For a particle in a box of length $L$, the stationary states are sinusoidal and the energies are $$E_n = \frac{n^2\pi^2\hbar^2}{2mL^2}, \quad n=1,2,3,\dots$$ The quantization arises from boundary conditions.

Each higher-energy state has more nodes and shorter wavelength. The ground state cannot have zero energy because the wavefunction cannot be both confined and completely flat.

In a finite potential well, the wavefunction extends into classically forbidden regions. Bound states still have discrete energies, but the tails outside the well are nonzero.

If a particle encounters a barrier with $E < V_0$, classical physics forbids passage. Quantum mechanically, the wavefunction decays exponentially in the barrier and can emerge on the other side with nonzero transmission probability.

Tunneling underlies scanning tunneling microscopes, alpha decay, Josephson junctions, and many semiconductor devices.