Simple Problems in One Dimension

Phase 2: Quantum Mechanics Core

Syllabus Goal

Master the Infinite Square Well (Particle in a Box).

Particle in a Box

Potential $V(x) = 0$ for $0 < x < L$, and $\infty$ otherwise. Schrödinger Equation: $$ -\frac{\hbar^2}{2m} \frac{d^2\psi}{dx^2} = E\psi $$

Solution

$$ \psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right) $$ $$ E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2} $$

• Energy is quantized ($n=1,2,3…$).
• Zero-point energy $E_1 > 0$.