Symmetric and Asymmetric Electron Wave Functions in a Proton-Electron System

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1. Homework Statement [/b]

1. Consider two protons, separated by a distance R, and one electron. Assume
that the electron wave function can be symmetric, s, or asymmetric, a, with:
ψs = Ns( ψ1 + ψ2);
ψa = Na( ψ1 - ψ2):
Here 1 and 2 are the electron wave functions for large R when bound with
proton 1 or proton 2, respectively
a) Derive a expression for the normalisation constants Ns and Na. [6 Marks]
b) Assuming the Hamiltonian is given by:

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where r1 and r2 are the distances to proton 1 and proton 2, respectively.
Derive an expression for the energies Es and Ea.

Homework Equations

for a) i found a solution here : http://farside.ph.utexas.edu/teaching/qmech/lectures/node129.html#e14.57

is this correct ?

im still confused with energues Ea and Eb

The Attempt at a Solution

 

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a) Ns = (1/√2)[(∫|ψ1|^2dτ + ∫|ψ2|^2dτ)^(-1/2)] Na = (1/√2)[(∫|ψ1|^2dτ - ∫|ψ2|^2dτ)^(-1/2)] b) Es = <ψs|H|ψs> = =N^2s[∫|ψ1|^2H1dτ + ∫|ψ2|^2H2dτ + 2∫ψ1*H12*ψ2dτ]Ea = <ψa|H|ψa> = =N^2a[∫|ψ1|^2H1dτ + ∫|ψ2|^2H2dτ - 2∫ψ1*H12*ψ2dτ]