Difference between expectation value and probabilty

[MGWorden]

Homework Statement

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

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

Homework 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

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 don't 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!