The dynamics of a nonrelativistic particle are governed by $$i\hbar\frac{\partial}{\partial t}\psi(\mathbf{r},t)=\hat{H}\psi(\mathbf{r},t).$$ For a particle in a potential $V(\mathbf{r},t)$, the Hamiltonian is often $\hat{H}=\hat{p}^2/2m+V$. This equation is linear, so superpositions evolve consistently.

When the Hamiltonian is time-independent, solutions can often be separated as $\psi(\mathbf{r},t)=\phi(\mathbf{r})e^{-iEt/\hbar}$. The spatial part satisfies an eigenvalue equation $\hat{H}\phi=E\phi$, producing discrete energy levels in bound systems.

The quantum harmonic oscillator is a cornerstone model because it approximates many systems near equilibrium. Its energy levels are $$E_n=\left(n+\frac{1}{2}\right)\hbar\omega, \qquad n=0,1,2,\dots.$$ The nonzero ground-state energy is a purely quantum effect called zero-point energy.

Quantum particles can penetrate classically forbidden regions where $E<V$. The wavefunction decays exponentially in such regions, leading to tunneling through barriers. This phenomenon explains alpha decay, scanning tunneling microscopy, and Josephson junction behavior.