Fourier Transform of a wavefunction

[children]

Why shud one take the Fourier transform of a wavefunction and multiply the result with its conjugate to get the probability? Why can't it be Fourier transform of the probability directly?

thank you

 

[Fredrik]

The wave function can't be a probability (or probability density) since it's complex. A probability must obviously be a real number between 0 and 1.

Also, you you only start by taking the Fourier transform if you're interested in the probability density of a certain value of the momentum. If you're interested in the probability density of a certain value of the position, you don't have to do a Fourier transform.

 

[G01]

As Fredrick said, you don't take Fourier transform of a wave function in the process of finding the probability density. The probability density is given by (in one dimension):

P(x)=\int\psi (x)^*\psi (x) dx

which does not involve a Fourier Transform.

Instead, the Fourier transform of a wave function will give the wave function in momentum space (call it \phi). Again, as Fredrick mentioned, we can use this to find the probability density for the momentum of the particle:

P(p)=\int\phi (p)^*\phi (p) dp