Page 226 - Principles and Applications of NanoMEMS Physics
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216 Appendix A
§ 1 mω 2 q ˆ 2 ·
H MLC = ¦ ¨ p ˆ n 2 + n ¸ , (A.7)
¸
¨
n © 2 m 2 ¹
then the eigenvalues (frequency dispersion curve) are,
ω = 4k sin § qa · ¸ , and the eigenfunctions (propagating modes),
¨
q
m © 2 ¹
−
q = ξ ⋅ exp i ( ( qna ω )) t [64]. Since, comparing (A.6) and (A.7), it is
n
obvious that the latter is the sum of the Hamiltonian of n “particles,” the
MLC may be visualized as consisting of a set of n particles vibrating
independently. In the context of the MLC, in which the vibrations represent
acoustic waves, such fictitious particles are, in fact, called phonons, and
(A.7) implies that the state of the MLC, in particular, its total energy, may be
specified by giving the number n of “particles” present.
A.4 Creation and Annihilation Operators
It turns out that making the association:
i − ω t 1
a e n = (m ω q ˆ + p iˆ ), (A.8)
n n n n n
=
2 m ω
n n
+ iω t 1
a e n = (m ω q ˆ − p iˆ ), (A.9)
n n n n n
=
2 m ω
n n
+
where a and a are new operators obeying the commutation relations
n n
=
+ + +
a
[a , a ]= δ , and [ ,a ] [a , a ] 0= , the quantized Hamiltonian
n n′ n n′ n ′ n n ′ n
(A.7) may be written as,
H ˆ = = ¦ ω § ¨a + a + 1 ·
¸ . (A.10)
MLC n n n
n © 2 ¹
Then, using the Hamiltonian expressed as in (A.11), and the commutation
relations for the new operators, it can be shown that following result is
follows,
ˆ
= =
H n > = = ¦ ω § ¨a + a + 1 · ¸ ¦ ω § ¨ +n 1 · ¸ n > = E n > . (A.11)
MLC n n n 2 n 2 MLC
n © ¹ n © ¹
This means that if the field contains n phonons, the result of measuring its
energy gives,