Page 235 - Principles and Applications of NanoMEMS Physics

A. QUANTUM MECHANICS PRIMER 225

where the latter simply expresses the completeness relation of the
wavefunctions.
For example, in terms of the field operators, the one-particle Hamiltonian
operator,

ˆ
H = ¦ α K β c α c , (A.54)
+
β
0
α β ,
where,
α K β = φ α ³ * () () ()dxxxKx φ β , (A.55)

is expressed as,
ˆ
H = ³ ψ + () () ()dxxxKx ψ . (A.56)
0
This is proven by substituting (A.53) and (A.54) into (A.59) to recover
(A.57):

+
+
*
³¦ φ α () Kcx α () x ¦ φ β () dxcx β = ¦ c α c β φ α ³ * () () ()dxxxKx φ β
α β α β ,
. (A.57)
+
= ¦ α K β c α c β
α β ,
An interpretation of the field operators is obtained by operating with them
on the vacuum state. For inatsnce, operating with ψ + () x on 0 , we obtain,

ψ + () x 0 = ¦ φ + ()cx α 0
+
α
α
, (A.58)
= ¦ φ α + () ( ) δφ xx α α = (x − x α )
α

since the operation of the creation operator of the state α on the vacuum
creates a particle there. This results indicates that ψ + () x behaves as the
ψ
creator or a particle at position x . Similarly, one obtains that () x destroys
a particle at position x .
In the context of this book, the second quantization formalism is key to
the presentation on the Luttinger liquid. This deals with the description of
electrons constrained to move in one dimension and described by the
Hamiltonian,
H = H + H , (A.59)
ˆ
ˆ
ˆ
0 Int