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Momentum representation, momentum space wave function

We study the relationship between the position space wave function and the momentum space wave function in quantum mechanics. We show that they are related by the Fourier transform, more specifically we show that a momentum wave function given by the Fourier transform of the space wave function satisfies the requirements to be a momentum wave function, that is, normalization (the integral of its magnitude square equals one) and gives the correct formula for the expected value of the momentum.

See the attached file.

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The momentum in quantum mechanics is defined as the operator


(this is in one variable, and p=-i*hbar*Gradient for more variables). The expected value of p, denoted by <p> is then

<p>=int_{-infinity}^{infinity} psi^*(x) (hbar/i) (d psi/dx)(x)dx,

and this is in space representation.

In the momentum representation, we look for g(p) that satisfies (15.60) and (15.61). As it is claimed there in the attachment, such g can be taken to be the Fourier transform of psi (with the hbar normalization in the definition of Fourier ...

Solution Summary

The solution discusses the momentum representation and the relationship between space wave function and momentum wave function.