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A. QUANTUM MECHANICS PRIMER                                   219


             phonons and photons, respectively, permeate many branches of physics, in
             particular, condensed  matter  physics. In the most  general case, when
             described in terms  of  these  discrete particles, the system  is  said to  be
             represented in the second quantization or number representation formalism.
             The term  second-quantization  derives from  the fact that in this theory the
             stuff the  systems  are made of, via this representation in terms of  discrete
             particles, become quantized, i.e., an aggregate of discrete particles. You will
             recall that in the  first quantization, it was the motion of  the  particles  that
             became quantized. A second-quantized system may exhibit particle creation
             and annihilation, and multi-body interactions, and the formalism of second
             quantization  (or  number  representation) has  been devised to deal with the
             complex dynamics of these systems, in particular, for keeping track of the
             large number, and the statistics, of the particles that may be involved. The
             formalism, thus, prescribes ways to  succintly  represent  pertinent
             wavefunctions and operators. The  mathematical space in which second-
             quantized operators and vectors reside is called Fock space.
               The simplest case occurs when the system has only one particle in, say,
             the state  α,  where this  state is completely specified by giving pertinent
             quantum numbers, e.g., particle momentum, spin and spin projection. In this
             case, the one-particle state is represented by the ket  1 α  , and is taken as
                                                          +
             produced by the operation of the creation operator  a   on the vacuum state
                                                          α
             0 α  , the state  of the system when  there are  no  particles present.
             Mathematically, this is expressed by,
                       +
                1 α  = a α  0 α  .                                                                                   (A.24)

               If the system can contain many noninteracting particles, where the state of
             each particle, say,  α, β , γ , δ , etc., respectively, is described by its
             respective set  of quantum numbers, then the state representation in the
             second quantization formalism would be given by,
                    +
             1 α  = a α  0 α  0 ,  β  0 ,  γ  0 ,  δ ,... =  1 α  0 ,  β  0 ,  γ  0 ,  δ ,... .                                 (A.25)
             If the system is in the vacuum state, i.e., there are no particles present in state
             α, then its state is represented by the ket  0  , which is taken as produced
                                                   α
             by the operation of the annihilation operator  a  on occupied single-particle
                                                     α
             state 1 α  . Mathematically, this is expressed as,


                0 α  =  a α  1 α  .                                                                                   (A.26)
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