- #1
BOAS
- 552
- 19
Homework Statement
A Hydrogen atom is in a homogeneous electric field. The field's interaction with the atom is described by the Hamiltonian ##\hat H = e E_0 r \cos \theta##.
Calculate the energy shift due to the linear stark effect in the following state of Hydrogen.
##\Psi = \frac{1}{\sqrt{2}} (\psi_{200} + \psi_{210})##
Hint: Use the fact that ##r \cos \theta = r \sqrt{\frac{4 \pi}{3}} Y_{1,0}## and the orthonormality of the spherical harmonics.
Homework Equations
The Attempt at a Solution
[/B]
From first order perturbation theory:
##\Delta E = \int_{dv} \Psi^* \hat H \Psi##
##\Delta E = \frac{1}{2} \int^{\infty}_0 \int^{2\pi}_0 \int^{\pi}_0 (\psi_{200}^* + \psi_{210}^*) \hat H (\psi_{200} + \psi_{210})##
Substituting the hint into the Hamiltonian, and using the fact that ##\psi_{nlm} = R_{nl} Y_{lm}## to separate the integral.
##I_{angular} = \int^{2\pi}_0 \int^{\pi}_0 (Y_{00}^* + Y_{10}^*)(Y_{00} + Y_{10}) Y_{10} \sin \theta d\theta d\phi##
I am confused about how to apply the argument of orthonormality to the product of three spherical harmonics. How do I proceed?
Thanks for any help you can give!