Page 61 - Principles and Applications of NanoMEMS Physics

48 Chapter 2

ˆ
from where a momentum operator P , given by:
~
ˆ
P = = (∇ + ∇ )= = ( − QQ ~ + ), (22)
i 2 q e q e 2 iq e

may be defined. This new momentum operator is related to p ˆ in that
p ˆ = lim P .
ˆ
e q → 0

2.2.1.1 Inductive Transmission Line Behavior

Inductive behavior is displayed by the so-called pure L-design, in which
the TL is considered to have very narrow width (high impedance). Its
mathematical description is given by:

= 2
H = − { ∇ − ∇ }, (23)
ˆ
0 2 e q e q
2 q e L
where the terms involving the line capacitance is neglected and the driving
voltage is set to zero. With this definition, and taking into account the
~ p =
/ ˆ
relationship Q p >=e iq e p > , the following relationships are obtained:

ˆ
P p >= = sin § q e p · > , (24)
¸ p
¨
q © = ¹
e
and

H ˆ p >= = 2 § ¨ − cos1 § q e p · · ¸ ¸ p > , (25)
¨
0 2 ¨ ¸
q L © © = ¹ ¹
e
These are the desired momentum eigenstates and the energy spectrum. What
is clear from (24) is that the current in a mesoscopic inductive line, given by
π
I = P ˆ L , is periodic, becomes zero whenever p = 2 = q ; q ≠ 0 , and
e e
that it is bounded by ( =− q e L = q e ) L . Similarly, from (25) it is determined
,
that the lowest energy state is degenerate at p = n= q .
e