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

In summary, we derived expressions for the normalization constants Ns and Na, as well as the energies Es and Ea, for a system of two protons and one electron separated by a distance R. We also provided the relevant equations and a link to a possible solution for the normalization constants.
  • #1
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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:

look image attachment

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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  • #2
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]
 

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Some examples of condensed matter systems include crystals, superconductors, magnets, semiconductors, and liquid crystals. These systems have unique physical properties that can be studied and manipulated.

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Condensed matter physics differs from other branches of physics in that it focuses on the collective behavior of large numbers of particles, rather than individual particles. It also deals with systems that are in a condensed state, such as solids and liquids, rather than the more extreme states of matter.

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