Undergrad Matrix element in problem with hydrogen atom

[Salmone]

I have a problem in calculate a matrix element in a problem with hydrogen atom.

I have an hydrogen atom and Hamiltonian eigenstates $|n,l,m>$ where $n$ are energy quantum numbers, $l$ are $L^2$ quantum numbers and $m$ are $L_z$ quantum numbers, I have to calculate the matrix element $<210|rsin(\theta)cos(\phi)|100>$ with $\theta \in [0,\pi]$, $\phi \in [0,2\pi]$, $r \in [0,+\infty]$ and the result I get is $\frac{4\pi}{27 \sqrt{2}}$, is it right?

 

[PeroK]

The hydrogen wavefunctions are here:

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydwf.html

$\psi_{210}$ is independent of $\phi$. Doesn't that make the integral $\int_0^{2\pi} \cos \phi \ d\phi$ zero?

 

[Salmone]

Yes you are right, I must have gotten the angles mixed up and didn't realize it.