Phy5646/Rayleigh Ritz Variational Principle Ex

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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.)