Page 227 - Principles and Applications of NanoMEMS Physics
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A. QUANTUM MECHANICS PRIMER 217
E = = ¦ ω § 1 ·
¨ +n
¸ . (A.12)
MLC n
n © 2 ¹
However, if the field contains no phonons (n=0), the energy is not zero, but
is given by,
E = 1 = ¦ ω . (A.13)
MLC n
2 n
This, n=0, state is called the vacuum state, and the corresponding energy, is
called zero-point energy. Notice that, since = ,0n , 1 , 2 3 ! ∞ , the zero-
point vacuum energy is, in principle, infinite! In practice, however, various
factors, such as, dielectric constant cutoff, preclude it from becoming
infinity, although still very large.
It we imagine the free-space in which a z-directed, x-polarized
electromagnetic wave propagates as being divided into cubes of volume
V = L , then, the solution to its associated electric field wave equation,
3
1 ∂ 2 E
∇ 2 E = x , (A.14)
x 2 2
c t ∂
may be obtained by separation of variables as,
E ( ) =tz, ¦ a q ( ) ( ) zft , (A.15)
x n n n
n
where, subject to the spatial boundary conditions
E x ( =z , 0 t ) = E x ( = Lz ,t ) 0= , one obtains,
ª d 2 2 º
« 2 + k n » f n () z = 0 → f n () z = sin ( zk n ), (A.16)
¬ dz ¼
where,
1
k = nπ / L ,n = , , 2 3 ! ∞ , (A.17)
n
and,
ª d 2 2 º
« 2 + ω n » q n () =t 0 → q n () =t cos (ω t − ) φ , (A.18)
n
¬ dt ¼
where ω = ck . Writing the electric field solution as
n n