Submitted by Team 1 (Joon-Il Kim, Jorge Barreda, Muhandis)
Questions:
Show that if an unperturbed Hamiltonian
where
is chosen, the first-order perturbation of the ground state of helium vanishes.
Solution:
If we choose
as above, from the eq. #4.5.3.1 the perturbed term can be written by
where
Then, the first-order perturbation energy of the ground state of helium is
The wave function is
where
The first and second terms in the integrand will give
The third term will give
Therefore, the first-order perturbation is
Therefore, if we put
and
into the above equation,
Thus, the first-order perturbation energy of the ground state of helium vanishes. Q.E.D
(Note: This problem is excerpted from Quantum Mechanics, 3nd edition, p479, which is written by Eugen Merzbacher.)