Page 59 - Principles and Applications of NanoMEMS Physics

46 Chapter 2

With q ˆ as the charge operator, instead of the conventional spatial
coordinate, the corresponding conjugate variable is taken as p ˆ , which then
represents the current operator, instead of the usual momentum operator. The
quantum mechanics of the TL then evolves from ( ) and the commutation
8
relation:
=
=
[q ˆ , ˆ p ] i . (9)
The fact that the eigenstates of q ˆ must be specified by an integer, n, allows
two consecutive states to be related to one another by the application of a
~
shift operator, in particular, Q ≡ e iq e / ˆ p = . By expanding the exponential, and
using ( ) and ( ), this shift operator may be shown to obey the
9
4
commutation relations:
~
~
[ ] −= q Q (10)
q, ˆ
Q
e
~ ~
+
[ Qq, ˆ + ]= q Q (11)
e
~ + ~ ~ ~ +
Q Q = Q = 1. (12)
Q
The shift operator, when applied to the number eigenstates defined by,
q ˆ n >= nq n > , produces the following new states:
e
~ + iα
Q n >= e n +1 n +1 > (13)

~ iα
Q n >= e n n −1 > (14)

where α s are undetermined phases. Therefore, (1 ) and (1 4) lead to the
3
n
~ + ~
interpretation of the shifter operators Q and Q as ladder operators that
increase and decrease the charge of the charge operator in its diagonal
representation.
The quantization apparatus is completed when the completeness and
orthogonality relations, and the inner product are stipulated, in this case as
given by (15)-(17), respectively,

¦ n n >< n = 1, (15)