Principles of Quantum Mechanics · Problem 10.2.1

Systems with N Degrees of Freedom — R. Shankar

Physics — Quantum Mechanics · particle-in-3d-box

Problem

(Particle in a Three-Dimensional Box). Recall that a particle in a one-dimensional box extending from

x=0

L

is confined to the region

0\leq x\leq L

; its wave function vanishes at the edges

x=0

and

L

(Exercise 5.2.5). Consider now a particle confined in a three-dimensional cubic box of volume

L^3

. Choosing as the origin one of its corners, and the

x

y

, and

z

axes along the three edges meeting there, show that the normalized energy eigenfunctions are

\psi_E(x,y,z)=\left(\frac{2}{L}\right)^{1/2}\sin\left(\frac{n_x\pi x}{L}\right)\left(\frac{2}{L}\right)^{1/2}\sin\left(\frac{n_y\pi y}{L}\right)\left(\frac{2}{L}\right)^{1/2}\sin\left(\frac{n_z\pi z}{L}\right)

where

E=\frac{\hbar^2\pi^2}{2ML^2}(n_x^2+n_y^2+n_z^2)

and

n_i

are positive integers.

Intuitive explanation

The key insight is that a 3D box problem separates into three independent 1D problems along each axis. Since the potential is zero inside and infinite at all six walls, the boundary condition (wave function vanishes at edges) applies independently to $x$ , $y$ , and $z$ . You can write $\psi(x,y,z)$ as a product of three 1D solutions, each with its own quantum number; the total energy is then the sum of three independent 1D energies. Normalization requires multiplying each 1D factor by the appropriate constant.

Intuitive explanation

Related theory

Particle On A Chain Particle In A Box

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