Integral help needed for Quantum Physics problem

In summary, the conversation discusses a quantum physics problem involving calculating psi(x) using the integral of A(k)eikx. The person asking for help is struggling with the integration and is directed towards the residue theorem and complex analysis to find the solution. The conversation ends with a brief explanation of the steps taken to solve the integral.
  • #1
Felicity
27
0

Homework Statement



integral from - infinity to + infinity of
N/(k2+a2) * eikx 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) ei(kx-wt) dk

which when t=0 goes to

psi(x,t) = integral from - infinity to + infinity A(k) eikx 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
 
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  • #2
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.
 
  • #3
Thank you so much! would the residue then be e-ax/2ai ?
 
  • #4
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?
 
  • #5
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?
 
  • #6
Felicity said:
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.
 
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1. What is integral help needed for Quantum Physics problem?

Integral help is needed for a Quantum Physics problem when the solution involves calculating the area under a curve or the sum of infinitely small values. This is a common technique used in Quantum Physics to solve problems involving continuous functions.

2. Why is integral help necessary for Quantum Physics?

Quantum Physics deals with the behavior of particles and energy at a very small scale, where classical mechanics does not apply. Integrals help in solving problems involving continuous functions, which are a fundamental concept in Quantum Physics.

3. How can I improve my understanding of integrals in Quantum Physics?

To improve your understanding of integrals in Quantum Physics, it is important to have a strong foundation in calculus and mathematical concepts. You can also practice solving problems and seek help from a tutor or online resources.

4. What are some common mistakes when using integrals in Quantum Physics?

One common mistake is not understanding the limits of integration, which can lead to incorrect solutions. Another mistake is not properly setting up the integral, such as using the wrong variable or not accounting for all terms in the equation.

5. Are there any tips for solving integrals in Quantum Physics?

One helpful tip is to break down the integral into smaller, manageable parts. You can also try using different techniques, such as substitution or integration by parts, to solve the integral. It is also important to carefully check your work and make sure your solution makes sense in the context of the problem.

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