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.