Subject: Summary: RPA and second quantization

From: sergiusz kwasniewski <[email-masked]>
Date: Thu, 26 Aug 1999 15:59:56 +0200 (MET DST)

Hi CCL'ers

I want to thank the many responses I received to my question:

I'm currently studying polarization propagator techniques like RPA and MBGF.
Does anyone know to what the following names are related to:
Random Phase and Second Quantization. I know more or less what it is about,
but I don't seem to find the connection between the theory and the name it
is given to.

-- 1 --

Dear Sergiusz:
   Try D. Pines and P. Nozieres beautiful book, The Theory of Quantum Liquids
(W. A. Benjamin, Menlo Park, CA, 1966).       Also Bohm's book on quantum
mechanics
has a nice treatment of second quantization (don't recall the exact title, but I
can get it for you
if you can't find it).
  - Susan

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 _/   Susan R. Atlas                                                     _/
 _/   Department of Physics and Astronomy                                _/
 _/   The University of New Mexico                                       _/
 _/   800 Yale NE                    email:  [email-masked] _/
 _/   Albuquerque, NM 87131          Phone:  (505) 277-1509              _/
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 _/   http://www.phys.unm.edu/CompMaterials                              _/
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-- 2 --

These concepts have originated in physics. Second quantization deals with
the algebra of creation and annihilation operators and connects quantum
field theory with many-body theories. A simple introduction to second 
quantization is given in Szabo and Ostlund: Modern Quantum Chemistry
(Dover 1996). More can be found out from books on quantum mechanics
(for instance, Baym's Lectures on quantum mechanics).

The random phase approximation (RPA) is, what I can see, not treated in
Szabo's and Ostlund's book, although it gives a good introduction to
Green's functions. RPA is an approximation to the two-body Green's function,
and was first used by in a paper by Bohm and Pines (Phys. Rev. 92, 609 (1953))
to describe plasma oscillations by assuming random phases. RPA is decribed
in books on many-body theories, for instance G.E. Brown: Many-body problems,
North-Holland, 1972 (probably there are some books on quantum chemistry that
have it, too).

          Greetings,
          
           TS
           
Tom Sundius
University of Helsinki, Department of Physics    phone +358-9-191 8339
P.O.Box 9, FIN-00014 Helsinki, Finland           fax   +358-9-191 8680

-- 3 --

My favorite source is Blazot and Ripka's "Quantum theory of finite systems".
This has a pretty comprehensive overview of RPA methods, second
quantization, etc. and (more importantly) what physical process is being
represented.

I'm not exactly sure of the origins of the "name" RPA.  However, I'm pretty
sure it's connected to the fact that in RPA, one assumes that the
excitations are collective and bosonic in form. In other words, the basic
RPA assumption is that the excitation operators, A and A^+, are the
operators  a boson (i.e harmionic oscillator) algebra, [A,A^+] \approx 1 and
hence generate coherent state excitations from the RPA vacuum defined as
A|0> = 0.
The A's themselves are related to elementary particle/hole excitations and
de-excitations via a cannonical Bogliobov transformation.
Again, most of this is in Blazot and Ripka.
If you're interested in a 2nd quantization derivation of all this, check out
one of my recent papers:
Car-Parrinello molecular dynamics on excited state surfaces. E. R. Bittner
and D. S. Kosov, J. Chem. Phys. 110, 6645 (1999)
To save your self a trip to the library, here's a link to the PDF version of
the paper:
http://k2.chem.uh.edu/bittner/Papers/CPMD.pdf

-Cheers!
ERB

--
Eric R Bittner, Asst. Professor                            ph: 713-743-2775
Dept. of Chemistry
fax:713-743-2709
University of Houston, Houston TX  77204       email: [email-masked]
"If we knew what it was we were doing, it would not be called
research, would it?" A. Einstein

-- 4 --

   I recommend that you have  a read of the following book,
  "second quantization-based methods in quantum chemistry"
  by Poul Jorgensen and Jack Simons (Academic Press , New York,1981)

Its pretty extensive and describes the RPA ( also known as Time-dependant 
Hartree-Fock TDHF)  in terms of second quantization.

As a very simple explanation I would say that first-quantization is the 
consideration of particles (electrons etc) as wavefunctions , and theory 
based on waves is thus in a first-quantized method. Second quantization  
essentailly converts wave ideas back to a particle description, thus one 
constructs wavefunctions from operators that either create or annihilate a 
particle from a given state.

noj

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			Sergiusz Kwasniewski
			LUC SBG/TS
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