Page 244 - Principles and Applications of NanoMEMS Physics

234 Appendix B

ˆ
i − 2 π § ¨ N η − 1 b δ · x ¸ () ()
ψ η () x = F η a −1 a / e L © 2 ¹ e − i η φ x ⎯ L→ ∞ F η a −1 a / e − i η φ x . (B.32)
⎯
⎯→

The derivation of this identity was undertaken by Delft and Schoeller [138]
in two steps. First, the demonstration that ψ () Nx is an eigenstate of the
η 0
bosonic operator b was undertaken, which guarantees that ψ η () Nx 0
η
q
may be expressed as a coherent state, and then the consequences of acting
with ψ on a general state were determined.
η
The relationship between ψ and b is captured by their commutation
η
η
q
relation which, in turn, derives from their respective definitions given in
(B.5) and (B.8). The pertinent commutation relations are,
x =
[b , ψ ()] δ α ψ () x , (B.33)
qη' η ηη' q η
+ *
[b , ψ ()] δ= α ψ () x , (B.34)
x
qη' η ηη' q η
α
where () x = i e iqx . Applying (A.91) on the ground state, we obtain,
q
n q
G G G
b qη ' ,ψ η () Nx − ψ η ()bx qη ' N = δ ηη ' α q ψ η () Nx . (B.35)
0 0 0
G
However, since b qη N = 0, the second term vanishes and we get the
0
result,
G G
b ψ () Nx = α ψ () Nx , (B.36)
η
qη
0 q η 0
G
which shows that ψ () Nx is an eigenvector of b , the boson
η
0 q η
annihilation operator, with eigenvalue α .
q
A well known result of quantum mechanics is that if a state is an
eigenvector of the annihilation operator, then this state is a coherent state
[139]. A coherent state has many useful properties. For instance, its
uncertainty relation is minimized, i.e., ∆x ∆p = = 2. Such a state may be
expressed in the form,
G ª º G
+
ψ () Nx = exp «¦ α ()bx qη » F λ () Nx , (B.37)
ˆ
η
0 q η η 0
q ¬ > 0 ¼