The infinite square well is the simplest bound quantum system with spatial structure. A particle is confined to $0<x<L$ by $$V(x)=0\quad(0<x<L),\qquad V=\infty\quad\text{outside}.$$ The wavefunction must vanish at the walls, so stationary states satisfy $$-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2}=E\psi,\qquad \psi(0)=\psi(L)=0.$$

Inside the well the solutions are sinusoidal. The boundary conditions allow only $$\psi_n(x)=\sqrt{\frac{2}{L}}\sin\left(\frac{n\pi x}{L}\right),\qquad n=1,2,3,\ldots$$ with energies $$E_n=\frac{n^2\pi^2\hbar^2}{2mL^2}.$$

Confinement therefore quantizes energy. There is no state with $n=0$ and $E=0$: the lowest energy is $$E_1=\frac{\pi^2\hbar^2}{2mL^2}.$$ A perfectly localized zero-momentum particle would violate the boundary conditions and the uncertainty relation.

The probability density $|\psi_n|^2$ contains $n-1$ internal nodes. Higher-energy states oscillate more rapidly and approach classical-looking spatial averages when $n$ is large, illustrating the correspondence between quantum and classical behavior.

The square well is idealized, but the structure it teaches is general: boundary conditions select allowed eigenfunctions, confinement creates discrete spectra, and the spatial shape of an eigenstate determines measurement probabilities.