How Do You Solve the Quantum Physics Integral Using the Residue Theorem?

[Felicity]

Homework Statement

integral from - infinity to + infinity of
N/(k²+a²) * e^ikx dk

Homework Equations

this is for a quantum physics problem (chapter 2 problem 1, gasiorowicz) where I am given A(k) = N/(k²+a²) and must calculate psi(x)

I am using the equation
psi(x,t) = integral from - infinity to + infinity A(k) e^i(kx-wt) dk

which when t=0 goes to

psi(x,t) = integral from - infinity to + infinity A(k) e^ikx dk

The Attempt at a Solution

I've tried integrating by parts, substitution and on my TI-89 however I am a little rusty with all these methods

Any help would be greatly appreciated

 

[Dick]

You need to do these sort of integrals in the complex k plane using contour integration. Review the residue theorem and some examples of how to use it and then take another look at the problem.

 

[Felicity]

Thank you so much! would the residue then be e^-ax/2ai ?

 

[Dick]

Something like that, yes. If you want more detailed help you should tell us how you got it. What did you get for the integral?

 

[Felicity]

well, as I am not well-versed in complex analysis I looked up residue theorem and found an example on wikipedia which I modified to fit my situation. The work is as follows

∫_-∞^∞dk (1/(k+ai)-1/(k-ai)) e^ikx

Which has a singularity at ai=k

so Res k=ai = (e^ikx)/2ai

so I multiply by 2*pi*I to get (pi*e^-ax)/a

and then put an absolute value on the "ax" which comes from integrating along the bottom of the arc of the line integral

Does this make sense? is there somewhere I can start to understand exactly how line integrals and residue theorem works?

 

[Dick]

well, as I am not well-versed in complex analysis I looked up residue theorem and found an example on wikipedia which I modified to fit my situation. The work is as follows

∫_-∞^∞dk (1/(k+ai)-1/(k-ai)) e^ikx

Which has a singularity at ai=k

so Res k=ai = (e^ikx)/2ai

so I multiply by 2*pi*I to get (pi*e^-ax)/a

and then put an absolute value on the "ax" which comes from integrating along the bottom of the arc of the line integral

Does this make sense? is there somewhere I can start to understand exactly how line integrals and residue theorem works?

You've left out a lot of the details, and in the first line the integrand should be exp(ikx)/((k+ia)(k-ia)) but yes that's it. I don't have any favorite references, but you can probably find a lot more examples on the web or in books on the subject of applied mathematics.