Why Energy Is Quantised: From Standing Waves to Coloured Molecules

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.