- #1
audreyh
- 11
- 0
We discussed this problem in class to some extent, and I'd just like to post it here so that I can continue the discussion on the conceptual physics of it as well as the algebra. I believe a lot can be learned from this problem.
"When an atom is placed in a uniform external electric field Eext, the energy levels are shifted -- a phenomenon known as the Stark effect (it is the electrical analog to the Zeeman effect). In this problem we analyze the Stark effect for the n=1 and n=2 states of hydrogen. Let the field point in the z direction, so the potential energy of the electron is
H'S = eEextz = eEextrcosθ
Treat this as a perturbation on the Bohr Hamiltonian ( [-hbar2/2m]del2 - e2/4πεr ) (thats epsilon knot not just epsilon). Spin is irrelevant to this problem, so ignore it, and neglect the fine structure.
a) Show that the ground state energy is not affected by this perturbation, in first order.
b) The first excited state is 4-fold degenerate: ψ200, ψ211, ψ21-1, ψ210. Using degenerate perturbation theory, determine the first-order corrections to the energy. Into how many levels does E2 split?
c) What are the "good" wave functions for part (b)? Find the expectation value of the electric dipole moment (pe = -er) in each of these "good" states.
Notice that the results are independent of the applied field -- evidently hydrogen in its first excited state can carry a permanent electric dipole moment.
Hint: There are a lot of integrals in this problem, but almost all of them are zero. So study each one carefully, before you do any calculations: If the φ integral vanishes, there's not much point in doing the r and θ integrals!
Partial answer: W13 = W31= -3eaEext; all other elements are zero."
A personal note on griffiths' notation. a is, of course, the bohr radius.
Wij is < ψi0 | H' | ψj0 >
I'm a bit exhausted from writing the problem out, so hopefully this will start some discussion, and then I will come back and post what I have so far.
my own hint :)
realize that H' ~ rcosθ and cosθ = P1 the legendre polynomial for l = 1. so many of the integrals will be zero since the Ylm(θ,φ)'s are orthogonal functions (the spherical harmonics defined by Griffiths, I don't know if that is how it's universally defined)
Thanks!
"When an atom is placed in a uniform external electric field Eext, the energy levels are shifted -- a phenomenon known as the Stark effect (it is the electrical analog to the Zeeman effect). In this problem we analyze the Stark effect for the n=1 and n=2 states of hydrogen. Let the field point in the z direction, so the potential energy of the electron is
H'S = eEextz = eEextrcosθ
Treat this as a perturbation on the Bohr Hamiltonian ( [-hbar2/2m]del2 - e2/4πεr ) (thats epsilon knot not just epsilon). Spin is irrelevant to this problem, so ignore it, and neglect the fine structure.
a) Show that the ground state energy is not affected by this perturbation, in first order.
b) The first excited state is 4-fold degenerate: ψ200, ψ211, ψ21-1, ψ210. Using degenerate perturbation theory, determine the first-order corrections to the energy. Into how many levels does E2 split?
c) What are the "good" wave functions for part (b)? Find the expectation value of the electric dipole moment (pe = -er) in each of these "good" states.
Notice that the results are independent of the applied field -- evidently hydrogen in its first excited state can carry a permanent electric dipole moment.
Hint: There are a lot of integrals in this problem, but almost all of them are zero. So study each one carefully, before you do any calculations: If the φ integral vanishes, there's not much point in doing the r and θ integrals!
Partial answer: W13 = W31= -3eaEext; all other elements are zero."
A personal note on griffiths' notation. a is, of course, the bohr radius.
Wij is < ψi0 | H' | ψj0 >
I'm a bit exhausted from writing the problem out, so hopefully this will start some discussion, and then I will come back and post what I have so far.
my own hint :)
realize that H' ~ rcosθ and cosθ = P1 the legendre polynomial for l = 1. so many of the integrals will be zero since the Ylm(θ,φ)'s are orthogonal functions (the spherical harmonics defined by Griffiths, I don't know if that is how it's universally defined)
Thanks!