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.

Figure 1. Eigenfunctions and energies for three 1D bound-state problems. Method: closed-form infinite-well and harmonic-oscillator solutions; bisection on the finite-well transcendental equation.

well

level n1

depth V015

a (finite)1.00

WHAT TO TRY

Switch the well between infinite, finite and harmonic: the whole stack of eigenfunctions rebuilds. The infinite well climbs as n squared, the harmonic oscillator is an evenly spaced ladder, and the finite well holds only a handful of bound states.
Drag the level n: the bright filled state sweeps up the ladder and gains one node at each step, and the highlighted stem in the bottom spectrum tracks which level you are on.
In the finite well, lower the depth V0 or narrow the width a: bound states pop out through the top one by one as the well can no longer hold them, and the bound-state count in the readout drops.