Infinite Square Well

The infinite square well (particle in a box) is the simplest exactly solvable quantum system. A particle is confined to 0 ≤ x ≤ L with infinite potential walls. The solutions reveal quantized energy levels — a fundamental quantum phenomenon.

Key Concepts

Boundary conditions force ψ(0) = ψ(L) = 0
Eigenstates: ψₙ(x) = √(2/L) sin(nπx/L)
Quantized energies: Eₙ = n²π²ℏ²/(2mL²) = n²E₁
Ground state energy E₁ ≠ 0 (zero-point energy)
Orthonormality: ⟨ψₘ|ψₙ⟩ = δₘₙ

Key Equations

Energy eigenvalues

E_n = \frac{n^2\pi^2\hbar^2}{2mL^2},\quad n=1,2,3,\ldots

Eigenstates

\psi_n(x) = \sqrt{\frac{2}{L}}\sin\!\left(\frac{n\pi x}{L}\right)

Ground state energy

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

Level spacing

E_n = n^2 E_1

Worked Example

Example Problem

Problem

Find the ground state energy of an electron in an infinite square well of width L = 0.1 nm.

Solution

E₁ = π²ℏ²/(2mL²). With ℏ=1.055×10⁻³⁴ J·s, m=9.11×10⁻³¹ kg, L=10⁻¹⁰ m: E₁ = (9.87×1.112×10⁻⁶⁸)/(2×9.11×10⁻³¹×10⁻²⁰) = 37.6 eV.

Practice

Exercises

7 problems

1 of 7

An electron (m = 9.11×10⁻³¹ kg) is trapped in an infinite square well of width L = 1.0 nm. Observe the n=1 standing wave and find the ground state energy E₁ in eV.

The animation shows the n=1 standing wave (blue) and its probability density |ψ₁|² (green dashes) inside an infinite square well of width L = 1 nm. The yellow dashed line marks the energy E₁. Enter E₁ in eV.

E₁ = eV

2 of 7

Using the energy level diagram for the same well (L = 1.0 nm), find E₃ in eV. Note that Eₙ = n²E₁.

The energy level diagram shows Eₙ = n²E₁. The green level (n=3) is highlighted. Use Eₙ = n²E₁ to find E₃ in eV. (A photon animates the n=3→n=1 emission.)

E₃ = eV

3 of 7

Find the energy of the photon emitted in the n=3 → n=1 transition for the electron in the 1.0 nm well, in eV.