Quantum Harmonic Oscillator
Ladder operators, equally-spaced energy levels, and the seed of QFT.
For $H = p^2/2m + \tfrac{1}{2}m\omega^2 x^2$ define $\hat a = \sqrt{m\omega/2\hbar}(\hat x + i\hat p/m\omega)$, $\hat a^\dagger$ its adjoint, satisfying $[\hat a, \hat a^\dagger]=1$. Then
$$\hat H = \hbar\omega\left(\hat a^\dagger \hat a + \tfrac{1}{2}\right), \quad E_n = \hbar\omega(n+\tfrac{1}{2}).$$States $|n\rangle = (\hat a^\dagger)^n |0\rangle / \sqrt{n!}$ are built by acting on the vacuum. The wavefunctions are Hermite polynomials times a Gaussian; classical limits emerge for coherent states $|\alpha\rangle = e^{-|\alpha|^2/2}\sum_n \alpha^n/\sqrt{n!}\,|n\rangle$, the eigenstates of $\hat a$.