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Old Feb 6th 2015, 02:41 PM   #1
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Probability density for momentum

I am having a little difficulty with problem 9 on the attached problem set.

Consider a particle moving in 1D, in an initial wave function of the form

psi(x,0)=psi(x)=A exp(-a(x-x0)^2) exp(ik0 x)

a) Determine the value of the normalization constant A so that the state is appropriately normalized

b) Determine the probability density rho(k) that a measurement on this state of the particle's momentum will yield the value p_k = hbar k. What is the probable value that would be obtained in such a measurement?

I hopefully calculated the value of A correctly with a gaussian integral. Please verify that A is correct.

I am stuck on part b. Perhaps I did not calculate dell correctly? How would I determine the most probable value?

Thanks for any hints on how to proceed or comments on where I might have gone wrong. I have not typed this problem formally but the equations are formatted.

My professor verified that I have the correct answer for part a, however my original approach to part b was incorrect.

I am attaching my latest effort but I really need help.

I need to multiply through by exp(-ikx) and take the Fourier transform, then integrate over all space.

However in the class notes, the original equation for psi was always in integral form over k-space.

My professor's comment was that the Fourier transform of |psi|^2 is not the same as |Fourier transform psi|^2

Do I need to differentiate the original equation first, before multiplying by exp(-ikx)?

Thank you.
Attached Files
File Type: pdf homework3.pdf (57.6 KB, 3 views)
File Type: pdf HW3_9b.pdf (301.2 KB, 2 views)

Last edited by roger; Feb 8th 2015 at 07:04 AM. Reason: New approach to solving problem
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Old Feb 11th 2015, 01:17 PM   #2
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Answers

Although I did not receive any replies, I am posting my solutions and my professors answers in case anyone is searching in the future.
Attached Files
File Type: pdf homework3.pdf (57.6 KB, 2 views)
File Type: pdf HW3.pdf (244.8 KB, 3 views)
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