Measurement Fourier Convolution Theorem

Consider a new function with the following Fourier transform,

 

For concreteness, imagine that this wavefunction represents the momentum of photons in a detector used to measure electrons in the state described by . After interacting with the photons in the detector, the electrons of initial momenta k will have new momenta given by K=k+k' where is k' is the momentum of the photon with which each electron interacted. After adding up all possible interactions, the new wavefunction of the electrons after measurement will be

This operation is known as the convolution and is written symbolically as . Confirm, within this example, the Fourier Convolution Theorem:

where A is a constant fixed for all functions f,g. Use your result to determine what the value of this constant must be. We have already descrbed the effects of the measurement process on the momentum distribution. Now, considering especially the limit W ;SPMlt;;SPMlt; D, explain in words the effect of the measurement process on the probability distribution of finding the electrons at different points in space.

Hint: This part will involve more Gaussian integration.