WEBVTT 00:00:00.035 --> 00:00:02.857 Why should energy ever come in separate amounts? 00:00:03.309 --> 00:00:06.177 I do not want to begin by declaring that it does. 00:00:06.629 --> 00:00:10.981 Let us begin with a familiar object instead: a stretched string, 00:00:11.064 --> 00:00:15.082 fixed at both ends, and the waves that can actually live on it. 00:00:16.734 --> 00:00:19.463 The string runs from zero to L. 00:00:19.567 --> 00:00:21.063 Each end is clamped, 00:00:21.146 --> 00:00:26.046 so neither endpoint can move, however strongly the rest of the string vibrates. 00:00:26.750 --> 00:00:29.839 That physical fact becomes a boundary condition. 00:00:30.035 --> 00:00:33.739 The displacement y must be zero at the left wall and zero again 00:00:33.739 --> 00:00:35.121 at the right wall. 00:00:35.918 --> 00:00:38.668 Try an arbitrary sinusoidal candidate. 00:00:38.751 --> 00:00:42.548 It begins correctly at zero, but look at its right end. 00:00:42.745 --> 00:00:47.296 The curve reaches the wall above the fixed point, so this shape cannot be a standing 00:00:47.296 --> 00:00:48.841 wave of this string. 00:00:49.545 --> 00:00:53.899 The simplest accepted shape places one half-wave between the walls. 00:00:54.096 --> 00:00:59.054 Both endpoints are nodes, and the entire string rises and falls as one arch. 00:00:59.850 --> 00:01:02.278 The next shape fits two half-waves. 00:01:02.474 --> 00:01:07.294 It still vanishes at both walls, and it gains another node exactly in the middle. 00:01:08.406 --> 00:01:11.019 A third shape fits three half-waves. 00:01:11.123 --> 00:01:14.895 Two interior nodes divide the string into three vibrating sections, 00:01:14.978 --> 00:01:18.357 with neighboring sections moving in opposite directions. 00:01:19.153 --> 00:01:22.591 We can continue forever, but not continuously. 00:01:22.694 --> 00:01:26.643 The length L must contain a whole number n of half-wavelengths. 00:01:27.439 --> 00:01:32.047 Solving for wavelength gives lambda n equals two L over n. 00:01:32.130 --> 00:01:38.353 The allowed wavelengths form a list: two L, L, two L over three, and so on. 00:01:38.895 --> 00:01:42.135 The same statement can be written using wave number k. 00:01:42.238 --> 00:01:49.170 Only n pi over L is allowed, with n equal to one, two, three, and onward. 00:01:49.874 --> 00:01:52.034 The important word is allowed. 00:01:52.138 --> 00:01:54.702 The material string could have many amplitudes, 00:01:54.785 --> 00:01:58.222 but its spatial patterns are selected by the two boundaries. 00:01:58.419 --> 00:02:02.065 Continuous guesses went in; a discrete family came out. 00:02:03.155 --> 00:02:04.258 Now replace the string 00:02:04.258 --> 00:02:08.323 by a particle trapped between two perfectly impenetrable walls. 00:02:08.519 --> 00:02:12.955 Quantum mechanics describes its state with a wavefunction, psi. 00:02:13.152 --> 00:02:15.544 What survives from the string argument? 00:02:16.596 --> 00:02:19.244 Here is the region available to the particle. 00:02:19.696 --> 00:02:25.141 Outside the two walls the particle cannot exist, so the wavefunction is zero there. 00:02:26.054 --> 00:02:27.955 At an ideal infinite wall, 00:02:28.039 --> 00:02:31.721 the wavefunction must meet that outside value continuously. 00:02:31.917 --> 00:02:37.711 Therefore psi is zero at x equals zero and again at x equals L. 00:02:38.195 --> 00:02:41.736 The first allowed state is one smooth arch. 00:02:41.933 --> 00:02:44.635 This is not a little particle following the curve. 00:02:44.719 --> 00:02:48.308 The curve is the wavefunction that will determine probabilities. 00:02:49.011 --> 00:02:52.759 The second state has two lobes and one interior zero. 00:02:52.843 --> 00:02:56.428 The third has three lobes and two interior zeros. 00:02:56.512 --> 00:02:59.903 They are the same spatial fitting patterns we met on the string. 00:03:00.606 --> 00:03:04.423 Write a sinusoidal candidate A sine k x. 00:03:04.507 --> 00:03:09.198 The left boundary is already satisfied because sine zero is zero. 00:03:09.995 --> 00:03:12.794 At the right wall, x becomes L. 00:03:12.897 --> 00:03:18.947 The amplitude there is A sine k L, and the wall requires that expression to vanish. 00:03:19.743 --> 00:03:22.995 Sine vanishes at whole multiples of pi. 00:03:23.099 --> 00:03:27.418 So k L must equal n pi, not an arbitrary number. 00:03:28.331 --> 00:03:31.466 Dividing by L gives the allowed wave numbers. 00:03:31.663 --> 00:03:34.310 Once again the boundaries have turned a continuous range 00:03:34.310 --> 00:03:36.656 of guesses into a numbered list. 00:03:37.360 --> 00:03:39.914 A fitting rule selects wave numbers. 00:03:40.111 --> 00:03:41.304 To find energies, 00:03:41.388 --> 00:03:45.382 we now use the stationary Schrödinger equation inside the box. 00:03:46.179 --> 00:03:49.698 Its left side measures the curvature of the wavefunction. 00:03:50.150 --> 00:03:55.525 The coefficient h squared over eight pi squared m is the familiar h bar squared 00:03:55.525 --> 00:03:59.369 over two m, written using Planck's constant h. 00:03:59.565 --> 00:04:02.991 The right side is energy times the same wavefunction. 00:04:03.695 --> 00:04:07.758 Differentiate sine twice and the original wave returns with a factor 00:04:07.758 --> 00:04:09.721 of minus k squared. 00:04:10.517 --> 00:04:13.281 Substitute that curvature into the equation. 00:04:13.385 --> 00:04:15.403 The common wavefunction cancels, 00:04:15.487 --> 00:04:20.526 leaving E equal to h squared k squared over eight pi squared m. 00:04:21.322 --> 00:04:24.330 Finally substitute the allowed k values. 00:04:24.527 --> 00:04:29.833 The energy of state n is n squared h squared over eight m L squared. 00:04:30.537 --> 00:04:33.347 That is where quantised energy enters. 00:04:33.544 --> 00:04:36.200 The equation relates energy to wavelength, 00:04:36.284 --> 00:04:39.311 while the walls permit only certain wavelengths. 00:04:39.395 --> 00:04:43.076 The discrete energies are the consequence of both facts together. 00:04:44.186 --> 00:04:47.263 The energy formula contains a powerful prediction. 00:04:47.367 --> 00:04:52.569 Keep the particle and its mass unchanged, but make the confining region wider. 00:04:53.621 --> 00:04:56.641 Beside the box are its first three energies. 00:04:56.837 --> 00:05:01.714 They are separate horizontal lines, and even the lowest state sits above zero. 00:05:02.510 --> 00:05:06.470 That nonzero ground-state energy is unavoidable. 00:05:06.783 --> 00:05:10.403 A wave trapped in a finite region cannot be perfectly flat, 00:05:10.486 --> 00:05:14.667 so it cannot have zero curvature and zero kinetic energy. 00:05:15.463 --> 00:05:18.855 The difference between neighboring levels is delta E n. 00:05:18.958 --> 00:05:22.488 Start with E n plus one minus E n. 00:05:23.401 --> 00:05:25.515 Insert the box energies. 00:05:25.712 --> 00:05:30.364 The numerator contains n plus one squared minus n squared, 00:05:30.448 --> 00:05:34.524 while every level carries the same factor one over L squared. 00:05:35.321 --> 00:05:38.572 The difference of the squares is two n plus one. 00:05:38.769 --> 00:05:42.427 So the gap is also proportional to one over L squared. 00:05:43.131 --> 00:05:45.012 Now widen the box. 00:05:45.116 --> 00:05:46.890 Watch both walls move apart 00:05:46.973 --> 00:05:50.492 while the three energy levels descend and crowd together. 00:05:51.544 --> 00:05:54.285 Nothing has changed about the integer labels. 00:05:54.389 --> 00:05:59.277 The allowed states are still numbered one, two, three, and onward. 00:05:59.729 --> 00:06:03.120 What changed is the energy scale attached to that list. 00:06:04.033 --> 00:06:08.968 A very wide box therefore has many levels inside any modest energy interval. 00:06:09.164 --> 00:06:11.438 The spectrum is discrete in principle, 00:06:11.521 --> 00:06:15.376 but the gaps may become too small for an experiment to distinguish. 00:06:16.289 --> 00:06:19.320 This is one route back toward classical physics. 00:06:19.516 --> 00:06:24.393 First, increasing L lowers the entire spectrum and shrinks every gap as one 00:06:24.393 --> 00:06:26.019 over L squared. 00:06:26.723 --> 00:06:29.684 There is a second effect at high quantum number. 00:06:29.788 --> 00:06:34.270 Divide a gap by the energy itself, and most constants cancel. 00:06:35.067 --> 00:06:40.698 The relative gap is two n plus one over n squared, which approaches two over n. 00:06:41.011 --> 00:06:46.398 At large n, neighboring states differ by a tiny fraction of their total energy. 00:06:47.195 --> 00:06:51.097 Finally, every real measurement has limited resolution. 00:06:51.293 --> 00:06:55.891 If thousands of allowed energies fall inside one unresolved band, changing 00:06:55.891 --> 00:07:00.791 energy appears continuous even though the microscopic list remains discrete. 00:07:01.704 --> 00:07:05.826 The classical world does not require the quantum rules to switch off. 00:07:06.023 --> 00:07:08.470 Large dimensions, high quantum numbers, 00:07:08.553 --> 00:07:11.991 and limited resolution make the steps too fine to notice. 00:07:13.088 --> 00:07:16.293 A wavefunction is not itself a probability. 00:07:16.490 --> 00:07:20.182 It can be positive, negative, or even complex. 00:07:20.495 --> 00:07:25.128 What predicts where the particle may be detected is its absolute square. 00:07:25.925 --> 00:07:27.725 Here is the second state. 00:07:27.922 --> 00:07:29.698 Its left lobe is positive 00:07:29.698 --> 00:07:34.679 and its right lobe is negative, but a probability density cannot be negative. 00:07:35.476 --> 00:07:38.089 Square the amplitude at every position. 00:07:38.402 --> 00:07:43.430 Both lobes rise above the axis, producing two regions where detection is likely. 00:07:44.226 --> 00:07:47.686 The total shaded area is one after normalization. 00:07:47.883 --> 00:07:50.772 That does not say the particle has a known position. 00:07:50.855 --> 00:07:55.279 It says some detection somewhere in the box has total probability one. 00:07:56.076 --> 00:07:59.583 At the middle, the wavefunction is exactly zero. 00:07:59.687 --> 00:08:04.668 Squaring zero still gives zero, so the particle is never detected at this node. 00:08:05.372 --> 00:08:08.286 Now compare the first three stationary states. 00:08:08.390 --> 00:08:12.420 The first density has one broad peak and no internal node. 00:08:12.961 --> 00:08:15.864 The probability still vanishes at both walls. 00:08:16.061 --> 00:08:20.044 Those zeros come from the confinement condition shared by every state. 00:08:20.840 --> 00:08:26.284 The second density has two peaks. Between them is one internal node at L over two, 00:08:26.367 --> 00:08:29.235 a position with exactly zero probability. 00:08:30.148 --> 00:08:33.629 The third density has three peaks and two internal nodes, 00:08:33.712 --> 00:08:36.824 at L over three and two L over three. 00:08:37.528 --> 00:08:40.640 A node is stronger than a low-probability region. 00:08:40.837 --> 00:08:45.121 At a node the wavefunction vanishes exactly, so an ideal position 00:08:45.121 --> 00:08:49.696 measurement never returns that point while the particle remains in that state. 00:08:50.493 --> 00:08:52.595 The pattern is systematic. 00:08:52.699 --> 00:08:58.168 State n has n lobes in its density and n minus one internal nodes. 00:08:58.964 --> 00:09:01.099 Higher states oscillate more rapidly 00:09:01.182 --> 00:09:03.899 because their allowed wavelengths are shorter. 00:09:04.212 --> 00:09:09.182 More oscillations produce more exact cancellations and therefore more nodes. 00:09:09.792 --> 00:09:11.244 These densities also warn 00:09:11.244 --> 00:09:15.157 us against imagining a classical bead bouncing between the walls. 00:09:15.261 --> 00:09:19.128 A classical bead can pass through every interior position. 00:09:19.231 --> 00:09:23.760 A stationary quantum state can forbid particular positions completely. 00:09:24.556 --> 00:09:28.214 The allowed energy tells us which stationary pattern we have. 00:09:28.318 --> 00:09:31.474 The square of that pattern tells us where detections can occur, 00:09:31.557 --> 00:09:34.100 and its nodes mark where they cannot. 00:09:35.169 --> 00:09:39.372 A toy model earns its keep only if it explains something real. 00:09:39.824 --> 00:09:42.226 Consider a conjugated dye molecule, 00:09:42.309 --> 00:09:46.663 built from a chain of neighboring atoms with overlapping p orbitals. 00:09:47.460 --> 00:09:49.409 The green skeleton marks the chain, 00:09:49.492 --> 00:09:53.532 while the blue region represents the shared pi-electron system. 00:09:53.729 --> 00:09:57.025 These electrons are not assigned to one particular bond; 00:09:57.108 --> 00:10:00.162 they are delocalised along much of the chain. 00:10:01.214 --> 00:10:07.124 That finite delocalised length acts, approximately, like a one-dimensional box. 00:10:07.321 --> 00:10:10.802 The detailed molecular potential is not perfectly square, 00:10:10.885 --> 00:10:14.078 but the familiar expression makes the dependence clear: 00:10:14.275 --> 00:10:17.376 every energy contains one over L squared. 00:10:18.288 --> 00:10:22.376 So the molecular orbitals form a numbered set of wave-like states. 00:10:22.572 --> 00:10:25.578 Their exact shapes are more complicated than sines, 00:10:25.661 --> 00:10:28.854 but confinement still produces separated energies. 00:10:30.251 --> 00:10:33.909 Electrons now fill the allowed levels from the bottom upward. 00:10:34.106 --> 00:10:38.170 Each spatial level can hold two electrons with opposite spin. 00:10:38.966 --> 00:10:43.481 If the conjugated system has N pi electrons and N is even, 00:10:43.564 --> 00:10:47.896 the highest filled level has index n equal to N over two. 00:10:48.692 --> 00:10:50.678 The next level is empty. 00:10:50.875 --> 00:10:55.333 Absorbing light can lift an electron across this frontier gap from level n 00:10:55.333 --> 00:10:57.424 to level n plus one. 00:10:58.336 --> 00:11:02.354 Begin with the difference E n plus one minus E n. 00:11:03.151 --> 00:11:05.857 Insert the particle-in-a-box energies. 00:11:06.309 --> 00:11:10.013 The two levels differ only in their squared quantum numbers. 00:11:10.926 --> 00:11:16.662 Simplifying gives a gap proportional to two n plus one divided by L squared. 00:11:17.575 --> 00:11:21.128 A photon is absorbed when its energy matches that gap. 00:11:21.325 --> 00:11:25.587 Delta E equals h f, or h c over wavelength. 00:11:26.383 --> 00:11:29.612 Solving for wavelength makes the trend explicit. 00:11:29.924 --> 00:11:34.349 A smaller energy gap corresponds to a longer absorbed wavelength. 00:11:35.146 --> 00:11:38.211 Compare two related conjugated systems. 00:11:38.408 --> 00:11:41.738 The shorter chain confines its pi electrons more tightly, 00:11:41.821 --> 00:11:44.701 so the frontier levels stand farther apart. 00:11:45.498 --> 00:11:49.609 The longer chain spreads the wavefunctions across a greater distance. 00:11:49.805 --> 00:11:52.079 Its relevant levels are closer together, 00:11:52.162 --> 00:11:55.367 so the transition needs a lower-energy photon. 00:11:56.070 --> 00:11:59.171 Lower photon energy means longer wavelength. 00:11:59.368 --> 00:12:02.663 As conjugation is extended through a family of dyes, 00:12:02.746 --> 00:12:06.834 the absorption band commonly shifts toward longer visible wavelengths. 00:12:07.630 --> 00:12:08.675 A dye looks coloured 00:12:08.675 --> 00:12:12.646 because it absorbs some visible wavelengths more strongly than others. 00:12:12.750 --> 00:12:15.049 The colour we observe comes from the light left 00:12:15.049 --> 00:12:20.042 over or transmitted, so it is not simply the colour of the absorbed photon. 00:12:20.838 --> 00:12:23.777 Now the complete chain of reasoning fits together. 00:12:23.973 --> 00:12:27.550 First, finite boundaries select standing-wave states. 00:12:28.253 --> 00:12:30.794 Second, electrons fill those states, 00:12:30.877 --> 00:12:35.661 leaving a particular gap between the highest filled and lowest empty levels. 00:12:36.574 --> 00:12:40.545 Third, light is absorbed when a photon matches that gap. 00:12:40.742 --> 00:12:45.223 Extending the conjugated region generally reduces the gap and moves the absorption 00:12:45.223 --> 00:12:47.047 toward longer wavelength. 00:12:47.843 --> 00:12:50.259 Finally, this remains a model. 00:12:50.455 --> 00:12:54.624 Real molecules have non-square potentials, electron interactions, 00:12:54.624 --> 00:12:57.678 vibrations, and environmental effects. 00:12:57.875 --> 00:13:01.753 Detailed spectra require more complete quantum chemistry. 00:13:02.550 --> 00:13:04.873 But the model has earned its keep. 00:13:05.185 --> 00:13:07.636 Fixed ends selected wavelengths; 00:13:07.832 --> 00:13:10.816 confined wavefunctions selected energies; 00:13:11.129 --> 00:13:14.183 molecular length controlled an optical gap. 00:13:14.798 --> 00:13:19.675 Energy is quantised because a bounded wave cannot fit in arbitrary ways.