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3. 1d Born approximation. For 1 dimensional scattering with potential V(x), it is also possible to write the solution in the form of 2m 6* (x) = Aeikz | dz'G(x, 2')V(x')e* (2²'), + %3D where G(x, x') is a Green's function satisfying an2 G(x, x') + k²G(x, x') = 6(x – x').

3. 1d Born approximation. For 1 dimensional scattering with potential V(x), it is also possible to write the solution in the form of 2m 6* (x) = Aeikz | dz'G(x, 2')V(x')e* (2²'), + %3D where G(x, x') is a Green's function satisfying an2 G(x, x') + k²G(x, x') = 6(x – x').

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Setup is from image 1.

Use the Green's function (see image 2) to derive the expression for 1d Born's approximation.

-eik(x-a') 2ik x > x' e-ik(a-a') x < x' G(2, a') = 1 2ik
Transcribed Image Text:-eik(x-a') 2ik x > x' e-ik(a-a') x < x' G(2, a') = 1 2ik
3. 1d Born approximation. For 1 dimensional scattering with potential V (x), it is also possible to write the solution in the form of 6*(x) = Ae" ikr + 2m / da'G(x,2')V(x')st(2'), where G(x, x') is a Green's function satisfying ar2 G(x, x') + k²G(x, x') = 8(x – x').
Transcribed Image Text:3. 1d Born approximation. For 1 dimensional scattering with potential V (x), it is also possible to write the solution in the form of 6*(x) = Ae" ikr + 2m / da'G(x,2')V(x')st(2'), where G(x, x') is a Green's function satisfying ar2 G(x, x') + k²G(x, x') = 8(x – x').
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