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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)