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220 Appendix A
In general, creation and annihilation operators are associated with each
specific particle. Thus, it would be imposible to annihilate a particle in the
state β with the annihilation operator for state α, i.e.,
0
a α 1 β = , β ≠ α . (A.27)
+
Since, using α to label a general state, 1 α = a α 0 α , one can express
(A.27) as,
+
a α 1 β = a α a β 0 β = δ αβ 0 β . (A.28)
When the system contains many particles in multiple states, say, three
particles in state γ , and one particle in state δ , following the above, the
state may be represented by,
+ + + +
3 γ 1 δ = a γ a γ a γ a δ 0 , (A.29)
where 0 represents the vacuum state.
The particles involved in second quantization may be identical or distinct.
Due to the specificity/correspondence of the creation and annihilation
operators with the state on which they operate, for any two single-particle
states α and β , describing the system, the states,
+ + + +
1 α 1 β ~ a α a β 0 ; 1 β 1 α ~ a β a α 0 , (A.30)
must be identical, except for a freely chosen phase factor. If the phase factor
is taken as real, then equating the two expressions gives,
+
+
1 α 1 β = 1 β 1 α a α a + β = a + β a . (A.31)
α
From knowldege that two identical boson are described by a symmetric
wavefunction, it is deduced that this expression gives the commutation
relation for bosons. On the other hand, from knowledge that two identical
fermions are described by an antisymmetric wavefunction, it is deduced that
+
+
+
1 α 1 β = − 1 β 1 α a α a + β = −a β a , (A.32)
α
gives the anticommutation relation for fermions. The anticommutation
relations for fermions embody the fact that two fermions cannot occupy the
+ + + +
same state (they obey Pauli’s exclusion principle), i.e., a α a α = −a α a α = 0.
