If a particle has a perfectly defined position wavefunction, what is its p wavefunc?by MadMike1986 Tags: dirac delta function, fourier transform, normalization, quantum 

#1
Aug711, 09:04 PM

P: 23

1. The problem statement, all variables and given/known data
The position and momentum wavefunctions are fourier transform pairs. If a particle has a perfectly defined position wavefunction Psi(x) = delta(x  x0), then what is its momentum wavefunction? Is this function normalizable? 2. Relevant equations Fourier transform relation (Can't figure out how to use the latex inputs to write this out) 3. The attempt at a solution I think I have to take the fourier transform of a dirac delta "function"... I believe this is just a single frequency sine or cosine wave, but I only think this based on intuition from what the fourier transform means. As far as if this function is normalizable, I'm not really sure what is meant by that. Does that mean you take the integral over all space (or in this case all momentums) of the wavefuntion squared and set it equal to 1? Any help/advice on this would be much appreciated. Its been a while since I've studied quantum. Thank you. 



#2
Aug711, 11:37 PM

P: 113

You're basically right. The fourier transform of the dirac delta will give a single momentum wave (an exponential with an imaginary argument), and the question as to whether it's normalizable is equivalent to asking whether it is squareintegrable.




#3
Aug811, 08:45 PM

P: 23

The integral of (sin(x))^2 dx from negative infinity to infinity does not converge, therefore it is not square integrable and not normalizable. My intuition is telling me that this means if you know the position exactly, then you cant know what the momentum is at all. Is this correct? 



#4
Aug811, 09:14 PM

Sci Advisor
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Thanks
P: 25,170

If a particle has a perfectly defined position wavefunction, what is its p wavefunc? 



#5
Aug911, 01:09 AM

P: 113

Dick is of course correct, and sin^2(p) isn't even the relevant function here. The fourier integral is trivial and you should do it out.



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