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Homework Help: Orthonormality and the Stark Effect

  1. Nov 15, 2016 #1
    1. The problem statement, all variables and given/known data
    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.

    2. Relevant equations

    3. The attempt at a solution

    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!
  2. jcsd
  3. Nov 15, 2016 #2

    Dr Transport

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    Science Advisor
    Gold Member

    In general the product of three spherical harmonics is called a Gaunt integral. In this case, we can make the simplification that [itex] Y_{00} = \frac{1}{\sqrt{4\pi}} [/itex]. Also you can make the case that since you have [itex] m = 0 [/itex] you can convert to Legendre polynomials and use https://en.wikipedia.org/wiki/Associated_Legendre_polynomials to reduce the integrals to product of two only.
  4. Nov 15, 2016 #3
    Oh that makes perfect sense, I should have seen that.

    Thank you for your help!
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