The box that allows only some energies

A particle trapped between two walls it cannot cross. Classically it has a speed, bounces back and forth forever, and can be given any energy at all — nudge it a little faster and it moves a little faster. Give it a wavelength instead and the answer changes completely.

Nothing has been assumed here beyond the wavelength. The discreteness is not an extra postulate: it is what happens when a wave is confined, and it is the identical argument that decides which notes a string will sound.

The whole derivation, in four lines #

A wall the particle cannot cross means the wave is zero there and beyond. Continuity requires it to reach zero at the wall rather than jumping, so ψ(0) = ψ(L) = 0.

A sine wave of wavelength λ vanishes at 0 and again at λ/2, λ, 3λ/2, and so on. Fitting one into a box of width L therefore requires

$$L = n\,\frac{\lambda}{2}, \qquad n = 1, 2, 3, \dots$$

which is the same condition a fixed string obeys. Substituting de Broglie’s relation λ = h/p gives p = nh/2L, and the kinetic energy p²/2m is then

$$E_n = \frac{n^2h^2}{8mL^2}.$$

That is the entire calculation. It contains no quantum postulate that is not already in λ = h/p, and the ladder it produces is fixed by the geometry: the ratios are 1 : 4 : 9 : 16, whatever the particle and whatever the box.

Why n² and not n #

The spacing is the part worth dwelling on, because three different quantisation problems in this field produce three different ladder shapes and the shape is the physics.

Fitting more half-wavelengths into a fixed length shortens the wavelength in inverse proportion, so the momentum rises linearly with n. The energy of a free particle goes as the square of momentum, so the energy rises as n². The consequence is that the levels get further apart as they climb — the gap between the first and second is 3E₁, between the ninth and tenth it is 19E₁ — and a box’s spectrum has no upper limit and no accumulation point.

The spacing is what makes the ladder unlike any other. Drawn to scale, the sixth level sits at thirty-six times the first and the rungs are still spreading — because the energy goes as $n^2$ rather than as $n$. Compare it with a hydrogen atom, whose levels crowd together as they rise and converge on a ceiling. A box confines forever and an atom lets go, and the difference is visible in the shape of the ladder before any arithmetic is done.

Three shapes, three potentials: a box gives n², a Coulomb well gives −1/n², an oscillator gives n + ½. Every quantisation argument in the field is one of those or a deformation of one, and knowing which is knowing what the confinement looks like without seeing it.

The same argument, in three places #

The reason this essay sits on the standing-waves ladder rather than opening one of its own is that nothing about it is new. It is the third rung of an argument the site has already made twice.

The same argument runs in three places and is worth seeing as one. A string fixed at both ends carries only the shapes that vanish at each end, so its wavelengths are $2L/n$ and its notes are a harmonic series. A box confines a wave the same way and gets the same list of wavelengths. What differs is only what is being computed from them — a frequency for the string, an energy for the particle — and the quantisation itself is the boundary condition, not the physics.

On a string the quantity that must vanish at the ends is the displacement, and what comes out discrete is the frequency. In an organ pipe closed at one end it is the pressure at one end and the displacement at the other, and what comes out is the same series with the even members missing. In a box it is the probability amplitude, and what comes out is the energy.

The differences are entirely in the dispersion relation — how frequency relates to wavelength for the wave in question. On a string, frequency is proportional to 1/λ, so the allowed frequencies go as n and a plucked string sounds a harmonic series. For a matter wave, energy goes as 1/λ², so the allowed energies go as n² and a confined particle’s levels spread apart. That single difference is why a guitar sounds musical and an atom does not.

The generalisation is worth stating as a rule, because it is used repeatedly from here on: confinement plus a dispersion relation gives a spectrum, and the shape of the spectrum is the dispersion relation read through the shape of the confinement. Every quantisation in this field, up to and including the band structure of a solid, is an instance of it.

The energy that cannot be removed #

The lowest state is n = 1, not n = 0. Setting n = 0 makes the wavefunction zero everywhere, which is not a state of a particle in a box — it is the absence of a particle.

So the particle’s least possible energy is E₁ = h²/8mL², and it cannot be cooled below it. Something confined is never at rest, and the tighter the confinement the more violently it is not at rest, because E₁ goes as 1/L².

This is not a technicality. It is why matter has volume: pressing atoms together squeezes their electrons into smaller boxes, the electrons’ zero-point energy rises steeply, and the energy cost is what pushes back. It is why helium stays liquid down to absolute zero at ordinary pressure — its zero-point motion exceeds the weak binding that would otherwise freeze it. And it is why a nucleus cannot contain an electron: confining one to 10⁻¹⁵ metres would give it a zero-point energy in the tens of MeV, far more than any nuclear binding can hold.

The arithmetic runs the other way too. An electron in a box one nanometre wide has E₁ = 0.376 eV, and the first gap is a little over an electronvolt — visible-light energies, which is why a nanometre-scale semiconductor crystal has a colour that depends on its size. That is a quantum dot, and its whole design principle is the 1/L² on this page.

Why a marble in a shoebox shows nothing #

Put the numbers in for a macroscopic object and the ladder disappears.

A one-gram marble in a ten-centimetre box has E₁ = 5.5 × 10⁻⁶⁴ joules. Rolling at one millimetre per second it has 5 × 10⁻¹⁰ joules of kinetic energy, which corresponds to n ≈ 3 × 10²⁶. The gap to the next level is a relative change of 7 × 10⁻²⁷ — smaller than any energy difference that could be measured, by a factor beyond absurdity.

That figure is the correspondence principle in one picture, and the honest form of it is worth stating carefully because the loose form is wrong. The quantum probability density does not converge to the classical one point by point — it oscillates between zero and twice the classical value at every n, for ever. What converges is the density averaged over any fixed window, and it converges as 1/n. So the classical answer is recovered not because the quantum wiggles die away but because nothing can resolve them.

Reading a size off a colour #

The 1/L² is not a curiosity; it is a manufacturing specification.

A crystal of cadmium selenide a few nanometres across confines its electrons in all three directions, and the gap between its lowest states — the energy of the light it absorbs and re-emits — is set by that size rather than by the material’s bulk properties. Grow the crystals two nanometres across and they fluoresce green; grow them six across and they fluoresce red; the chemistry is identical in both flasks and only the growth time differs.

The application is now ordinary. Quantum-dot displays use the effect to convert a blue backlight into precisely placed red and green, with linewidths narrower than a phosphor can manage, which is what a wide colour gamut requires. Biological labelling uses it because a set of dots excited by one wavelength emits at several, so several things can be tracked at once.

What makes it a good illustration for this page is that the design variable is a length. Not a composition, not a doping level, not a temperature — a length, entering as its inverse square, exactly as the four-line derivation above says it must.

What a stationary state does not do #

The states drawn above are called stationary, and the word means something precise that the picture actively hides.

A single one of these states has a probability density that does not change with time at all. The wavefunction rotates in the complex plane at a rate E/ħ, but its magnitude is fixed, so nothing about the particle’s distribution moves. An electron in the ground state of a box is not sloshing back and forth slowly; it is not doing anything.

Motion requires a superposition. Put the particle in a mixture of the n = 1 and n = 2 states and the two rotate at different rates, their relative phase changes, and the probability density genuinely oscillates from one side of the box to the other — at a frequency (E₂ − E₁)/h, which is exactly the frequency of the photon a transition between them would emit.

A stationary state does not do anything, and the word means exactly that. Its probability density is constant in time, so an electron in a single level is not orbiting, not oscillating, and not moving in any sense a picture could show. What does move is a superposition: two levels together have a relative phase that advances at a rate set by their energy difference, and the density sloshes at that frequency. Nothing radiates from a stationary state because nothing about it changes.

That correspondence is not a coincidence and it is the beginning of the answer to how a quantum system radiates: an atom in a single stationary state has no oscillating charge distribution and therefore no reason to emit, and an atom in a superposition of two states has one, oscillating at the frequency of the line.

The same answer from the other direction #

The zero-point energy was introduced above as a consequence of $n$ starting at one rather than nought, which is correct and reads as a technicality about counting. There is a second route to it that makes it look inevitable, and the two agreeing is worth having.

A wave confined to a region of width $L$ cannot be a single wavelength. Cutting a sine wave off at two points requires a spread of wavenumbers to build the cut, and the narrower the region the wider that spread has to be — a purely classical statement about Fourier transforms, true of a pulse of sound or a burst of radio and having nothing to do with quantum mechanics.

The quantum content is one substitution: momentum is $\hbar$ times wavenumber. So a spread in wavenumber is a spread in momentum, and a particle confined to $L$ has a momentum uncertain by at least of order $\hbar/L$. A nonzero spread in momentum means a nonzero mean square momentum even when the mean is zero, and mean square momentum is kinetic energy:

$$E \gtrsim \frac{\hbar^2}{2mL^2}.$$

That is the ground-state energy to within a factor of $\pi^2$, obtained without solving anything.

Checking the exact numbers is worth doing because the box is one of the few cases where both quantities can be computed in closed form. The ground state has a position spread of about $0.181L$ and a momentum spread of exactly $\pi\hbar/L$, so their product is $0.568\,\hbar$ — comfortably above the bound of $\hbar/2$ and not at it. The state that achieves the minimum is a Gaussian, and a sine cut off at two walls is not one.

The reframing matters because it removes the last appearance of a special assumption. Quantisation came out of a boundary condition; the zero-point energy comes out of a Fourier transform; and the only quantum input anywhere is $p = h/\lambda$. Everything else in this essay is a fact about waves.

What the momentum of a confined particle is #

There is a question the ladder invites and answers wrongly if it is answered casually. The derivation fixed $p = nh/2L$, so it is natural to say the particle in the $n$-th state has that momentum. It does not, and the reason exposes something the drawing cannot.

A stationary state of a box is a sine, and a sine is the sum of two travelling waves going opposite ways. So the state contains $+nh/2L$ and $-nh/2L$ in equal measure, and the mean momentum is exactly zero — which it must be, since a particle in a stationary state is not going anywhere. What the derivation fixed is the magnitude, and the mean square momentum is $2mE_n$ exactly.

Worse, the two values are not the only possible outcomes of a momentum measurement. The wavefunction is a sine inside the box and zero outside, and truncating a sine spreads its spectrum: its momentum-space description is not two spikes but two peaks of finite width, centred on $\pm nh/2L$ and with tails reaching to every momentum. A momentum measurement on a particle in the ground state can, with small probability, return any value at all.

That is not a defect of the measurement; it is what confinement means. Position and momentum cannot both be sharp, the state has a definitely bounded position, and the price is paid in momentum.

The infinite well takes the price further than a real system does, and the excess is a good illustration of what the idealisation costs. Because the wavefunction has a kink at each wall — its slope jumps discontinuously — its momentum tail falls off only as the square of the momentum, which is slow enough that the mean fourth power of the momentum is infinite. A perfectly confined particle has a finite kinetic energy and an infinite mean square kinetic energy, which is a nonsense produced entirely by the wall.

A finite well removes it. There the wavefunction bends smoothly through the boundary rather than kinking, its momentum tail falls off fast enough for every moment to exist, and the pathology disappears. It is the same lesson the section below draws about the level spacings, arriving in a place nobody looks: an infinite wall is not a strong wall but an impossible one, and it produces impossible answers to questions the drawn figure does not think to ask.

What an infinite wall costs #

Every result above assumes walls of infinite height, and that assumption is doing more work than it looks.

A real confinement is a well of finite depth, and the wave does not vanish at its edge — it decays into the wall exponentially, which is the same leakage that lets a particle tunnel. The consequences are three.

The wave extends slightly beyond the nominal walls, so the effective box is wider than the drawn one and every level sits lower than n²h²/8mL² predicts. The correction is small for the ground state of a deep well and large for the highest bound states.

A finite well holds only a finite number of bound states. Above the well’s depth the particle is free, the spectrum becomes continuous, and the ladder simply stops — the infinite well’s endless tower of levels is an artefact of a wall no material provides.

And the sharpest states are the ones the approximation is worst for. Near the top of a real well the levels are compressed relative to n², not spread, which is the opposite of what the drawn figure shows.

An infinite wall is an idealisation, and what it costs is worth naming. A wall of finite height does not stop the wave at the boundary — the amplitude decays through it, and if the wall is thin enough something emerges on the far side. So a real box has levels slightly lower than the idealised ones and a lifetime rather than a permanent confinement. The infinite well is the case where the leakage has been set to zero by hand, which is why it gives exact answers to a problem nobody has.

Where the walls come from #

In one dimension the box is a fiction. In practice, confinement in the sense of this page happens in three settings and each is worth a sentence.

A semiconductor quantum well. A thin layer of one material between two of a wider gap presents electrons with a genuine square well, tens of nanometres wide and a few tenths of an electronvolt deep. The levels are measurable, the 1/L² is confirmed by growing wells of different thickness, and the whole of laser-diode design rests on placing them where they are wanted.

A molecule’s delocalised electrons. A chain of conjugated bonds is, to a first approximation, a one-dimensional box for the π electrons, and the “free electron model” predicts the colour of a dye from the length of its chain — badly, but well enough to get the trend right and to explain why longer chains absorb further to the red.

A nucleus. The shell model treats nucleons as occupying levels in a rounded well, and the magic numbers of exceptional stability come out of the level ordering. The well is not square and the correction matters, which is what makes the model interesting rather than a curiosity.

What the picture cannot show #

The figure draws the wavefunction as a curve riding on the energy level, and both of those choices are conventions rather than facts.

The height of the curve above its level line has no physical meaning at all — the vertical scale of ψ and the vertical scale of energy are different quantities, and drawing them on one axis is a device for showing which state has which energy. A reader who measures the amplitude against the energy axis is measuring an arbitrary choice made by the person who drew it.

The curve is also real, and a wavefunction generally is not. For the infinite well the stationary states can be written as real sines, and that hides the time dependence entirely: each state actually rotates in the complex plane at a rate set by its energy, which is invisible in a static drawing and is the whole reason superposed states move while single states do not.

And the drawing shows one particle. Two particles in the same box do not simply occupy two of these curves — what they are allowed to do together is an additional rule that this picture cannot express and that decides the structure of every atom.

Where the ladder goes next #

The rungs from here: the finite well solved properly, with the transcendental matching condition that decides how many bound states fit; the harmonic oscillator, whose evenly spaced ladder is the one Planck needed for the blackbody; the box in three dimensions, where degeneracy appears and different states share an energy; the ring, where the boundary condition is periodicity rather than a zero and the ladder changes shape again; and the same counting applied to an enormous number of coupled boxes, which turns a ladder into a band.

The claim to carry forward is what the confinement does. A free particle can have any energy; a confined one cannot, and nothing was added to make that true. Quantisation is a boundary condition, not a postulate — and the shape of the resulting ladder is a fingerprint of the shape that did the confining.