The quantum harmonic oscillator has potential $$V(x)=\frac{1}{2}m\omega^2 x^2.$$ Its energies are $$E_n = \left(n+\frac{1}{2}\right)\hbar\omega, \quad n=0,1,2,\dots$$

Even in the ground state, the oscillator has energy $\frac{1}{2}\hbar\omega$. This is a direct consequence of the uncertainty principle.

Define $\hat a$ and $\hat a^\dagger$ so they lower and raise the energy level index by one. They simplify the algebra and reveal the evenly spaced spectrum.

Oscillator eigenfunctions are Gaussian times Hermite polynomials. The ground state is a pure Gaussian, while excited states have more nodes.

The harmonic oscillator appears in molecular vibrations, lattice phonons, cavity modes of light, and the quantization of fields.