Library
Physics

Quantum Particles in a Box

An exploration of why quantum particles confined to a one-dimensional space exhibit discrete energy levels, derived from the Schrödinger equation and boundary conditions.

Scene 1 of 4
The Infinite Well
LV = ∞V = ∞
V(x)=∞,  x≤0 or x≥LV(x) = \infty, \; x \le 0 \text{ or } x \ge L
Imagine a particle trapped in a box of width L. The walls are infinitely high, meaning the particle cannot exist outside, forcing it to settle into specific patterns.
Step-by-step solver
1

(a) Potential Setup

Define the infinite well potential boundaries.

V(x)=0 for 0<x<L, else ∞V(x) = 0 \text{ for } 0 < x < L, \text{ else } \infty
2

(b) Boundary Conditions

The wavefunction must be zero at the infinite walls.

ψ(0)=0,ψ(L)=0\psi(0) = 0, \psi(L) = 0
3

(c) Schrödinger Equation

Set up the time-independent equation within the well.

−ℏ22md2ψdx2=Eψ-\frac{\hbar^2}{2m} \frac{d^2\psi}{dx^2} = E\psi
4

(d) Solving for k

Solve the second-order ODE and apply boundary conditions to find valid wavenumbers.

kn=nπL,n=1,2,…k_n = \frac{n\pi}{L}, n=1, 2, \dots
5

(e) Energy Levels

Relate wavenumber to energy eigenvalue.

En=n2π2ℏ22mL2E_n = \frac{n^2\pi^2\hbar^2}{2mL^2}

Original question

Derive the quantised energy levels for a particle of mass m in a one-dimensional infinite potential well of width L, by solving the time-independent Schrodinger equation with appropriate boundary conditions.

Follow-up chat

Ask me anything about this lesson — I'll answer using what we just covered.