Degenerate Electron Gas

1
Degenerate Electron Gas
Unrealistic model
Uniform positive charge density n
Very large volume VL3
System is neutral
L
Eventually take limit
Positive charge is a uniform background not
quantum mechanical.
Particle in a box
Wave functions of electrons
2
Total Hamiltonian
Electron-background piece
Electron piece
Background piece
to handle infinities.
We will need a convergence factor
At end, need to take limit
3
Background Pieces
Background piece
Electron-background piece
Translational invariance
Note last term is infinite when we go to desired
limits.
4
Electron Piece
All of the physical effects are in He. Evaluate
in second quantization form.
Kinetic Piece
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Kinetic Energy Operator
Number operator
6
Now Start on P.E.
Helicity conservation Coulomb interaction does
not flip spins
Momentum conservation
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Potential Operator
Let
Conservation of momentum automatically
8
No Scattering Term
Anticommutation relations
Number Operators
Still an infinity as µ?0, but its just a number
and and we can redefine zero of energy.
Cancels remaining N2/V piece
9
New Hamiltonian
Define dimensionless parameters
Perturbation in high density limit (r0 ltlt a0).
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Notes on High Density Perturbation Expansion
In high density limit
See problems 5 6
Term diverges as rs?0. Have to handle carefully
at second order.
We will evaluate a (kinetic energy term) and b
(potential energy term) and leave the rest to
more powerful machinery that we develop later (I
hope).
In high density limit
Kinetic energy of ground state plane waves.
Exchange perturbation
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Ground State Fermi Level
In ground state, two electron (one spin up, on
spin down) fill each momentum state starting at
the lowest and going up until all of the
electrons are used up. The energy where they stop
is call the Fermi level.
Fermi Level
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Sum over states below Fermi level
Note we are still integrating in rectangular
coordinates.
Fermi Level Number Operator
Spherical integral
Sum over ?
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Ground state energy First Term
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Ground state energy Second Term
Makes vacancy in two states
Must fill the same two vacancies
Thus, either
This is the non-scattering case q0 Already
subtracted.
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Working on the Matrix Element
Intersection of two spheres
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Important Integral
Take zero of coordinate system at k and z along
q do k integral first in cylindrical coordinates
for half of volume (then double).
Volume of Integration
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Finish the Integration
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Expansion First Two Terms
Kinetic Term
Minimum at 4.82 a0
Exchange Term