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Hartree-Fock Theory
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Question:
Thanks to all those who answered to "Suggestions on Hartree-Fock."
I am an undergraduate working on a problem in Quantum Hadrondynamics.
I am trying to understand how the Hartree-Fock theory is formulated
in terms of the infinite dimensional Grassmannian. If anyone is able,
please explain this to me in detail. Or, please suggest some titles that
I may want to read. Please understand that this is only my third semester
out of high school. My mathematical background is composed of the usual
calculus sequence. I have also completed a one year equivalent in Fouriere
analysis and complex analysis. And recently I have completed a semester of
study in advance differential equations(systems and stability theory
terms of Lyapunov functions). Please suggest what other mathematical
knowledge I must obtain to understand the Grassmannian formulation of
the Hartree-Fock theory. Thank you in advance.
Replies:
Grassmannians??? That sounds un-necessary. You can find descriptions
of them in quantum field theory books though. But Hartree-Fock is
much simpler than that and can be derived from basic quantum theory.
Although I did not think even simple quantum theory was generally
taught before the junior year in college...
Basically the mathematical knowledge you need is a thorough understanding
of the Schrodinger and Heisenberg formulations of quantum mechanics,
the concept of a quantum operator, Hilbert spaces, -- and the physics
behind the Pauli exclusion principle which is what the Fock part
of Hartree-Fock is needed for... plus the concept of Grassman variables...
I just noticed the beginning of your note again. Are you doing
research work with a professor at your university? Did your professor
refer to infinite dimensional Grassmannians? Also, I am not sure
what quantum hadron-dynamics is - is this another name for the
"chromodynamics" of the strong force with quarks and gluons? I am
no expert in this area, and you may need to get hold of a real one...
However, some other suggestions on areas of mathematics you probably
ought to investigate to understand what you are doing:
1. Group theory, symmetries, particularly the applications to physics
2. The concept and methods of "second quantization" (assuming you already
know something about regular quantization).
3. Perturbation theory
4. The mathematics of arbitrary kinds of spaces: topology, differential
topology, etc.
A book that might be readable for you is Itzhikson and Zuber's book
on quantum field theory (which I am pretty sure does discuss the
Grassman techniques). Good luck!
Arthur Smith
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Update: June 2012
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