3-Dimension Expectation Values (QM)

[moo5003]

Hello, I have a problem that wants me to find the expectation value of <r> <r^2> for the ground state of hydrogen (part a.). My friend and I already completed the exercise but I'm concerned about how we found the expectation value. Since the ground state of hydrogen is only dependent on r do we only integrate over r? I notice that if we integrate over psi and phi we will add an extra 2pi^2 multiplied with what we had previously. Any help would be appreciated:

Recap- Do you integrate over all three dimensions if the wave function is only dependent on one?

 

[OlderDan]

Hello, I have a problem that wants me to find the expectation value of <r> <r^2> for the ground state of hydrogen (part a.). My friend and I already completed the exercise but I'm concerned about how we found the expectation value. Since the ground state of hydrogen is only dependent on r do we only integrate over r? I notice that if we integrate over psi and phi we will add an extra 2pi^2 multiplied with what we had previously. Any help would be appreciated:

Recap- Do you integrate over all three dimensions if the wave function is only dependent on one?

You need to make sure the wave functions are normalized and that you are accounting for the variation of spatial volume in the vecinity of any given r. If you are using the full wave function with its normalization constant then you need the angular integrals to get the normalization correct. There are also factors of r in the volume element dV that are important. People often look at the function u(r) = rR(r) as a better representation of the radial wave function because the amount of 3-D space associated with any given dr is proportional to r². This all comes together naturally if you use the full normalized wave function integrated over all space.