A harmonic oscillator is a device for which the restoring force on a particle mass is proportional to its displacement from equilibrium; i.e.,

F = −kx
and thus
m(d²x/dt²) = −kx

where m is the mass of the particle and k is a constant, usually called the stiffness coefficient.

The potential energy function is then V(x)=½kx².

The Hamiltonian H for the harmonic oscillator is then

H = ½p²/m + ½kx²

where p is the momentum of the particle.

This means that the Hamiltonian operator for a harmonic oscillator is

H^φ = −(h²/2m)(d²φ/dt²) + ½kx²φ
and thus the
time independent
Schrödinger equation is
−(h²/2m)(d²φ/dt²) + ½kx²φ = Eφ

where φ is the wave function and h is Planck's constant.

The energy E is an eigenvalue of the equation and is equal to (n+½)h.

The wave function is a complex-valued function such that its squared value is the probability density.

The solutions give the probability density functions in terms of the dimensionless variable ζ=x/σ

φ_n²(ζ) = (1/(2ⁿn!√π)H_n²(ζ)exp(−ζ²)

where H_n(ζ) is the Hermite polynomial of order n.

The probability density function in terms of the displacement x is then given by

P_n(x) = φ_n²(x/σ)/σ

where

σ² = hω/k = h/(mω)
where the frequency ω is
ω = (k/m)^½

It can be shown that in the limit as n→∞ the squared values of the Hermite polynomials H_n² approach

2(2n/e)ⁿcos²(x(2n)^½−nπ/2)exp(x²)/(1−x²/2n)^½

where e=2.7218....

This means that the probability density functions for the harmonic oscillators asymptotically approach

(φ(x))² = 2(1/(2ⁿn!√π))(2n/e)ⁿcos²(x(2n)^½−nπ/2)/(1−x²/2n)^½

The average of cos²(z) is 1/2. So the spatial average of (φ(x))² is essentially

(Φ(x))² = (1/(2ⁿn!√π))(2n/e)ⁿ/(1−x²/2n)^½

The classical time-spent probability density function for a hamrmonic oscillator is

P(x) = 1/(π(x_max² − x²)^½
= (x_maxπ)/(1 − ½(x/x_max)²)^½

where x_max is the maximum deviation. The energy of the oscillator is equal to ½kx_max². Therefore x_max=(2E/k)^½. By the appropriate choice of units P(x) can be made proportional to (Φ(x))². For probability distributions constant factors do not matter because they cancel out in the normalization process.

The conclusion is then therefore that the spatial average of the quantum theoretic probability distribution for a harmonic oscillator is asymptotically equal to the classical time-spent probability distribution for such an oscillator. ` `