Introduction: The Most Important Problem in Physics
The quantum harmonic oscillator describes any system near a stable equilibrium: molecules vibrating, electromagnetic field modes, phonons in solids. It's exactly solvable and serves as the foundation for quantum field theory.
The Potential
\[V(x) = \frac{1}{2}m\omega^2 x^2 = \frac{1}{2}kx^2\]Parabolic well with natural frequency \(\omega = \sqrt{k/m}\).
The TISE
\[-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + \frac{1}{2}m\omega^2 x^2 \psi = E\psi\]Defining \(\xi = \sqrt{m\omega/\hbar}\,x\) and \(\epsilon = 2E/\hbar\omega\):
\[\frac{d^2\psi}{d\xi^2} + (\epsilon - \xi^2)\psi = 0\]Key Features
- No bound state "edge"—potential rises forever
- All states are bound
- Classically forbidden region exists for all finite \(E\)
- Solution involves Gaussian times polynomial
The Quantum Connection
Near any equilibrium, the potential is approximately parabolic (Taylor expand!). So the harmonic oscillator is the universal model for small vibrations. Its equally spaced energy levels are unique among quantum systems and lead to the concept of quantized excitations (quanta).