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A Fourier Analysis of the Schrödinger Equation for a Harmonic Oscillator

A harmonic oscillator is a system in which a restoring force on a mass is proportional to its deviation x from equilibrium; i.e., −kx. Its
potential energy is thus ½kx² and thus its total energy E is given by

E = ½mv² + ½kx²

where m is the particle mass and v is its velocity. The parameter k is called the stiffness coefficient.

The Hamiltonian function for the system is then

H = ½p²/m + ½kx²

where p is the momentum of the particle.

The time-independent Schrödinger equation for the system is then

−½(h²/m)(d²ψ/dx²) + ½kx²ψ = Eψ

where h is Planck's constant divided by 2π and ψ is called the wave function for the system.

The above equation can be rearranged to

(d²ψ/dx²) = −(m/h²)[2E − kx²]ψ

This is not a particularly hard differential equation to solve.The It is a special case of a
Helmholz equation.
Its notable characteristic is its oscillation between extremes. What is shown below is
the solution over a range including x=0.

The square of the wave function is the probability density function,
as shown below.

The peaks in the probability density function can be interpreted as the allowable states of the system.

The Fourier Transform

The Fourier transform for a function f(x) is

F(ω) = ∫ exp(−iωx)f(x)dx

where i is the square root of −1 and the integration is
from −∞ to +∞.

The Fourier transform has two significant properties for the analysis. The first is that transform of a derivative
is

F_{f'}(ω) = iωF(ω)
and hence
F_{f"}(ω) = −ω²F(ω)

The second is that

dF/dω = −iF_{xf}(ω)
and hence
d²F/dω² = −F_{x²f}(ω)

When the Fourier transform is applied to the time-independent Schrödinger equation
for a harmonic oscillator the result is

½(h²/m)ω²F(ω) −½k(d²F/dω²) = EF(ω)

where F(ω) is now the Fourier transform of ψ, the wave function.

The above equation can be rearranged to

(d²F/dω²) = −[2E/k − (h²/(km))ω²]F(ω)

This is essentially the same structure as the Schrödinger equation from which it was derived and thus has the same oscillatory type of solution.
In this case however the peaks in the probability density function can be interpreted as the spectrum of the system.