Caption: A diagram illustrating key features of the 1-dimensional quantum harmonic oscillator.
Features:
Note the ideal harmonic oscillator potential extends to ±∞, and so all particles are bound.
Realistic harmonic oscillator potentials only extend a finite distance, and so particles can escape if they have enough energy.
The allowed energy levels obey the formula
Only the first 8 energy levels are displayed.
The ideal quantum harmonic oscillators has an infinite number of energy levels.
Realistic quantum harmonic oscillators transition to non-harmonic oscillator potentials as one moves away from the equilibrium point (here x=0 point) and eventually the potentials stop growing. Thus, eventually ones reaches unbounded states as energy increases to the point where the particle can escape the potential well and go off to infinity.
No other simple potential gives equal spacing.
The supersymmetric (SUSY) harmonic oscillator potentials also give equally spaced energy levels, but they arn't so simple.
The wave functions do NOT extend in the energy dimension. They have there own axis unit. Their superposition here is to concretely illustrate their behavior and their energy simultaneously.
The diagram doesn't clarify if the wave functions are relatively to-scale or not.
The wave functions are positive above their respective energy level line and negative below.
Nevertheless, the boundary provided by harmonic oscillator potential enforces quantization.
Thus, only an integer number of wavelengths of the wave function can be fit with the harmonic oscillator potential's potential well.
The wavelengths of the standing waves do vary though as the diagram shows.