures/the-particle-in-a-box?t=579.0754166666668): carbon\_5 is shown on the screen, written out. - [09:39.075](https://academa.ai/lectures/the-particle-in-a-box?t=579.0754166666668): carbon\_6 is shown on the screen, written out. - [09:39.075](https://academa.ai/lectures/the-particle-in-a-box?t=579.0754166666668): bond\_1 is shown on the screen, written out. - [09:39.075](https://academa.ai/lectures/the-particle-in-a-box?t=579.0754166666668): bond\_2 is shown on the screen, written out. - [09:39.075](https://academa.ai/lectures/the-particle-in-a-box?t=579.0754166666668): bond\_3 is shown on the screen, written out. - [09:39.075](https://academa.ai/lectures/the-particle-in-a-box?t=579.0754166666668): bond\_4 is shown on the screen, written out. - [09:39.075](https://academa.ai/lectures/the-particle-in-a-box?t=579.0754166666668): bond\_5 is shown on the screen, written out. - [09:39.075](https://academa.ai/lectures/the-particle-in-a-box?t=579.0754166666668): molecular\_length is shown on the screen, written out. ##### [09:44.667](https://academa.ai/lectures/the-particle-in-a-box?t=584.6674166666668) Narration: The green skeleton marks the chain, while the blue region represents the shared pi-electron system. These electrons are not assigned to one particular bond; they are delocalised along much of the chain. Board: molecule — a Figure (x\_range=(0.0, 10.0), y\_range=(0.0, 4.0)); heading\_molecule — a Heading that says "A Molecular Box"; cloud — a Polygon \[blue\] labelled "upright("delocalised pi electrons")" drawn in molecule (vertices=((0.6, 1.05), (9.4, 1.05), (9.4, 2.95), (0.6, 2.95)), fill\_opacity=0.14); carbon\_1 — a Point \[yellow\] drawn in molecule (location=(1.0, 2.0)); carbon\_2 — a Point \[yellow\] drawn in molecule (location=(2.5, 2.45)); carbon\_3 — a Point \[yellow\] drawn in molecule (location=(4.0, 2.0)); carbon\_4 — a Point \[yellow\] drawn in molecule (location=(5.5, 2.45)); carbon\_5 — a Point \[yellow\] drawn in molecule (location=(7.0, 2.0)); carbon\_6 — a Point \[yellow\] drawn in molecule (location=(8.5, 2.45)); bond\_1 — a Line \[green\] drawn in molecule (start=(1.0, 2.0), end=(2.5, 2.45)); bond\_2 — a Line \[green\] drawn in molecule (start=(2.5, 2.45), end=(4.0, 2.0)); bond\_3 — a Line \[green\] drawn in molecule (start=(4.0, 2.0), end=(5.5, 2.45)); bond\_4 — a Line \[green\] drawn in molecule (start=(5.5, 2.45), end=(7.0, 2.0)); bond\_5 — a Line \[green\] drawn in molecule (start=(7.0, 2.0), end=(8.5, 2.45)); molecular\_length — a Brace \[yellow\] labelled "L" drawn in molecule (x\_start=1.0, x\_end=8.5, y=1.0) Actions: - [09:45.318](https://academa.ai/lectures/the-particle-in-a-box?t=585.3184166666667): bond\_3 is indicated — a transient flash. - [09:47.523](https://academa.ai/lectures/the-particle-in-a-box?t=587.5234166666668): cloud is indicated — a transient flash. - [09:55.43](https://academa.ai/lectures/the-particle-in-a-box?t=595.4304166666668): molecule moves to a new place on the board. - [09:55.43](https://academa.ai/lectures/the-particle-in-a-box?t=595.4304166666668): concept is shown on the screen, written out. - [09:55.43](https://academa.ai/lectures/the-particle-in-a-box?t=595.4304166666668): box\_energy is shown on the screen, written out. ##### [09:58.456](https://academa.ai/lectures/the-particle-in-a-box?t=598.4564166666668) Narration: That finite delocalised length acts, approximately, like a one-dimensional box. The detailed molecular potential is not perfectly square, but the familiar expression makes the dependence clear: every energy contains one over L squared. Board: concept — a Panel that says "In a conjugated chain, neighboring p orbitals overlap. Their pi electrons are spread over much of the molecular length rather than confined to one bond."; box\_energy — a Math \[text\] that says "$E\_n=frac(n^2 h^2, 8 m L^2)$"; molecule — a Figure (x\_range=(0.0, 10.0), y\_range=(0.0, 4.0)); heading\_molecule — a Heading that says "A Molecular Box"; cloud — a Polygon \[blue\] labelled "upright("delocalised pi electrons")" drawn in molecule (vertices=((0.6, 1.05), (9.4, 1.05), (9.4, 2.95), (0.6, 2.95)), fill\_opacity=0.14); carbon\_1 — a Point \[yellow\] drawn in molecule (location=(1.0, 2.0)); carbon\_2 — a Point \[yellow\] drawn in molecule (location=(2.5, 2.45)); carbon\_3 — a Point \[yellow\] drawn in molecule (location=(4.0, 2.0)); carbon\_4 — a Point \[yellow\] drawn in molecule (location=(5.5, 2.45)); carbon\_5 — a Point \[yellow\] drawn in molecule (location=(7.0, 2.0)); carbon\_6 — a Point \[yellow\] drawn in molecule (location=(8.5, 2.45)); bond\_1 — a Line \[green\] drawn in molecule (start=(1.0, 2.0), end=(2.5, 2.45)); bond\_2 — a Line \[green\] drawn in molecule (start=(2.5, 2.45), end=(4.0, 2.0)); bond\_3 — a Line \[green\] drawn in molecule (start=(4.0, 2.0), end=(5.5, 2.45)); bond\_4 — a Line \[green\] drawn in molecule (start=(5.5, 2.45), end=(7.0, 2.0)); bond\_5 — a Line \[green\] drawn in molecule (start=(7.0, 2.0), end=(8.5, 2.45)); molecular\_length — a Brace \[yellow\] labelled "L" drawn in molecule (x\_start=1.0, x\_end=8.5, y=1.0) Actions: - [10:0.476](https://academa.ai/lectures/the-particle-in-a-box?t=600.4764166666668): molecular\_length is indicated — a transient flash. - [10:14.176](https://academa.ai/lectures/the-particle-in-a-box?t=614.1764166666668): box\_energy (the "L^2" part) is indicated — a transient flash. ##### [10:15.844](https://academa.ai/lectures/the-particle-in-a-box?t=615.8439166666667) Narration: So the molecular orbitals form a numbered set of wave-like states. Their exact shapes are more complicated than sines, but confinement still produces separated energies. Board: Unchanged from the preceding beat in this scene. Actions: - [10:27.304](https://academa.ai/lectures/the-particle-in-a-box?t=627.3039166666667): box\_energy is hidden from the screen — left the board. - [10:27.304](https://academa.ai/lectures/the-particle-in-a-box?t=627.3039166666667): concept is hidden from the screen — left the board. - [10:27.304](https://academa.ai/lectures/the-particle-in-a-box?t=627.3039166666667): heading\_molecule is hidden from the screen — left the board. - [10:27.304](https://academa.ai/lectures/the-particle-in-a-box?t=627.3039166666667): molecule is hidden from the screen — left the board. - [10:27.304](https://academa.ai/lectures/the-particle-in-a-box?t=627.3039166666667): cloud is hidden from the screen — molecule left the board. - [10:27.304](https://academa.ai/lectures/the-particle-in-a-box?t=627.3039166666667): carbon\_1 is hidden from the screen — molecule left the board. - [10:27.304](https://academa.ai/lectures/the-particle-in-a-box?t=627.3039166666667): carbon\_2 is hidden from the screen — molecule left the board. - [10:27.304](https://academa.ai/lectures/the-particle-in-a-box?t=627.3039166666667): carbon\_3 is hidden from the screen — molecule left the board. - [10:27.304](https://academa.ai/lectures/the-particle-in-a-box?t=627.3039166666667): carbon\_4 is hidden from the screen — molecule left the board. - [10:27.304](https://academa.ai/lectures/the-particle-in-a-box?t=627.3039166666667): carbon\_5 is hidden from the screen — molecule left the board. - [10:27.304](https://academa.ai/lectures/the-particle-in-a-box?t=627.3039166666667): carbon\_6 is hidden from the screen — molecule left the board. - [10:27.304](https://academa.ai/lectures/the-particle-in-a-box?t=627.3039166666667): bond\_1 is hidden from the screen — molecule left the board. - [10:27.304](https://academa.ai/lectures/the-particle-in-a-box?t=627.3039166666667): bond\_2 is hidden from the screen — molecule left the board. - [10:27.304](https://academa.ai/lectures/the-particle-in-a-box?t=627.3039166666667): bond\_3 is hidden from the screen — molecule left the board. - [10:27.304](https://academa.ai/lectures/the-particle-in-a-box?t=627.3039166666667): bond\_4 is hidden from the screen — molecule left the board. - [10:27.304](https://academa.ai/lectures/the-particle-in-a-box?t=627.3039166666667): bond\_5 is hidden from the screen — molecule left the board. - [10:27.304](https://academa.ai/lectures/the-particle-in-a-box?t=627.3039166666667): molecular\_length is hidden from the screen — molecule left the board. ##### [10:28.504](https://academa.ai/lectures/the-particle-in-a-box?t=628.5039166666668) Narration: Electrons now fill the allowed levels from the bottom upward. Each spatial level can hold two electrons with opposite spin. Board: Empty. Actions: - [10:28.504](https://academa.ai/lectures/the-particle-in-a-box?t=628.5039166666668): heading\_levels is shown on the screen, written out. - [10:30.454](https://academa.ai/lectures/the-particle-in-a-box?t=630.4544166666667): levels is shown on the screen, written out. - [10:30.454](https://academa.ai/lectures/the-particle-in-a-box?t=630.4544166666667): level\_1 is shown on the screen, written out. - [10:30.454](https://academa.ai/lectures/the-particle-in-a-box?t=630.4544166666667): level\_2 is shown on the screen, written out. - [10:30.454](https://academa.ai/lectures/the-particle-in-a-box?t=630.4544166666667): level\_3 is shown on the screen, written out. - [10:30.454](https://academa.ai/lectures/the-particle-in-a-box?t=630.4544166666667): level\_4 is shown on the screen, written out. - [10:30.454](https://academa.ai/lectures/the-particle-in-a-box?t=630.4544166666667): electron\_1a is shown on the screen, written out. - [10:30.454](https://academa.ai/lectures/the-particle-in-a-box?t=630.4544166666667): electron\_1b is shown on the screen, written out. - [10:30.454](https://academa.ai/lectures/the-particle-in-a-box?t=630.4544166666667): electron\_2a is shown on the screen, written out. - [10:30.454](https://academa.ai/lectures/the-particle-in-a-box?t=630.4544166666667): electron\_2b is shown on the screen, written out. - [10:30.454](https://academa.ai/lectures/the-particle-in-a-box?t=630.4544166666667): electron\_3a is shown on the screen, written out. - [10:30.454](https://academa.ai/lectures/the-particle-in-a-box?t=630.4544166666667): electron\_3b is shown on the screen, written out. - [10:34.297](https://academa.ai/lectures/the-particle-in-a-box?t=634.2974166666668): levels moves to a new place on the board. - [10:34.297](https://academa.ai/lectures/the-particle-in-a-box?t=634.2974166666668): fill\_1 is shown on the screen, written out. ##### [10:37.231](https://academa.ai/lectures/the-particle-in-a-box?t=637.2309166666668) Narration: If the conjugated system has N pi electrons and N is even, the highest filled level has index n equal to N over two. Board: fill\_1 — a Math \[text\] that says "$upright("two electrons per level")$"; levels — a Figure (x\_range=(0.0, 5.0), y\_range=(0.0, 5.2)); heading\_levels — a Heading that says "Filling the Allowed Levels"; level\_1 — a Line \[gray\] labelled "E\_1" drawn in levels (start=(0.7, 0.8), end=(4.3, 0.8)); level\_2 — a Line \[gray\] labelled "E\_2" drawn in levels (start=(0.7, 1.55), end=(4.3, 1.55)); level\_3 — a Line \[green\] labelled "E\_n" drawn in levels (start=(0.7, 2.55), end=(4.3, 2.55)); level\_4 — a Line \[yellow\] labelled "E\_(n+1)" drawn in levels (start=(0.7, 4.15), end=(4.3, 4.15)); electron\_1a — a Point \[blue\] drawn in levels (location=(1.8, 0.8)); electron\_1b — a Point \[blue\] drawn in levels (location=(2.2, 0.8)); electron\_2a — a Point \[blue\] drawn in levels (location=(1.8, 1.55)); electron\_2b — a Point \[blue\] drawn in levels (location=(2.2, 1.55)); electron\_3a — a Point \[blue\] drawn in levels (location=(1.8, 2.55)); electron\_3b — a Point \[blue\] drawn in levels (location=(2.2, 2.55)) Actions: - [10:42.408](https://academa.ai/lectures/the-particle-in-a-box?t=642.4084166666668): level\_3 is indicated — a transient flash. - [10:45.45](https://academa.ai/lectures/the-particle-in-a-box?t=645.4504166666668): fill\_2 is shown on the screen, written out. ##### [10:47.339](https://academa.ai/lectures/the-particle-in-a-box?t=647.3389166666668) Narration: The next level is empty. Absorbing light can lift an electron across this frontier gap from level n to level n plus one. Board: fill\_1 — a Math \[text\] that says "$upright("two electrons per level")$"; fill\_2 — a Math \[text\] that says "$n=frac(N,2)$"; levels — a Figure (x\_range=(0.0, 5.0), y\_range=(0.0, 5.2)); heading\_levels — a Heading that says "Filling the Allowed Levels"; level\_1 — a Line \[gray\] labelled "E\_1" drawn in levels (start=(0.7, 0.8), end=(4.3, 0.8)); level\_2 — a Line \[gray\] labelled "E\_2" drawn in levels (start=(0.7, 1.55), end=(4.3, 1.55)); level\_3 — a Line \[green\] labelled "E\_n" drawn in levels (start=(0.7, 2.55), end=(4.3, 2.55)); level\_4 — a Line \[yellow\] labelled "E\_(n+1)" drawn in levels (start=(0.7, 4.15), end=(4.3, 4.15)); electron\_1a — a Point \[blue\] drawn in levels (location=(1.8, 0.8)); electron\_1b — a Point \[blue\] drawn in levels (location=(2.2, 0.8)); electron\_2a — a Point \[blue\] drawn in levels (location=(1.8, 1.55)); electron\_2b — a Point \[blue\] drawn in levels (location=(2.2, 1.55)); electron\_3a — a Point \[blue\] drawn in levels (location=(1.8, 2.55)); electron\_3b — a Point \[blue\] drawn in levels (location=(2.2, 2.55)) Actions: - [10:47.861](https://academa.ai/lectures/the-particle-in-a-box?t=647.8614166666667): level\_4 is indicated — a transient flash. - [10:50.822](https://academa.ai/lectures/the-particle-in-a-box?t=650.8224166666668): excitation is shown on the screen, written out. ##### [10:56.85](https://academa.ai/lectures/the-particle-in-a-box?t=656.8499166666668) Narration: Begin with the difference E n plus one minus E n. Board: fill\_1 — a Math \[text\] that says "$upright("two electrons per level")$"; fill\_2 — a Math \[text\] that says "$n=frac(N,2)$"; levels — a Figure (x\_range=(0.0, 5.0), y\_range=(0.0, 5.2)); heading\_levels — a Heading that says "Filling the Allowed Levels"; level\_1 — a Line \[gray\] labelled "E\_1" drawn in levels (start=(0.7, 0.8), end=(4.3, 0.8)); level\_2 — a Line \[gray\] labelled "E\_2" drawn in levels (start=(0.7, 1.55), end=(4.3, 1.55)); level\_3 — a Line \[green\] labelled "E\_n" drawn in levels (start=(0.7, 2.55), end=(4.3, 2.55)); level\_4 — a Line \[yellow\] labelled "E\_(n+1)" drawn in levels (start=(0.7, 4.15), end=(4.3, 4.15)); electron\_1a — a Point \[blue\] drawn in levels (location=(1.8, 0.8)); electron\_1b — a Point \[blue\] drawn in levels (location=(2.2, 0.8)); electron\_2a — a Point \[blue\] drawn in levels (location=(1.8, 1.55)); electron\_2b — a Point \[blue\] drawn in levels (location=(2.2, 1.55)); electron\_3a — a Point \[blue\] drawn in levels (location=(1.8, 2.55)); electron\_3b — a Point \[blue\] drawn in levels (location=(2.2, 2.55)); excitation — a Vector \[red\] labelled "Delta E" drawn in levels (start=(3.1, 2.55), end=(3.1, 4.15)) Actions: - [10:57.911](https://academa.ai/lectures/the-particle-in-a-box?t=657.9114166666668): gap\_work is shown on the screen, written out. ##### [11:2.146](https://academa.ai/lectures/the-particle-in-a-box?t=662.1459166666667) Narration: Insert the particle-in-a-box energies. The two levels differ only in their squared quantum numbers. Board: Unchanged from the preceding beat in this scene. Actions: - [11:2.494](https://academa.ai/lectures/the-particle-in-a-box?t=662.4944166666668): gap\_work is shown on the screen, written out. - [11:7.196](https://academa.ai/lectures/the-particle-in-a-box?t=667.1964166666668): gap\_work (the "(n+1)^2-n^2" part) is indicated — a transient flash. ##### [11:9.503](https://academa.ai/lectures/the-particle-in-a-box?t=669.5029166666668) Narration: Simplifying gives a gap proportional to two n plus one divided by L squared. Board: Unchanged from the preceding beat in this scene. Actions: - [11:9.851](https://academa.ai/lectures/the-particle-in-a-box?t=669.8514166666668): gap\_work is shown on the screen, written out. - [11:11.94](https://academa.ai/lectures/the-particle-in-a-box?t=671.9404166666668): gap\_work (the "2n+1" part) is indicated — a transient flash. - [11:13.972](https://academa.ai/lectures/the-particle-in-a-box?t=673.9724166666667): gap\_work (the "L^2" part) is indicated — a transient flash. ##### [11:15.698](https://academa.ai/lectures/the-particle-in-a-box?t=675.6984166666667) Narration: A photon is absorbed when its energy matches that gap. Delta E equals h f, or h c over wavelength. Board: Unchanged from the preceding beat in this scene. Actions: - [11:16.232](https://academa.ai/lectures/the-particle-in-a-box?t=676.2324166666667): photon\_work is shown on the screen, written out. - [11:18.961](https://academa.ai/lectures/the-particle-in-a-box?t=678.9614166666668): photon\_work (the "Delta E" part) is indicated — a transient flash. - [11:23.628](https://academa.ai/lectures/the-particle-in-a-box?t=683.6284166666668): photon\_work (the "frac(h c, lambda)" part) is indicated — a transient flash. ##### [11:25.18](https://academa.ai/lectures/the-particle-in-a-box?t=685.1799166666667) Narration: Solving for wavelength makes the trend explicit. A smaller energy gap corresponds to a longer absorbed wavelength. Board: Unchanged from the preceding beat in this scene. Actions: - [11:25.528](https://academa.ai/lectures/the-particle-in-a-box?t=685.5284166666668): photon\_work is shown on the screen, written out. - [11:31.287](https://academa.ai/lectures/the-particle-in-a-box?t=691.2874166666668): photon\_work (the "frac(h c, Delta E)" part) is indicated — a transient flash. - [11:33.342](https://academa.ai/lectures/the-particle-in-a-box?t=693.3419166666667): fill\_1 is hidden from the screen — left the board. - [11:33.342](https://academa.ai/lectures/the-particle-in-a-box?t=693.3419166666667): fill\_2 is hidden from the screen — left the board. - [11:33.342](https://academa.ai/lectures/the-particle-in-a-box?t=693.3419166666667): gap\_work is hidden from the screen — left the board. - [11:33.342](https://academa.ai/lectures/the-particle-in-a-box?t=693.3419166666667): heading\_levels is hidden from the screen — left the board. - [11:33.342](https://academa.ai/lectures/the-particle-in-a-box?t=693.3419166666667): levels is hidden from the screen — left the board. - [11:33.342](https://academa.ai/lectures/the-particle-in-a-box?t=693.3419166666667): level\_1 is hidden from the screen — levels left the board. - [11:33.342](https://academa.ai/lectures/the-particle-in-a-box?t=693.3419166666667): level\_2 is hidden from the screen — levels left the board. - [11:33.342](https://academa.ai/lectures/the-particle-in-a-box?t=693.3419166666667): level\_3 is hidden from the screen — levels left the board. - [11:33.342](https://academa.ai/lectures/the-particle-in-a-box?t=693.3419166666667): level\_4 is hidden from the screen — levels left the board. - [11:33.342](https://academa.ai/lectures/the-particle-in-a-box?t=693.3419166666667): electron\_1a is hidden from the screen — levels left the board. - [11:33.342](https://academa.ai/lectures/the-particle-in-a-box?t=693.3419166666667): electron\_1b is hidden from the screen — levels left the board. - [11:33.342](https://academa.ai/lectures/the-particle-in-a-box?t=693.3419166666667): electron\_2a is hidden from the screen — levels left the board. - [11:33.342](https://academa.ai/lectures/the-particle-in-a-box?t=693.3419166666667): electron\_2b is hidden from the screen — levels left the board. - [11:33.342](https://academa.ai/lectures/the-particle-in-a-box?t=693.3419166666667): electron\_3a is hidden from the screen — levels left the board. - [11:33.342](https://academa.ai/lectures/the-particle-in-a-box?t=693.3419166666667): electron\_3b is hidden from the screen — levels left the board. - [11:33.342](https://academa.ai/lectures/the-particle-in-a-box?t=693.3419166666667): excitation is hidden from the screen — levels left the board. - [11:33.342](https://academa.ai/lectures/the-particle-in-a-box?t=693.3419166666667): photon\_work is hidden from the screen — left the board. ##### [11:33.942](https://academa.ai/lectures/the-particle-in-a-box?t=693.9419166666668) Narration: Compare two related conjugated systems. The shorter chain confines its pi electrons more tightly, so the frontier levels stand farther apart. Board: Empty. Actions: - [11:33.942](https://academa.ai/lectures/the-particle-in-a-box?t=693.9419166666668): heading\_compare is shown on the screen, written out. - [11:37.831](https://academa.ai/lectures/the-particle-in-a-box?t=697.8314166666668): short is shown on the screen, written out. - [11:37.831](https://academa.ai/lectures/the-particle-in-a-box?t=697.8314166666668): short\_cloud is shown on the screen, written out. - [11:37.831](https://academa.ai/lectures/the-particle-in-a-box?t=697.8314166666668): short\_chain is shown on the screen, written out. - [11:37.831](https://academa.ai/lectures/the-particle-in-a-box?t=697.8314166666668): short\_low is shown on the screen, written out. - [11:37.831](https://academa.ai/lectures/the-particle-in-a-box?t=697.8314166666668): short\_high is shown on the screen, written out. - [11:37.831](https://academa.ai/lectures/the-particle-in-a-box?t=697.8314166666668): short\_jump is shown on the screen, written out. - [11:42.568](https://academa.ai/lectures/the-particle-in-a-box?t=702.5684166666667): short\_jump is indicated — a transient flash. ##### [11:44.387](https://academa.ai/lectures/the-particle-in-a-box?t=704.3869166666667) Narration: The longer chain spreads the wavefunctions across a greater distance. Its relevant levels are closer together, so the transition needs a lower-energy photon. Board: short — a Figure (x\_range=(0.0, 8.0), y\_range=(0.0, 5.0)); heading\_compare — a Heading that says "Longer Conjugation, Smaller Gap"; short\_cloud — a Polygon \[blue\] drawn in short (vertices=((0.8, 3.4), (5.0, 3.4), (5.0, 4.4), (0.8, 4.4)), fill\_opacity=0.14); short\_chain — a Line \[green\] drawn in short (start=(1.0, 3.9), end=(4.8, 3.9)); short\_low — a Line \[green\] drawn in short (start=(1.1, 0.9), end=(5.1, 0.9)); short\_high — a Line \[yellow\] drawn in short (start=(1.1, 2.8), end=(5.1, 2.8)); short\_jump — a Vector \[red\] labelled "Delta E\_(upright("short"))" drawn in short (start=(3.1, 0.9), end=(3.1, 2.8)) Actions: - [11:44.932](https://academa.ai/lectures/the-particle-in-a-box?t=704.9324166666668): short moves to a new place on the board. - [11:44.932](https://academa.ai/lectures/the-particle-in-a-box?t=704.9324166666668): long is shown on the screen, written out. - [11:44.932](https://academa.ai/lectures/the-particle-in-a-box?t=704.9324166666668): long\_cloud is shown on the screen, written out. - [11:44.932](https://academa.ai/lectures/the-particle-in-a-box?t=704.9324166666668): long\_chain is shown on the screen, written out. - [11:44.932](https://academa.ai/lectures/the-particle-in-a-box?t=704.9324166666668): long\_low is shown on the screen, written out. - [11:44.932](https://academa.ai/lectures/the-particle-in-a-box?t=704.9324166666668): long\_high is shown on the screen, written out. - [11:44.932](https://academa.ai/lectures/the-particle-in-a-box?t=704.9324166666668): long\_jump is shown on the screen, written out. - [11:50.191](https://academa.ai/lectures/the-particle-in-a-box?t=710.1914166666668): long\_jump is indicated — a transient flash. ##### [11:55.215](https://academa.ai/lectures/the-particle-in-a-box?t=715.2154166666667) Narration: Lower photon energy means longer wavelength. As conjugation is extended through a family of dyes, the absorption band commonly shifts toward longer visible wavelengths. Board: short — a Figure (x\_range=(0.0, 8.0), y\_range=(0.0, 5.0)); long — a Figure (x\_range=(0.0, 8.0), y\_range=(0.0, 5.0)); heading\_compare — a Heading that says "Longer Conjugation, Smaller Gap"; short\_cloud — a Polygon \[blue\] drawn in short (vertices=((0.8, 3.4), (5.0, 3.4), (5.0, 4.4), (0.8, 4.4)), fill\_opacity=0.14); short\_chain — a Line \[green\] drawn in short (start=(1.0, 3.9), end=(4.8, 3.9)); short\_low — a Line \[green\] drawn in short (start=(1.1, 0.9), end=(5.1, 0.9)); short\_high — a Line \[yellow\] drawn in short (start=(1.1, 2.8), end=(5.1, 2.8)); short\_jump — a Vector \[red\] labelled "Delta E\_(upright("short"))" drawn in short (start=(3.1, 0.9), end=(3.1, 2.8)); long\_cloud — a Polygon \[blue\] drawn in long (vertices=((0.5, 3.4), (7.5, 3.4), (7.5, 4.4), (0.5, 4.4)), fill\_opacity=0.14); long\_chain — a Line \[green\] drawn in long (start=(0.7, 3.9), end=(7.3, 3.9)); long\_low — a Line \[green\] drawn in long (start=(1.1, 1.25), end=(6.9, 1.25)); long\_high — a Line \[yellow\] drawn in long (start=(1.1, 2.35), end=(6.9, 2.35)); long\_jump — a Vector \[red\] labelled "Delta E\_(upright("long"))" drawn in long (start=(4.0, 1.25), end=(4.0, 2.35)) Actions: - [11:55.563](https://academa.ai/lectures/the-particle-in-a-box?t=715.5634166666667): long\_jump is indicated — a transient flash. - [11:57.293](https://academa.ai/lectures/the-particle-in-a-box?t=717.2934166666668): short\_jump is indicated — a transient flash. ##### [12:6.995](https://academa.ai/lectures/the-particle-in-a-box?t=726.9954166666668) Narration: A dye looks coloured because it absorbs some visible wavelengths more strongly than others. The colour we observe comes from the light left over or transmitted, so it is not simply the colour of the absorbed photon. Board: Unchanged from the preceding beat in this scene. Actions: - [12:7.529](https://academa.ai/lectures/the-particle-in-a-box?t=727.5294166666667): short\_cloud is indicated — a transient flash. - [12:9.294](https://academa.ai/lectures/the-particle-in-a-box?t=729.2944166666667): long\_cloud is indicated — a transient flash. - [12:20.324](https://academa.ai/lectures/the-particle-in-a-box?t=740.3239166666667): heading\_compare is hidden from the screen — left the board. - [12:20.324](https://academa.ai/lectures/the-particle-in-a-box?t=740.3239166666667): long is hidden from the screen — left the board. - [12:20.324](https://academa.ai/lectures/the-particle-in-a-box?t=740.3239166666667): long\_cloud is hidden from the screen — long left the board. - [12:20.324](https://academa.ai/lectures/the-particle-in-a-box?t=740.3239166666667): long\_chain is hidden from the screen — long left the board. - [12:20.324](https://academa.ai/lectures/the-particle-in-a-box?t=740.3239166666667): long\_low is hidden from the screen — long left the board. - [12:20.324](https://academa.ai/lectures/the-particle-in-a-box?t=740.3239166666667): long\_high is hidden from the screen — long left the board. - [12:20.324](https://academa.ai/lectures/the-particle-in-a-box?t=740.3239166666667): long\_jump is hidden from the screen — long left the board. - [12:20.324](https://academa.ai/lectures/the-particle-in-a-box?t=740.3239166666667): short is hidden from the screen — left the board. - [12:20.324](https://academa.ai/lectures/the-particle-in-a-box?t=740.3239166666667): short\_cloud is hidden from the screen — short left the board. - [12:20.324](https://academa.ai/lectures/the-particle-in-a-box?t=740.3239166666667): short\_chain is hidden from the screen — short left the board. - [12:20.324](https://academa.ai/lectures/the-particle-in-a-box?t=740.3239166666667): short\_low is hidden from the screen — short left the board. - [12:20.324](https://academa.ai/lectures/the-particle-in-a-box?t=740.3239166666667): short\_high is hidden from the screen — short left the board. - [12:20.324](https://academa.ai/lectures/the-particle-in-a-box?t=740.3239166666667): short\_jump is hidden from the screen — short left the board. ##### [12:20.924](https://academa.ai/lectures/the-particle-in-a-box?t=740.9239166666667) Narration: Now the complete chain of reasoning fits together. First, finite boundaries select standing-wave states. Board: Empty. Actions: - [12:20.924](https://academa.ai/lectures/the-particle-in-a-box?t=740.9239166666667): heading\_summary is shown on the screen, written out. - [12:26.078](https://academa.ai/lectures/the-particle-in-a-box?t=746.0784166666667): summary\_1 is shown on the screen, written out. ##### [12:29.174](https://academa.ai/lectures/the-particle-in-a-box?t=749.1744166666667) Narration: Second, electrons fill those states, leaving a particular gap between the highest filled and lowest empty levels. Board: summary\_1 — a Text \[text\] that says "Boundaries select molecular standing-wave states."; heading\_summary — a Heading that says "What the Toy Model Earns" Actions: - [12:30.892](https://academa.ai/lectures/the-particle-in-a-box?t=750.8924166666668): summary\_2 is shown on the screen, written out. - [12:32.378](https://academa.ai/lectures/the-particle-in-a-box?t=752.3784166666667): summary\_2 (the "specific energy" part) is indicated — a transient flash. ##### [12:37.008](https://academa.ai/lectures/the-particle-in-a-box?t=757.0079166666667) Narration: Third, light is absorbed when a photon matches that gap. Extending the conjugated region generally reduces the gap and moves the absorption toward longer wavelength. Board: summary\_1 — a Text \[text\] that says "Boundaries select molecular standing-wave states."; summary\_2 — a Text \[text\] that says "Electron filling makes the first empty state a specific energy away."; heading\_summary — a Heading that says "What the Toy Model Earns" Actions: - [12:43.741](https://academa.ai/lectures/the-particle-in-a-box?t=763.7414166666667): summary\_3 is shown on the screen, written out. - [12:46.295](https://academa.ai/lectures/the-particle-in-a-box?t=766.2954166666667): summary\_3 (the "longer wavelength" part) is indicated — a transient flash. ##### [12:48.242](https://academa.ai/lectures/the-particle-in-a-box?t=768.2424166666667) Narration: Finally, this remains a model. Real molecules have non-square potentials, electron interactions, vibrations, and environmental effects. Detailed spectra require more complete quantum chemistry. Board: summary\_1 — a Text \[text\] that says "Boundaries select molecular standing-wave states."; summary\_2 — a Text \[text\] that says "Electron filling makes the first empty state a specific energy away."; summary\_3 — a Text \[text\] that says "A smaller gap absorbs a longer wavelength of light."; heading\_summary — a Heading that says "What the Toy Model Earns" Actions: - [12:49.891](https://academa.ai/lectures/the-particle-in-a-box?t=769.8914166666667): summary\_4 is shown on the screen, written out. - [12:59.423](https://academa.ai/lectures/the-particle-in-a-box?t=779.4234166666668): summary\_4 (the "not exact" part) is indicated — a transient flash. ##### [13:3.831](https://academa.ai/lectures/the-particle-in-a-box?t=783.8309166666668) Narration: But the model has earned its keep. Fixed ends selected wavelengths; confined wavefunctions selected energies; molecular length controlled an optical gap. Energy is quantised because a bounded wave cannot fit in arbitrary ways. Board: summary\_1 — a Text \[text\] that says "Boundaries select molecular standing-wave states."; summary\_2 — a Text \[text\] that says "Electron filling makes the first empty state a specific energy away."; summary\_3 — a Text \[text\] that says "A smaller gap absorbs a longer wavelength of light."; summary\_4 — a Text \[text\] that says "The model predicts trends, not exact spectra or perceived colour."; heading\_summary — a Heading that says "What the Toy Model Earns" Actions: - [13:6.849](https://academa.ai/lectures/the-particle-in-a-box?t=786.8494166666667): summary\_1 is indicated — a transient flash. - [13:11.516](https://academa.ai/lectures/the-particle-in-a-box?t=791.5164166666667): summary\_2 is indicated — a transient flash. - [13:14.918](https://academa.ai/lectures/the-particle-in-a-box?t=794.9184166666668): summary\_3 is indicated — a transient flash. - [13:18.378](https://academa.ai/lectures/the-particle-in-a-box?t=798.3784166666667): summary\_4 is indicated — a transient flash. - [13:21.457](https://academa.ai/lectures/the-particle-in-a-box?t=801.456625): heading\_summary is hidden from the screen — left the board. - [13:21.457](https://academa.ai/lectures/the-particle-in-a-box?t=801.456625): summary\_1 is hidden from the screen — left the board. - [13:21.457](https://academa.ai/lectures/the-particle-in-a-box?t=801.456625): summary\_2 is hidden from the screen — left the board. - [13:21.457](https://academa.ai/lectures/the-particle-in-a-box?t=801.456625): summary\_3 is hidden from the screen — left the board. - [13:21.457](https://academa.ai/lectures/the-particle-in-a-box?t=801.456625): summary\_4 is hidden from the screen — left the board.