Page 60 - Principles and Applications of NanoMEMS Physics

2. NANOMEMS PHYSICS: Quantum Wave-Particle Phenomena 47

< n m >= δ , (16)
nm

φ
φ
< ψ >= ¦ n ∈Z < n >< nψ >= ¦ n ∈Z φ * ( n)ψ ( n) , (17)
where n belongs to the set of non-negative integers Z.
These relationships permit obtaining the fundamental quantum
mechanical properties of the TL, namely, the eigenfunctions of the
“momentum” operator p ˆ , i.e., the nature of the current, and the energy
spectrum.
Assuming the usual relations [53], p ˆ p >= p p > and
f p) ˆ ( p >= f ( p) p > , Li [62] expands the momentum states in terms of the

number states, p >= ¦ n ∈Z c ( p) n > together with the shifting operation
n
~ p =
/ ˆ
Q p >=e iq e p > , to obtain the relationship
c c = exp (iq p = + iα ). This, in turn, yields the momentum
n + 1 n e n + 1
expansion in terms of the number states as,

p >= ¦ n ∈Z κ n e inq e p =/ n > (18)

n n
i ¦ α j −i ¦ α − j
where κ = e j =1 and κ = e = j 1 for n>0. Making the substitution
n −n
p → p = ( π q e ) in the exponential of (18) yields the same state p > ,
+
2
from where it is determined that the momentum operator p ˆ is periodic.
Further progress towards obtaining the eigenstates and dispersion is attained
by noticing that, if one defines new discrete derivative operators by:
−
ψ (n +1 ) ψ ( ) n
∇ ψ () n = , (19)
q e
q e
ψ () n −ψ (n 1− )
∇ ψ () n = , (20)
q e
q e

then Schrödinger’s equation (8), may be expressed as:
− = 2 ­ { ∇ − ∇ }+ ­ q ˆ 2 + q ˆ ½ ½ ¾ ¾ ψ = εψ , (21)
V
®
®
2 q 2 L e q e q ¯ 2 C ¿ ¿
e ¯