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Fourier Transforms of
the Time Independent
Schrödinger Equation

Consider a particle of mass m moving a potential field V(x) in a one dimensional space.
The Hamiltonian of this system is

H = ½p²/m + V(x)

where p is the particle's momentum.

Its time indepent Schrödinger equation is

−(½h²/m)(d²ψ/dx²) + V(x)ψ = Eψ

where h is Planck's constant divided by 2π, ψ(x) is the system's wave
function and E is the system's energy.

Consider the Fourier transform of the equation

(½h²/m)ω²Ψ(ω) + ∫_{−∞}^{∞}U(ω−ν)Ψ(ν) = EΨ(ω)

where Ψ(ω) and U(ω) are the Fourier transforms of ψ(x) and V(x), respectively. Ψ(ω) and U(ω) can be considered
vectors in an infinite dimensional complex space. Let S be an infinite dimensional diagonal square matrix with ω² as the diagonal element of each row.
Let W infinite dimensional square matrix with U(ω−ν) as the elements on each row.
The above equation can then be represented as the
operator (matrix) equation

kSΨ + WΨ = EΨ
or, equivalently
(kS + W)Ψ = EΨ

where k is the scalar (½h²/m). Ψ is an eigenvector of (kS+W) and E is its eigenvalue.

The term kS can be considered a perturbation of the system represented by WΨ=EΨ.
If h→0 or m→∞ the quantum analysis should asymptotically approach the classical solution.
These limits correspond to k→0.