Page 58 - Principles and Applications of NanoMEMS Physics

2. NANOMEMS PHYSICS: Quantum Wave-Particle Phenomena 45

q ˆ q >= nq e q > (4)

In other words, the result of measuring the charge in the TL must be n times
the fundamental electron charge, q e, where n is a positive integer. Since,
from a comparison with the MLC description, charge adopts the role of a
“coordinate” operator in the quantized Hamiltonian, the form of the
corresponding “momentum” operator p ˆ , and in particular,

ˆ p = § = ∂ · 2 = −= 2 ∂ 2 (5)
2
¸
¨
¨
¸
© i q ∂ ¹ q ∂ 2
must reflect this new situation. This is accomplished by replacing the partial
derivative by its finite-difference approximation in charge coordinate space
[65], i.e.,

∂ ψ = ψ (n + ) 1 − 2ψ (n ) +ψ (n − ) 1
2
q ∂ 2 q 2 (6)
e

where q e is the fundamental unit discretizing the charge “axis” and ψ is the
electron wavefunction in the charge representation. Assuming the line is
driven by a voltage source V, Schrödinger’s for the TL is given by Eq.(7)
[61, 62]:

½
­
− = 2 { ψ − 2 ψ + ψ } ® q ˆ 2 + ˆ V ψ = εψ (7)
+
q ¾
2 q 2 L n+1 n n−1 ¯ 2 C ¿ n n
e
or, using Eq. (4):
½
n ψ =
− = 2 { ψ − 2 ψ + ψ }+ ­ q e 2 n 2 + Vq e ¾ εψ . (8)
®
2 q 2 L n+1 n n−1 ¯ 2 C ¿ n n
e
Imposing charge discreteness, thus, turns Schrödinger’s equation for a TL
into a discrete, instead of a partial, differential equation.
The implications of charge discreteness are gauged from the nature of the
corresponding eigenvalues and eigenvectors for this equation. Obtaining
these becomes more transparent upon developing the quantum theory of
mesoscopic TLs [61, 62], which we outline below following Li [61].