The Schroedinger equation in the frequency domain

An older version of this with some extra integrals is FreqSeqn.htm. The file ONE-DS.htm considers the one- dimensional version of the Schroedinger equation in frequency space.

Begin with

Equation 1

Expand

Equation 2

Thus

Equation 3

So that the Schroedinger equation becomes

Equation 4

Use the fact that to make the last term

_{Equation 5}

_{Then let f = f’’ + f’ to get}

Equation 6

_{So that the Schroedinger equation becomes}

_{Equation 7}

_{The last term is a convolution integral (sum). This is the second term on the right in equation 11 in the paper. Convolution integrals are usually seen in the form}

_{Here this is in the form – see Convol3.doc for details about the Born-Von Karmon sum.}

_{Equation 9}

_{The Born-Von Karman sum works as long as both phi and v originate as sums in frequency space.}

_Define

_{Equation 10}

_{The Gaussian transform is used in the evaluation of.}

 Finally that the Schoedinger equation become

_{Equation 11}

_{So that at each iteration (more detail in Cterm.htm)}

_{Equation 12}

_{Two center potential}

_{Suppose Equation 13}

_Then

_{Equation 14}

_{This is equations 12 and 15 in the paper. Note that the vhat(r) multiplied by phi is the integral done to evaluate the second term on the left in equation 11. The w(q) which is arrived at as the back transform of of W(p) is the v(r) of equation 12 complete with the singularities at r=R1 and r=R2.}