3.3 Evaluation of the integrals in helium

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3.3 Evaluation of the integrals in helium
Calculating the direct and exchange integrals to
make quantitative predictions for some of the
energy levels in the helium atom, based on the
theory described in the previous sections. This
provides an example of the use of atomic
wavefunctions to carry out a calculation where
there are no corresponding classical orbits and
gives an indication of the complexities that
arise in systems with more than one electron..
The important point to be learnt from this
section, however, is not the mathematical
techniques but rather to see that the integrals
arise from the Coulomb interaction between
electrons treated by straightforward quantum
mechanics.
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3.3.1 Ground State_1
To calculate the energy of the 1s2 configuration
we need to find the expectation value of
e2/4??0r12 in eqn 3.1this calculation is the
same as the evaluation of the mutual repulsion
between two charge distributions in classical
electrostatics, as in eqn3.15 with ?1s(r1) and
?nl(r2) ?1s(r2). The integral can be considered
in different ways. We could calculate the energy
of the charge distribution of electron 1 in the
potential created by electron 2, or the other way
around. This section does neither it uses a
method that treats each electron symmetrically
(as in Appendix B), but of course each approach
gives the same numerical result. Electron 1
produces an electrostatic potential at radial
distance r2 given by

(3.21)
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3.3.1 Ground State_2
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3.3.1 Ground State_3
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3.3.2 Excited states the direct integral_1
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3.3.2 Excited states the direct integral_2
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3.3.2 Excited states the direct integral_3
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3.3.2 Excited states the direct integral_4
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3.3.2 Excited states the direct integral_5
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3.3.3 Excited states the exchange integral_1
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3.3.3 Excited states the exchange integral_2
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3.3.3 Excited states the exchange integral_3
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3.3.3 Excited states the exchange integral_4
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Exercises
3.1 3.2 (a)(b) 3.4 3.5 (a) 3.6 3.7