The Infinite Square Well

Take a single electron and trap it. Not loosely — box it into a tiny region a few nanometres wide, with walls so steep the electron simply cannot get out. Physicists do exactly this inside a quantum dot, a speck of semiconductor so small that the electrons rattling around inside it behave like a textbook problem come to life. And the textbook problem is the one on this page: the infinite square well, the very first fully solvable model in quantum mechanics, and the one every physicist meets on day one of confinement.

The setup is deliberately austere. A particle of mass m is confined to a one-dimensional stretch of length L. Inside, it feels nothing — the potential is flat and zero. At the two edges the potential leaps to +\infty, an unclimbable wall. That is the whole apparatus. Yet from this bare box tumble out three of the most important ideas in all of quantum theory: the particle's energy can only take certain discrete values, its lowest possible energy is not zero, and the states it lives in are standing waves. One box, taught thoroughly, and you understand why the quantum world comes in rungs.

V(x) = \begin{cases} 0, & 0 < x < L \\ +\infty, & x \le 0 \ \text{or}\ x \ge L. \end{cases}

Boundary conditions force standing waves

Inside the well the potential is zero, so the time-independent Schrödinger equation is just that of a free particle:

-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} = E\,\psi, \qquad 0 < x < L.

The general solution is a mix of sines and cosines, \psi(x) = A\sin(kx) + B\cos(kx), with k = \sqrt{2mE}/\hbar. On its own that allows any energy at all. What pins everything down are the walls. The wavefunction must be continuous, and outside the well — where the potential is infinite — the particle simply cannot be, so \psi = 0 there. Continuity then demands that \psi vanish exactly at both edges:

\psi(0) = 0 \qquad\text{and}\qquad \psi(L) = 0.

The first condition kills the cosine (\cos(0) = 1 \ne 0), leaving \psi(x) = A\sin(kx). The second condition, \sin(kL) = 0, is the crucial squeeze: a sine is zero only at whole multiples of \pi, so kL is not free to be anything — it must be n\pi for a whole number n:

k_n = \frac{n\pi}{L}, \qquad n = 1, 2, 3, \dots

This is exactly the same maths as a guitar string clamped at both ends: only those wavelengths that fit a whole number of half-humps between the walls are allowed. The particle can only exist as one of a discrete ladder of standing waves. Normalising so the total probability is 1 (setting \int_0^L |\psi|^2\,dx = 1) fixes A = \sqrt{2/L}, and we have the stationary states:

Seeing the standing waves

The formulae are clean, but the picture is cleaner still. Below is the wavefunction \psi_n(x) plotted across the box. Drag the slider to climb the ladder of states and watch what happens: each step up in n squeezes in one more half-wave, and the curve wiggles faster and faster while always returning obediently to zero at both walls.

Two counts are worth burning into memory. A node is an interior point where \psi_n = 0 (not counting the walls); state n has exactly n - 1 of them. An antinode is a peak of the wave; state n has n of them. The ground state n = 1 is the gentlest possible wave that still fits: one hump, zero interior nodes.

What the particle actually does is governed not by \psi_n but by its square, the probability density |\psi_n(x)|^2. Switch to that picture below.

This is where quantum intuition parts ways with classical. A marble rattling in a box would be found with equal probability anywhere inside it. The quantum particle is nothing of the sort: in the ground state it strongly prefers the middle and avoids the walls, and in excited states it clumps into a landscape of n hills and n - 1 forbidden valleys where it is never found at all.

The energy ladder: En ∝ n²

Because every allowed energy is E_n = n^2 E_1, the whole spectrum is fixed once you know the single number

E_1 = \frac{\pi^2\hbar^2}{2mL^2}.

Every other level is just a whole-number-squared multiple of it: E_2 = 4E_1, E_3 = 9E_1, E_4 = 16E_1, and so on. Because of that n^2, the rungs do not sit evenly spaced; they spread apart as you climb. The gap between neighbouring levels is

E_{n+1} - E_n = \big[(n+1)^2 - n^2\big]E_1 = (2n+1)\,E_1,

which grows without limit. The picture below draws the ladder to scale — notice the widening gaps, the opposite of the crowding levels you saw in the hydrogen atom.

The dependence on L is just as striking. Because E_1 \propto 1/L^2, shrinking the box shoots every energy up. Halve the width and every level quadruples. This is not a mathematical accident but a real, measurable effect: it is precisely why smaller quantum dots glow with bluer (higher-energy) light than larger ones. Squeezing a quantum particle costs energy — a kind of "quantum pressure" that resists confinement, and the ultimate reason matter doesn't simply collapse.

Worked examples

Example 1 — energy ratios. An electron in a box sits in the ground state with energy E_1. It absorbs a photon and jumps to the n = 3 state. How much larger is its energy now? Since E_n = n^2 E_1,

\frac{E_3}{E_1} = \frac{3^2 E_1}{E_1} = 9.

Its energy is nine times the ground-state value. And the energy of the absorbed photon is the gap, E_3 - E_1 = 9E_1 - E_1 = 8E_1.

Example 2 — the standing wavelength. A box is L = 1.0\ \text{nm} wide and the particle is in the n = 4 state. What is the wavelength of the standing wave? Using \lambda_n = 2L/n,

\lambda_4 = \frac{2 \times 1.0\ \text{nm}}{4} = 0.5\ \text{nm}.

Exactly four half-wavelengths (4 \times 0.25\ \text{nm} = 1.0\ \text{nm}) fit inside the box — the standing-wave condition made visible.

Example 3 — counting nodes. How many interior nodes does the n = 5 state have, and how many antinodes? Nodes: n - 1 = 4. Antinodes (peaks): n = 5. The wavefunction crosses zero four times between the walls and bulges into five humps.

Example 4 — shrinking the box. A particle's box is squeezed from width L to width L/2. By what factor does the ground-state energy change? Since E_1 \propto 1/L^2, halving L multiplies E_1 by (1/(L/2)^2)/(1/L^2) = 4. Every level jumps to four times its old value — the quantum pressure of tighter confinement.

Because a particle sitting perfectly still would violate the uncertainty principle. Confining it to a region of width L pins down its position to within \Delta x \sim L, which forces its momentum to be uncertain by at least \Delta p \gtrsim \hbar/L. A spread in momentum means the particle must carry some kinetic energy — roughly (\Delta p)^2/2m \sim \hbar^2/2mL^2, which is exactly the scale of E_1. This irreducible leftover is the zero-point energy. It is why liquid helium never freezes solid under its own vapour pressure however cold you make it, and why matter has a size at all: the electrons in an atom cannot collapse to the nucleus, because squeezing them into a smaller box costs more energy than they can lose. Absolute stillness is simply not on the quantum menu.

Two classic traps. The first: it is tempting to start counting states at n = 0, the way you might for many things in physics. But n = 0 gives \psi_0(x) = \sqrt{2/L}\,\sin(0) = 0 everywhere — a wavefunction that is zero across the whole box describes no particle at all (the total probability would be zero, not one). So n starts at 1, the lowest energy is E_1 \ne 0, and there is no state of zero energy. (Negative n just flips the sign of the same wave and adds nothing new.)

The second trap: assuming the particle is equally likely to be found anywhere in the box, the way a classical marble would be. Not so. The probability density |\psi_n|^2 = (2/L)\sin^2(n\pi x/L) is bumpy: peaks at the antinodes, and exact zeros at the nodes where the particle is never found. The flat classical distribution only emerges as an average in the limit of very large n — a glimpse of the correspondence principle, quantum weirdness fading into classical smoothness only when the quantum numbers get huge.