Simple Harmonic Oscillator Eigenfunctions

[Screen shot of the SHO ground state.]

A simple harmonic oscillator (SHO) with a potential energy V(x) = ½mω²x²   has energy eigenfunctions ψ_n(x) that are expressed in terms of a Gaussian times a Hermite polynomial H_n(x)
 

The angular frequency ω=(K/m)^½ is that of a classical mass on a spring with spring constant K.  Substituting these SHO eigenfunctions ψ_n(x) into the time-independent Schrödinger equation shows that they have energy eigenvalues E_n that scale as the principal quantum number n
 

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Note that unlike the infinite square well model, the simple harmonic oscillator energy eigenvalues are evenly spaced.

In order to compare ISW and SHO eigenfunctions, the spring constant is chosen so that the ground state energy eigenvalues of these two systems are equal.