Quantum harmonic oscillator

The quantum harmonic oscillator is the quantum version of a mass attached to a spring. Its potential energy is

$$V(x)=\frac{1}{2}m{\omega }^2{x}^2.$$

The Hamiltonian operator is

$$\hat{H}=\frac{{\hat{p}}^2}{2m}+\frac{1}{2}m{\omega }^2{\hat{x}}^2.$$

Solving the Schrödinger equation gives discrete energy levels:

$$E_n=\hslash \omega \left(n+\frac{1}{2}\right),n=0,1,2,\dots$$

The lowest state still has energy $\frac{1}{2}\hslash \omega$, called zero-point energy.

Why it matters

This model appears everywhere because many stable systems look approximately quadratic near equilibrium. It is the local model for vibrations, field modes, and many perturbative calculations.

Classical bridge

Classically, the oscillator traces closed curves in Phase space and can have any nonnegative energy. Quantum mechanically, states live in Hilbert space and energy comes in equally spaced levels.

Common pitfalls

• The ground state is not a particle sitting motionless at $x=0$.
• Quantised energy levels are not equally spaced for all potentials; equal spacing is special to the harmonic oscillator.
• Zero-point energy does not mean you can extract free mechanical work from the ground state.

Source trail: Susskind The Theoretical Minimum index; reference book: Quantum Mechanics: The Theoretical Minimum by Leonard Susskind and Art Friedman.