Fourier Transform of a wavefunction

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

Answers and Replies

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Fredrik
Staff Emeritus
Science Advisor
Gold Member
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
Homework Helper
Gold Member
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:

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