Particle in a well: a quantum zoo

What you are seeing: three canonical 1D quantum bound-state problems. In each one, the particle is trapped by a potential $V(x)$. Solving the time-independent Schrodinger equation $-\tfrac{\hbar^2}{2m}\psi''(x) + V(x)\psi(x) = E\psi(x)$ gives a discrete ladder of energy levels $E_n$ and their wavefunctions $\psi_n$.

The well sets the level ladder:

The potential is drawn in grey; horizontal lines mark each bound level; the wavefunction $\psi_n(x)$ for the selected level is drawn in blue (filled area). For the harmonic oscillator we shift each wavefunction vertically to its eigenenergy to suggest the level structure.