{"version":1,"lectureId":"01M14TZZSX238T89EBZHGVYW3N","attempt":0,"publication":{"slug":"the-particle-in-a-box","title":"Why Energy Is Quantised: From Standing Waves to Coloured Molecules","subject":"physics","summary":"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.","metaDescription":"Why confined waves admit discrete energies, how box size controls spacing, and how the same model explains trends in conjugated dye colours.","transcript":"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. 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. 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. 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. 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.","watch":{"version":1,"scenes":[{"title":"Only Certain Waves Fit","start":0,"end":125.43370833333334,"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=<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=<function>)","mode_2":"a ParametricCurve [green] labelled \"n=2\" drawn in string (function=<function>)","mode_3":"a ParametricCurve [red] labelled \"n=3\" drawn in string (function=<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":[{"start":0,"say":"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.","live":[],"does":[[0,"card is shown on the screen, written out."],[1.5,"card: enter:write-left-to-right."],[14.4665,"card is hidden from the screen — left the board."]]},{"start":15.6665,"say":"The string runs from zero to L. Each end is clamped, so neither endpoint can move, however strongly the rest of the string vibrates.","live":null,"does":[[15.6665,"heading is shown on the screen, written out."],[16.212,"string is shown on the screen, written out."],[16.212,"rest_line is shown on the screen, written out."],[19.66,"left_wall is shown on the screen, written out."],[19.66,"right_wall is shown on the screen, written out."],[20.891000000000002,"left_end is shown on the screen, written out."],[20.891000000000002,"right_end is shown on the screen, written out."]]},{"start":25.682,"say":"That physical fact becomes a boundary condition. The displacement y must be zero at the left wall and zero again at the right wall.","live":["string","heading","left_wall","right_wall","rest_line","left_end","right_end"],"does":[[27.690999999999995,"string moves to a new place on the board."],[27.690999999999995,"boundary is shown on the screen, written out."],[31.881999999999998,"boundary (the \"y(0)=0\" part) is indicated — a transient flash."],[33.971,"boundary (the \"y(L)=0\" part) is indicated — a transient flash."]]},{"start":35.558,"say":"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.","live":["boundary","string","heading","left_wall","right_wall","rest_line","left_end","right_end"],"does":[[37.799,"miss is shown on the screen, drawn."],[41.781,"missed_end is shown on the screen, written out."],[44.567,"missed_end is indicated — a transient flash."],[49.211,"missed_end is hidden from the screen."]]},{"start":49.811,"say":"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.","live":["boundary","string","heading","left_wall","right_wall","rest_line","left_end","right_end","miss"],"does":[[49.811,"miss is hidden from the screen."],[52.551,"mode_1 is shown on the screen, drawn."],[56.336,"left_end is indicated — a transient flash."],[56.336,"right_end is indicated — a transient flash."]]},{"start":60.988,"say":"The next shape fits two half-waves. It still vanishes at both walls, and it gains another node exactly in the middle.","live":["boundary","string","heading","left_wall","right_wall","rest_line","left_end","right_end","mode_1"],"does":[[60.988,"mode_1 is hidden from the screen."],[62.358000000000004,"mode_2 is shown on the screen, drawn."],[66.99100000000001,"middle_node is shown on the screen, written out."]]},{"start":69.4045,"say":"A third shape fits three half-waves. Two interior nodes divide the string into three vibrating sections, with neighboring sections moving in opposite directions.","live":["boundary","string","heading","left_wall","right_wall","rest_line","left_end","right_end","mode_2","middle_node"],"does":[[69.4045,"mode_2 is hidden from the screen."],[69.4045,"middle_node is hidden from the screen."],[70.78600000000002,"mode_3 is shown on the screen, drawn."],[72.57400000000001,"third_node_1 is shown on the screen, written out."],[72.57400000000001,"third_node_2 is shown on the screen, written out."],[77.76400000000001,"mode_3 is indicated — a transient flash."]]},{"start":79.8495,"say":"We can continue forever, but not continuously. The length L must contain a whole number n of half-wavelengths.","live":["boundary","string","heading","left_wall","right_wall","rest_line","left_end","right_end","mode_3","third_node_1","third_node_2"],"does":[[85.33,"fit_work is shown on the screen, written out."],[85.887,"fit_work (the \"n\" part) is indicated — a transient flash."],[86.305,"fit_work (the \"frac(lambda, 2)\" part) is indicated — a transient flash."]]},{"start":88.345,"say":"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.","live":null,"does":[[88.693,"fit_work is shown on the screen, written out."],[91.259,"fit_work (the \"frac(2L, n)\" part) is indicated — a transient flash."]]},{"start":100.1135,"say":"The same statement can be written using wave number k. 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Continuous guesses went in; a discrete family came out.","live":["boundary","allowed","string","heading","left_wall","right_wall","rest_line","left_end","right_end","mode_3","third_node_1","third_node_2"],"does":[[122.56300000000003,"A box is drawn around allowed."],[124.39204166666667,"allowed is hidden from the screen — left the board."],[124.39204166666667,"boundary is hidden from the screen — left the board."],[124.39204166666667,"fit_work is hidden from the screen — left the board."],[124.39204166666667,"heading is hidden from the screen — left the board."],[124.39204166666667,"string is hidden from the screen — left the board."],[124.39204166666667,"left_wall is hidden from the screen — string left the board."],[124.39204166666667,"right_wall is hidden from the screen — string left the board."],[124.39204166666667,"rest_line is hidden from the screen — string left the board."],[124.39204166666667,"left_end is hidden from the screen — string left the board."],[124.39204166666667,"right_end is hidden from the screen — string left the board."],[124.39204166666667,"mode_3 is hidden from the screen — string left the board."],[124.39204166666667,"third_node_1 is hidden from the screen — string left the board."],[124.39204166666667,"third_node_2 is hidden from the screen — string left the board."]]}]},{"title":"A Particle as a Confined Wave","start":125.43370833333334,"end":284.93758333333335,"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=<function>, x_range=(0.0, 1.0))","wave_2":"a FunctionPlot [green] labelled \"psi_2\" drawn in box (function=<function>, x_range=(0.0, 1.0))","wave_3":"a FunctionPlot [red] labelled \"psi_3\" drawn in box (function=<function>, x_range=(0.0, 1.0))"},"beats":[{"start":125.43370833333334,"say":"Now replace the string by a particle trapped between two perfectly impenetrable walls. 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Therefore psi is zero at x equals zero and again at x equals L.","live":["box","heading_fit","left_wall","right_wall","baseline"],"does":[[153.27870833333336,"box moves to a new place on the board."],[153.27870833333336,"boundary is shown on the screen, written out."],[155.10170833333333,"boundary (the \"psi(0)=0\" part) is indicated — a transient flash."],[157.22570833333336,"boundary (the \"psi(L)=0\" part) is indicated — a transient flash."]]},{"start":159.85770833333333,"say":"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.","live":["boundary","box","heading_fit","left_wall","right_wall","baseline"],"does":[[160.41470833333335,"wave_1 is shown on the screen, drawn."],[167.17170833333336,"wave_1 is indicated — a transient flash."]]},{"start":170.34920833333334,"say":"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.","live":["boundary","box","heading_fit","left_wall","right_wall","baseline","wave_1"],"does":[[170.34920833333334,"wave_1 is hidden from the screen."],[170.76770833333336,"wave_2 is shown on the screen, drawn."],[174.37770833333335,"wave_2 is hidden from the screen."],[174.37770833333335,"wave_3 is shown on the screen, drawn."]]},{"start":182.31520833333335,"say":"Write a sinusoidal candidate A sine k x. The left boundary is already satisfied because sine zero is zero.","live":["boundary","box","heading_fit","left_wall","right_wall","baseline","wave_3"],"does":[[183.90570833333334,"fit_work is shown on the screen, written out."],[184.89270833333336,"fit_work (the \"sin(k x)\" part) is indicated — a transient flash."]]},{"start":190.98470833333334,"say":"At the right wall, x becomes L. The amplitude there is A sine k L, and the wall requires that expression to vanish.","live":null,"does":[[191.58870833333333,"fit_work is shown on the screen, written out."],[195.69870833333334,"fit_work (the \"sin(k L)\" part) is indicated — a transient flash."]]},{"start":200.55920833333334,"say":"Sine vanishes at whole multiples of pi. So k L must equal n pi, not an arbitrary number.","live":null,"does":[[202.03370833333335,"fit_work is shown on the screen, written out."],[205.70270833333336,"fit_work (the \"n pi\" part) is indicated — a transient flash."]]},{"start":208.64770833333336,"say":"Dividing by L gives the allowed wave numbers. Once again the boundaries have turned a continuous range of guesses into a numbered list.","live":null,"does":[[208.99570833333334,"fit_work is shown on the screen, written out."],[215.94970833333335,"A box is drawn around fit_work."],[217.11070833333332,"boundary is hidden from the screen — left the board."],[217.11070833333332,"fit_work is hidden from the screen — left the board."],[217.11070833333332,"heading_fit is hidden from the screen — left the board."]]},{"start":217.71070833333334,"say":"A fitting rule selects wave numbers. To find energies, we now use the stationary Schrödinger equation inside the box.","live":["box","left_wall","right_wall","baseline","wave_3"],"does":[[217.71070833333334,"heading_energy is shown on the screen, written out."],[223.66670833333336,"energy_work is shown on the screen, written out."]]},{"start":226.58920833333335,"say":"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.","live":["box","left_wall","right_wall","baseline","wave_3","heading_energy"],"does":[[228.40070833333334,"energy_work (the \"psi''(x)\" part) is indicated — a transient flash."],[241.70570833333335,"energy_work (the \"E\" part) is indicated — a transient flash."]]},{"start":244.70870833333333,"say":"Differentiate sine twice and the original wave returns with a factor of minus k squared.","live":null,"does":[[245.05670833333335,"energy_work is shown on the screen, written out."],[249.20170833333336,"energy_work (the \"-k^2\" part) is indicated — a transient flash."]]},{"start":251.40420833333334,"say":"Substitute that curvature into the equation. The common wavefunction cancels, leaving E equal to h squared k squared over eight pi squared m.","live":null,"does":[[257.20870833333333,"energy_work is shown on the screen, written out."],[259.27570833333334,"energy_work (the \"k^2\" part) is indicated — a transient flash."]]},{"start":262.56920833333334,"say":"Finally substitute the allowed k values. The energy of state n is n squared h squared over eight m L squared.","live":null,"does":[[262.91770833333334,"energy_work is shown on the screen, written out."],[267.58470833333337,"energy_work (the \"n^2\" part) is indicated — a transient flash."],[269.97670833333336,"energy_work (the \"L^2\" part) is indicated — a transient flash."]]},{"start":271.7262083333334,"say":"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.","live":null,"does":[[280.54970833333334,"energy_result is shown on the screen, written out."],[282.85970833333334,"A box is drawn around energy_work."],[283.8959166666667,"box is hidden from the screen — left the board."],[283.8959166666667,"left_wall is hidden from the screen — box left the board."],[283.8959166666667,"right_wall is hidden from the screen — box left the board."],[283.8959166666667,"baseline is hidden from the screen — box left the board."],[283.8959166666667,"wave_3 is hidden from the screen — box left the board."],[283.8959166666667,"energy_result is hidden from the screen — left the board."],[283.8959166666667,"energy_work is hidden from the screen — left the board."],[283.8959166666667,"heading_energy is hidden from the screen — left the board."]]}]},{"title":"How the Energy Gaps Shrink","start":284.93758333333335,"end":432.26925000000006,"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":[{"start":284.93758333333335,"say":"The energy formula contains a powerful prediction. Keep the particle and its mass unchanged, but make the confining region wider.","live":[],"does":[[284.93758333333335,"heading is shown on the screen, written out."],[285.54158333333334,"base_formula is shown on the screen, written out."],[292.13558333333333,"left_wall is shown on the screen, written out."],[292.13558333333333,"right_wall is shown on the screen, written out."],[292.13558333333333,"box_floor is shown on the screen, written out."],[292.49558333333334,"picture is shown on the screen, written out."],[292.49558333333334,"width_brace is shown on the screen, written out."]]},{"start":294.03608333333335,"say":"Beside the box are its first three energies. They are separate horizontal lines, and even the lowest state sits above zero.","live":["base_formula","picture","heading","left_wall","right_wall","box_floor","width_brace"],"does":[[295.7775833333333,"level_1 is shown on the screen, written out."],[296.13758333333334,"level_2 is shown on the screen, written out."],[296.2875833333334,"level_3 is shown on the screen, written out."],[301.55858333333333,"zero_line is shown on the screen, written out."]]},{"start":303.02958333333333,"say":"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.","live":["base_formula","picture","heading","left_wall","right_wall","box_floor","width_brace","zero_line","level_1","level_2","level_3"],"does":[[304.23758333333336,"level_1 is indicated — a transient flash."],[308.03358333333335,"left_wall is indicated — a transient flash."],[308.03358333333335,"right_wall is indicated — a transient flash."]]},{"start":315.47208333333333,"say":"The difference between neighboring levels is delta E n. Start with E n plus one minus E n.","live":null,"does":[[315.98258333333337,"gap_1 is shown on the screen, written out."],[321.0565833333334,"spacing_work is shown on the screen, written out."]]},{"start":323.06108333333333,"say":"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.","live":["base_formula","picture","heading","left_wall","right_wall","box_floor","width_brace","zero_line","level_1","level_2","level_3","gap_1"],"does":[[323.28158333333334,"spacing_work is shown on the screen, written out."],[332.8715833333333,"spacing_work (the \"L^2\" part) is indicated — a transient flash."]]},{"start":334.58608333333336,"say":"The difference of the squares is two n plus one. So the gap is also proportional to one over L squared.","live":null,"does":[[336.36258333333336,"spacing_work is shown on the screen, written out."],[336.36258333333336,"spacing_work (the \"2n+1\" part) is indicated — a transient flash."],[340.73958333333337,"spacing_work (the \"L^2\" part) is indicated — a transient flash."]]},{"start":342.48858333333334,"say":"Now widen the box. Watch both walls move apart while the three energy levels descend and crowd together.","live":null,"does":[[343.24358333333333,"left_wall is redrawn as the numbers it depends on change."],[343.24358333333333,"right_wall is redrawn as the numbers it depends on change."],[343.24358333333333,"box_floor is redrawn as the numbers it depends on change."],[343.24358333333333,"width_brace is redrawn as the numbers it depends on change."],[343.24358333333333,"level_1 is redrawn as the numbers it depends on change."],[343.24358333333333,"level_2 is redrawn as the numbers it depends on change."],[343.24358333333333,"level_3 is redrawn as the numbers it depends on change."],[343.24358333333333,"gap_1 is redrawn as the numbers it depends on change."],[343.24358333333333,"width ticks to 4.2."]]},{"start":351.12258333333335,"say":"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.","live":null,"does":[[356.46358333333336,"level_1 is indicated — a transient flash."],[357.04358333333334,"level_2 is indicated — a transient flash."],[357.69458333333336,"level_3 is indicated — a transient flash."]]},{"start":364.11058333333335,"say":"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.","live":null,"does":[[372.20258333333334,"gap_1 is indicated — a transient flash."],[375.55808333333334,"base_formula is hidden from the screen — left the board."],[375.55808333333334,"heading is hidden from the screen — left the board."],[375.55808333333334,"picture is hidden from the screen — left the board."],[375.55808333333334,"left_wall is hidden from the screen — picture left the board."],[375.55808333333334,"right_wall is hidden from the screen — picture left the board."],[375.55808333333334,"box_floor is hidden from the screen — picture left the board."],[375.55808333333334,"width_brace is hidden from the screen — picture left the board."],[375.55808333333334,"zero_line is hidden from the screen — picture left the board."],[375.55808333333334,"level_1 is hidden from the screen — picture left the board."],[375.55808333333334,"level_2 is hidden from the screen — picture left the board."],[375.55808333333334,"level_3 is hidden from the screen — picture left the board."],[375.55808333333334,"gap_1 is hidden from the screen — picture left the board."],[375.55808333333334,"spacing_work is hidden from the screen — left the board."]]},{"start":376.15808333333337,"say":"This is one route back toward classical physics. First, increasing L lowers the entire spectrum and shrinks every gap as one over L squared.","live":[],"does":[[376.15808333333337,"heading_limit is shown on the screen, written out."],[380.50058333333334,"item_1 is shown on the screen, written out."],[384.92358333333334,"item_1 (the \"1/L^2\" part) is indicated — a transient flash."]]},{"start":386.61458333333337,"say":"There is a second effect at high quantum number. Divide a gap by the energy itself, and most constants cancel.","live":["item_1","heading_limit"],"does":[[389.98158333333333,"relative is shown on the screen, written out."]]},{"start":394.62208333333336,"say":"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.","live":["item_1","relative","heading_limit"],"does":[[396.02658333333335,"relative (the \"frac(2n+1, n^2)\" part) is indicated — a transient flash."],[399.1265833333334,"relative (the \"frac(2, n)\" part) is indicated — a transient flash."],[401.23958333333337,"item_2 is shown on the screen, written out."]]},{"start":406.5300833333334,"say":"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.","live":["item_1","relative","item_2","heading_limit"],"does":[[409.1185833333334,"item_3 is shown on the screen, written out."],[413.6235833333334,"item_3 (the \"cannot resolve\" part) is indicated — a transient flash."]]},{"start":420.86458333333337,"say":"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.","live":["item_1","relative","item_2","item_3","heading_limit"],"does":[[429.82758333333334,"A box is drawn around relative."],[431.22758333333337,"heading_limit is hidden from the screen — left the board."],[431.22758333333337,"item_1 is hidden from the screen — left the board."],[431.22758333333337,"item_2 is hidden from the screen — left the board."],[431.22758333333337,"item_3 is hidden from the screen — left the board."],[431.22758333333337,"relative is hidden from the screen — left the board."]]}]},{"title":"Probability and the Places Never Found","start":432.26925000000006,"end":573.0964166666668,"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=<function>, x_range=(0.0, 1.0))","curve_2":"a FunctionPlot [green] drawn in case_2 (function=<function>, x_range=(0.0, 1.0))","curve_3":"a FunctionPlot [red] drawn in case_3 (function=<function>, x_range=(0.0, 1.0))","density":"a FunctionPlot [green] drawn in density_axes (function=<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=<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":[{"start":432.26925000000006,"say":"A wavefunction is not itself a probability. 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More oscillations produce more exact cancellations and therefore more nodes.","live":["case_1","caption_1","case_2","caption_2","case_3","caption_3","node_count","heading_states","curve_1","area_1","curve_2","area_2","node_2","curve_3","area_3","node_3a","node_3b"],"does":[[536.4402500000001,"curve_1 is indicated — a transient flash."],[545.4272500000001,"curve_3 is indicated — a transient flash."]]},{"start":547.18775,"say":"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.","live":null,"does":[[559.2162500000001,"node_2 is indicated — a transient flash."],[559.41625,"node_3a is indicated — a transient flash."],[559.61625,"node_3b is indicated — a transient flash."]]},{"start":562.40475,"say":"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.","live":null,"does":[[571.0312500000001,"A box is drawn around node_count."],[572.05475,"caption_1 is hidden from the screen — left the board."],[572.05475,"caption_2 is hidden from the screen — left the board."],[572.05475,"caption_3 is hidden from the screen — left the board."],[572.05475,"case_1 is hidden from the screen — left the board."],[572.05475,"curve_1 is hidden from the screen — case_1 left the board."],[572.05475,"area_1 is hidden from the screen — case_1 left the board."],[572.05475,"case_2 is hidden from the screen — left the board."],[572.05475,"curve_2 is hidden from the screen — case_2 left the board."],[572.05475,"area_2 is hidden from the screen — case_2 left the board."],[572.05475,"node_2 is hidden from the screen — case_2 left the board."],[572.05475,"case_3 is hidden from the screen — left the board."],[572.05475,"curve_3 is hidden from the screen — case_3 left the board."],[572.05475,"area_3 is hidden from the screen — case_3 left the board."],[572.05475,"node_3a is hidden from the screen — case_3 left the board."],[572.05475,"node_3b is hidden from the screen — case_3 left the board."],[572.05475,"heading_states is hidden from the screen — left the board."],[572.05475,"node_count is hidden from the screen — left the board."]]}]},{"title":"Why Conjugated Molecules Have Colour","start":573.0964166666668,"end":802.4982916666668,"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":[{"start":573.0964166666668,"say":"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.","live":[],"does":[[573.0964166666668,"heading_molecule is shown on the screen, written out."],[579.0754166666668,"molecule is shown on the screen, written out."],[579.0754166666668,"cloud is shown on the screen, written out."],[579.0754166666668,"carbon_1 is shown on the screen, written out."],[579.0754166666668,"carbon_2 is shown on the screen, written out."],[579.0754166666668,"carbon_3 is shown on the screen, written out."],[579.0754166666668,"carbon_4 is shown on the screen, written out."],[579.0754166666668,"carbon_5 is shown on the screen, written out."],[579.0754166666668,"carbon_6 is shown on the screen, written out."],[579.0754166666668,"bond_1 is shown on the screen, written out."],[579.0754166666668,"bond_2 is shown on the screen, written out."],[579.0754166666668,"bond_3 is shown on the screen, written out."],[579.0754166666668,"bond_4 is shown on the screen, written out."],[579.0754166666668,"bond_5 is shown on the screen, written out."],[579.0754166666668,"molecular_length is shown on the screen, written out."]]},{"start":584.6674166666668,"say":"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.","live":["molecule","heading_molecule","cloud","carbon_1","carbon_2","carbon_3","carbon_4","carbon_5","carbon_6","bond_1","bond_2","bond_3","bond_4","bond_5","molecular_length"],"does":[[585.3184166666667,"bond_3 is indicated — a transient flash."],[587.5234166666668,"cloud is indicated — a transient flash."],[595.4304166666668,"molecule moves to a new place on the board."],[595.4304166666668,"concept is shown on the screen, written out."],[595.4304166666668,"box_energy is shown on the screen, written out."]]},{"start":598.4564166666668,"say":"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.","live":["concept","box_energy","molecule","heading_molecule","cloud","carbon_1","carbon_2","carbon_3","carbon_4","carbon_5","carbon_6","bond_1","bond_2","bond_3","bond_4","bond_5","molecular_length"],"does":[[600.4764166666668,"molecular_length is indicated — a transient flash."],[614.1764166666668,"box_energy (the \"L^2\" part) is indicated — a transient flash."]]},{"start":615.8439166666667,"say":"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.","live":null,"does":[[627.3039166666667,"box_energy is hidden from the screen — left the board."],[627.3039166666667,"concept is hidden from the screen — left the board."],[627.3039166666667,"heading_molecule is hidden from the screen — left the board."],[627.3039166666667,"molecule is hidden from the screen — left the board."],[627.3039166666667,"cloud is hidden from the screen — molecule left the board."],[627.3039166666667,"carbon_1 is hidden from the screen — molecule left the board."],[627.3039166666667,"carbon_2 is hidden from the screen — molecule left the board."],[627.3039166666667,"carbon_3 is hidden from the screen — molecule left the board."],[627.3039166666667,"carbon_4 is hidden from the screen — molecule left the board."],[627.3039166666667,"carbon_5 is hidden from the screen — molecule left the board."],[627.3039166666667,"carbon_6 is hidden from the screen — molecule left the board."],[627.3039166666667,"bond_1 is hidden from the screen — molecule left the board."],[627.3039166666667,"bond_2 is hidden from the screen — molecule left the board."],[627.3039166666667,"bond_3 is hidden from the screen — molecule left the board."],[627.3039166666667,"bond_4 is hidden from the screen — molecule left the board."],[627.3039166666667,"bond_5 is hidden from the screen — molecule left the board."],[627.3039166666667,"molecular_length is hidden from the screen — molecule left the board."]]},{"start":628.5039166666668,"say":"Electrons now fill the allowed levels from the bottom upward. Each spatial level can hold two electrons with opposite spin.","live":[],"does":[[628.5039166666668,"heading_levels is shown on the screen, written out."],[630.4544166666667,"levels is shown on the screen, written out."],[630.4544166666667,"level_1 is shown on the screen, written out."],[630.4544166666667,"level_2 is shown on the screen, written out."],[630.4544166666667,"level_3 is shown on the screen, written out."],[630.4544166666667,"level_4 is shown on the screen, written out."],[630.4544166666667,"electron_1a is shown on the screen, written out."],[630.4544166666667,"electron_1b is shown on the screen, written out."],[630.4544166666667,"electron_2a is shown on the screen, written out."],[630.4544166666667,"electron_2b is shown on the screen, written out."],[630.4544166666667,"electron_3a is shown on the screen, written out."],[630.4544166666667,"electron_3b is shown on the screen, written out."],[634.2974166666668,"levels moves to a new place on the board."],[634.2974166666668,"fill_1 is shown on the screen, written out."]]},{"start":637.2309166666668,"say":"If the conjugated system has N pi electrons and N is even, the highest filled level has index n equal to N over two.","live":["fill_1","levels","heading_levels","level_1","level_2","level_3","level_4","electron_1a","electron_1b","electron_2a","electron_2b","electron_3a","electron_3b"],"does":[[642.4084166666668,"level_3 is indicated — a transient flash."],[645.4504166666668,"fill_2 is shown on the screen, written out."]]},{"start":647.3389166666668,"say":"The next level is empty. Absorbing light can lift an electron across this frontier gap from level n to level n plus one.","live":["fill_1","fill_2","levels","heading_levels","level_1","level_2","level_3","level_4","electron_1a","electron_1b","electron_2a","electron_2b","electron_3a","electron_3b"],"does":[[647.8614166666667,"level_4 is indicated — a transient flash."],[650.8224166666668,"excitation is shown on the screen, written out."]]},{"start":656.8499166666668,"say":"Begin with the difference E n plus one minus E n.","live":["fill_1","fill_2","levels","heading_levels","level_1","level_2","level_3","level_4","electron_1a","electron_1b","electron_2a","electron_2b","electron_3a","electron_3b","excitation"],"does":[[657.9114166666668,"gap_work is shown on the screen, written out."]]},{"start":662.1459166666667,"say":"Insert the particle-in-a-box energies. The two levels differ only in their squared quantum numbers.","live":null,"does":[[662.4944166666668,"gap_work is shown on the screen, written out."],[667.1964166666668,"gap_work (the \"(n+1)^2-n^2\" part) is indicated — a transient flash."]]},{"start":669.5029166666668,"say":"Simplifying gives a gap proportional to two n plus one divided by L squared.","live":null,"does":[[669.8514166666668,"gap_work is shown on the screen, written out."],[671.9404166666668,"gap_work (the \"2n+1\" part) is indicated — a transient flash."],[673.9724166666667,"gap_work (the \"L^2\" part) is indicated — a transient flash."]]},{"start":675.6984166666667,"say":"A photon is absorbed when its energy matches that gap. Delta E equals h f, or h c over wavelength.","live":null,"does":[[676.2324166666667,"photon_work is shown on the screen, written out."],[678.9614166666668,"photon_work (the \"Delta E\" part) is indicated — a transient flash."],[683.6284166666668,"photon_work (the \"frac(h c, lambda)\" part) is indicated — a transient flash."]]},{"start":685.1799166666667,"say":"Solving for wavelength makes the trend explicit. A smaller energy gap corresponds to a longer absorbed wavelength.","live":null,"does":[[685.5284166666668,"photon_work is shown on the screen, written out."],[691.2874166666668,"photon_work (the \"frac(h c, Delta E)\" part) is indicated — a transient flash."],[693.3419166666667,"fill_1 is hidden from the screen — left the board."],[693.3419166666667,"fill_2 is hidden from the screen — left the board."],[693.3419166666667,"gap_work is hidden from the screen — left the board."],[693.3419166666667,"heading_levels is hidden from the screen — left the board."],[693.3419166666667,"levels is hidden from the screen — left the board."],[693.3419166666667,"level_1 is hidden from the screen — levels left the board."],[693.3419166666667,"level_2 is hidden from the screen — levels left the board."],[693.3419166666667,"level_3 is hidden from the screen — levels left the board."],[693.3419166666667,"level_4 is hidden from the screen — levels left the board."],[693.3419166666667,"electron_1a is hidden from the screen — levels left the board."],[693.3419166666667,"electron_1b is hidden from the screen — levels left the board."],[693.3419166666667,"electron_2a is hidden from the screen — levels left the board."],[693.3419166666667,"electron_2b is hidden from the screen — levels left the board."],[693.3419166666667,"electron_3a is hidden from the screen — levels left the board."],[693.3419166666667,"electron_3b is hidden from the screen — levels left the board."],[693.3419166666667,"excitation is hidden from the screen — levels left the board."],[693.3419166666667,"photon_work is hidden from the screen — left the board."]]},{"start":693.9419166666668,"say":"Compare two related conjugated systems. The shorter chain confines its pi electrons more tightly, so the frontier levels stand farther apart.","live":[],"does":[[693.9419166666668,"heading_compare is shown on the screen, written out."],[697.8314166666668,"short is shown on the screen, written out."],[697.8314166666668,"short_cloud is shown on the screen, written out."],[697.8314166666668,"short_chain is shown on the screen, written out."],[697.8314166666668,"short_low is shown on the screen, written out."],[697.8314166666668,"short_high is shown on the screen, written out."],[697.8314166666668,"short_jump is shown on the screen, written out."],[702.5684166666667,"short_jump is indicated — a transient flash."]]},{"start":704.3869166666667,"say":"The longer chain spreads the wavefunctions across a greater distance. Its relevant levels are closer together, so the transition needs a lower-energy photon.","live":["short","heading_compare","short_cloud","short_chain","short_low","short_high","short_jump"],"does":[[704.9324166666668,"short moves to a new place on the board."],[704.9324166666668,"long is shown on the screen, written out."],[704.9324166666668,"long_cloud is shown on the screen, written out."],[704.9324166666668,"long_chain is shown on the screen, written out."],[704.9324166666668,"long_low is shown on the screen, written out."],[704.9324166666668,"long_high is shown on the screen, written out."],[704.9324166666668,"long_jump is shown on the screen, written out."],[710.1914166666668,"long_jump is indicated — a transient flash."]]},{"start":715.2154166666667,"say":"Lower photon energy means longer wavelength. As conjugation is extended through a family of dyes, the absorption band commonly shifts toward longer visible wavelengths.","live":["short","long","heading_compare","short_cloud","short_chain","short_low","short_high","short_jump","long_cloud","long_chain","long_low","long_high","long_jump"],"does":[[715.5634166666667,"long_jump is indicated — a transient flash."],[717.2934166666668,"short_jump is indicated — a transient flash."]]},{"start":726.9954166666668,"say":"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.","live":null,"does":[[727.5294166666667,"short_cloud is indicated — a transient flash."],[729.2944166666667,"long_cloud is indicated — a transient flash."],[740.3239166666667,"heading_compare is hidden from the screen — left the board."],[740.3239166666667,"long is hidden from the screen — left the board."],[740.3239166666667,"long_cloud is hidden from the screen — long left the board."],[740.3239166666667,"long_chain is hidden from the screen — long left the board."],[740.3239166666667,"long_low is hidden from the screen — long left the board."],[740.3239166666667,"long_high is hidden from the screen — long left the board."],[740.3239166666667,"long_jump is hidden from the screen — long left the board."],[740.3239166666667,"short is hidden from the screen — left the board."],[740.3239166666667,"short_cloud is hidden from the screen — short left the board."],[740.3239166666667,"short_chain is hidden from the screen — short left the board."],[740.3239166666667,"short_low is hidden from the screen — short left the board."],[740.3239166666667,"short_high is hidden from the screen — short left the board."],[740.3239166666667,"short_jump is hidden from the screen — short left the board."]]},{"start":740.9239166666667,"say":"Now the complete chain of reasoning fits together. First, finite boundaries select standing-wave states.","live":[],"does":[[740.9239166666667,"heading_summary is shown on the screen, written out."],[746.0784166666667,"summary_1 is shown on the screen, written out."]]},{"start":749.1744166666667,"say":"Second, electrons fill those states, leaving a particular gap between the highest filled and lowest empty levels.","live":["summary_1","heading_summary"],"does":[[750.8924166666668,"summary_2 is shown on the screen, written out."],[752.3784166666667,"summary_2 (the \"specific energy\" part) is indicated — a transient flash."]]},{"start":757.0079166666667,"say":"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.","live":["summary_1","summary_2","heading_summary"],"does":[[763.7414166666667,"summary_3 is shown on the screen, written out."],[766.2954166666667,"summary_3 (the \"longer wavelength\" part) is indicated — a transient flash."]]},{"start":768.2424166666667,"say":"Finally, this remains a model. Real molecules have non-square potentials, electron interactions, vibrations, and environmental effects. Detailed spectra require more complete quantum chemistry.","live":["summary_1","summary_2","summary_3","heading_summary"],"does":[[769.8914166666667,"summary_4 is shown on the screen, written out."],[779.4234166666668,"summary_4 (the \"not exact\" part) is indicated — a transient flash."]]},{"start":783.8309166666668,"say":"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.","live":["summary_1","summary_2","summary_3","summary_4","heading_summary"],"does":[[786.8494166666667,"summary_1 is indicated — a transient flash."],[791.5164166666667,"summary_2 is indicated — a transient flash."],[794.9184166666668,"summary_3 is indicated — a transient flash."],[798.3784166666667,"summary_4 is indicated — a transient flash."],[801.456625,"heading_summary is hidden from the screen — left the board."],[801.456625,"summary_1 is hidden from the screen — left the board."],[801.456625,"summary_2 is hidden from the screen — left the board."],[801.456625,"summary_3 is hidden from the screen — left the board."],[801.456625,"summary_4 is hidden from the screen — left the board."]]}]}]},"durationSeconds":802,"chapters":[{"title":"Only Certain Waves Fit","startSeconds":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. 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."},{"title":"A Particle as a Confined Wave","startSeconds":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? 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."},{"title":"How the Energy Gaps Shrink","startSeconds":284.93758333333335,"narration":"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."},{"title":"Probability and the Places Never Found","startSeconds":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. 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."},{"title":"Why Conjugated Molecules Have Colour","startSeconds":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. 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."}]}}