Harmonic Oscillator

Phase 2: Quantum Mechanics Core

Syllabus Goal

Master the quantum harmonic oscillator.

Importance

Approximates any stable equilibrium potential near the minimum. $$ V(x) = \frac{1}{2}m\omega^2 x^2 $$

Methods

1. Analytic: Power series method (Hermite polynomials).
2. Algebraic (Ladder Operators): $$ \hat{a}\pm = \frac{1}{\sqrt{2\hbar m\omega}} (\mp i\hat{p} + m\omega\hat{x}) $$ $$ H = \hbar\omega(\hat{a}+ \hat{a}_- + 1/2) $$

Spectrum

$$ E_n = \hbar\omega(n + 1/2) $$

• Equidistant energy levels.