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Old Jan 25th 2015, 01:06 PM   #1
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Hilbert space norm of vector invariant in different coordinate systems

Thank you for the help so far. I feel I am learning a great deal.

The fourth problem of the attached problem set states let Psi be a complex-valued wave function for a particle in 1-D and gives its Fourier transform. The question asks to prove that the integral from negative infinity to positive infinity of the norm squared Fourier transpose is equal to the integral from negative infinity to positive infinity of the norm squared of the wave function.

Please see the attachments.

I am not sure that I have the correct approach to proving this. The hint is to use the plane wave expansion of the delta function at some point.

Could someone give me a hint as to how I should be approaching this or at what point I may have gone wrong?

Thank you.

Roger
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File Type: pdf homework1.pdf (69.2 KB, 1 views)
File Type: pdf HW1_4.pdf (196.0 KB, 2 views)
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Old Jan 25th 2015, 04:33 PM   #2
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I usually prefer to do this kind of problem by just pluggin' n' chuggin'.

Refer to the attachment below.

In lines 1 and 2 I have simply listed the FT of the wavefunction and its complex conjugate. On line 3 I have plugged these expressions into the k integral. It looks like a real mess this way, but after doing a bit of re-arranging (I'm not going to bother with the Mathematical justification to do it) and you get line 4. Note the delta function integral at the end. I leave the rest to you.

I don't know what the rest of your course is going to require, but this sort of derivation crops up in a number of useful situations.

-Dan
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Hilbert space norm of vector invariant in different coordinate systems-codecogseqn-2-.gif  
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Old Jan 26th 2015, 07:09 PM   #3
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Thank you

Thank you once again. I was able to follow your explanation and work forward to complete this exercise. Your help is invaluable to my understanding the concepts and learning the material.
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