Canonical variables obey $[\hat q,\hat p]=i\hbar$. Formally, this replaces Poisson brackets by $\{f,g\}\to \dfrac{1}{i\hbar}[\hat f,\hat g]$, though the correspondence is subtle and constrained by ordering issues and no-go theorems.

Defining $a=\sqrt{\dfrac{m\omega}{2\hbar}}\,\hat x + \dfrac{i}{\sqrt{2m\hbar\omega}}\,\hat p$ gives $H=\hbar\omega(a^\dagger a+\tfrac12)$. Eigenstates $|n\rangle$ are generated algebraically and are central to quantum field theory.

A particle confined to $0<x<L$ with $\psi(0)=\psi(L)=0$ has energies $E_n=\dfrac{n^2\pi^2\hbar^2}{2mL^2}$. This model illustrates quantization from boundary conditions rather than from algebra alone.

The attractive delta potential $V(x)=-\lambda\delta(x)$ supports a single bound state, while the repulsive delta acts as a singular scatterer. Such models are valuable idealizations in low-dimensional physics and cold-atom theory.