Bound States in One Dimension: Boxes and Wells

The infinite square well, solved

Take a particle trapped between two impenetrable walls: V=0 for $0<x<L and V=\infty$ outside. The particle simply cannot be in the walls, so \psi=0 there. Inside, with V=0, the time-independent Schrödinger equation reduces to a free-particle equation, and the wall positions supply the boundary conditions that quantize it.

\psi(x) = A\sin kx + B\cos kx, \qquad k = \frac{\sqrt{2mE}}{\hbar}

General solution inside the well; the boundary conditions fix the constants and quantize k.

E_n = \frac{n^2\pi^2\hbar^2}{2mL^2}, \qquad \psi_n(x)=\sqrt{\frac{2}{L}}\,\sin\!\left(\frac{n\pi x}{L}\right), \quad n=1,2,3,\dots

The complete spectrum and eigenstates of the infinite square well.

Reading the solution

The lowest state, n=1, has energy E_1 = \pi^2\hbar^2/2mL^2, which is not zero. A confined quantum particle can never sit perfectly still. This irreducible zero-point energy is demanded by the uncertainty principle: pinning the particle to a region of size L forces a spread in momentum, hence a minimum kinetic energy.

Two features guide all your intuition. Energy grows as n^2, so levels spread apart as you climb. And energy scales as 1/L^2: squeeze the box and the levels shoot up. This quantum confinement is why a smaller quantum dot glows bluer than a larger one — real, tunable, and sold in televisions. As a number, an electron in a box of L=0.1\,\text{nm} has E_1\approx 38\,\text{eV}, comparable to atomic energies; the same electron in a micron-sized box has levels far too close to resolve, and it behaves classically.

The states are mutually orthogonal, the n-th state has n-1 interior nodes, and any wavefunction in the well can be built as a Fourier sine series of them. Computing an expectation value such as \langle x\rangle=L/2 or \langle x^2\rangle is now a routine integral.

The finite square well: leaking out

Make the walls finite — depth V_0 instead of infinity — and something new happens. For a bound state with E<V_0, the region outside is classically forbidden, yet the wavefunction there is not zero. It decays exponentially. The finite square well is the first place you see a particle's wavefunction bleed into a forbidden zone.

\psi_{\text{outside}}(x) \propto e^{-\kappa |x|}, \qquad \kappa = \frac{\sqrt{2m(V_0-E)}}{\hbar}

Outside a finite well the wavefunction decays with penetration depth 1/κ, not oscillates.

You solve it by matching \psi and its slope \psi' at each wall (both must be continuous). This gives a transcendental equation with no closed form — you solve it graphically or numerically. The result: only a finite number of bound states exist, they alternate even and odd in symmetry, and each sits slightly below its infinite-well counterpart because the wavefunction now has a little more room.

k\tan\!\left(\tfrac{kL}{2}\right) = \kappa \quad (\text{even states}), \qquad -k\cot\!\left(\tfrac{kL}{2}\right) = \kappa \quad (\text{odd states})

The transcendental matching conditions for a symmetric finite well of width L.

Lessons for every bound state

These two wells teach the general rules. A bound particle has a discrete spectrum. The ground state is nodeless; each higher state adds one node (the node theorem). And wherever the walls are finite, the wavefunction tunnels a little into the forbidden region — the exponential tail that becomes the whole story of the next guide.

The same recipe handles the quantum harmonic oscillator you met in Quantum Mechanics I, where ladder operators give evenly spaced levels E_n=(n+\tfrac12)\hbar\omega — again with a nonzero zero-point energy. Wells and oscillators are the building blocks of quantum dots, molecular vibrations and the modes of every quantum field.