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Fourier transform with complex conjugate

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Show that integral from - infinity to + infinity of psi_1(x) times psi_2*(x) dx is equal to integral from - infinity to + infinity of phi_1(k) times phi_2*(k) dk.

* indicates complex conjugate

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Problem: Show that

where [ ]* indicates complex conjugate.


This problem is related to the Fourier transform of a function or distribution.
First of all, we need to define the Fourier transform, since there are several ways to define this transform, although the properties are the same.
According to the statement to be proved, the Fourier transform of a function (x) will be defined as:
( 1)
Its inverse is

Solution Summary

The following posting helps with Fourier transform problems. Complex conjugates are indicated.

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Expected value of momentum and Fourier transform

The expected value of the momentum operator in quantum mechanics can be calculated using the spatial wave function. Using that the momentum wave function is the Fourier transform of the spatial wave function we obtain an expression for this expected value in term of the momentum wave function, that is, we prove the Equation 15.64 in the attached file.

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