Quantum · undergrad

Quantum Harmonic Oscillator

The exactly solvable quantum system underlying vibrations and field modes.

Published Apr 26, 2026 · 1 min

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Hamiltonian

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

The allowed energies are

E_n=\hbar\omega\left(n+\frac{1}{2}\right), \qquad n=0,1,2,\dots.

The ground state has nonzero zero-point energy.

The noncommutativity measure that organizes quantum observables.

The master equation governing the time evolution of the wavefunction.

Linearized motion near stable equilibrium and the normal-mode structure.