Difference between expectation value and probabilty

[MGWorden]

1. The problem statement, all variables and given/known data

Psi(x) = Ax -a<x<a

I am trying to find the probability that my measured momentum is between h/a and 2h/a


2. Relevant equations

I have normalized A= sqrt(3/(2a^3))

I know that if I was finding the expected momentum I would use
\int\Psi * p \Psi dx



3. The attempt at a solution


so far I have done
\int \Psi * p \Psi dx
with bounds from from h/a and 2h/a


but I know this is the expected value of momentum and I dont think that is the same thing as the probability of momentum.

Could someone please explain the difference in finding the probability of momentum and the expected value of momentum. Or if they are the same thing let me know there is no difference.

Thanks for your time.

[fzero]

The probability that the momentum lies between p1 and p2 is not related to the expected value of momentum. The expected value of momentum is the most likely momentum value we'd find in the state \Psi(x). The probability to find the particle with a given momentum p is | \psi(p) |^2, where \psi(p) is the normalized wavefunction in momentum space. To find the probability of finding the momentum in a range of values, we have to integrate this.

[MGWorden]

Thanks for your help. I converted over to momentum space using a fourier transform then integrated.

Your explanation saved me much time searching.

Cheers!