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218 Appendix A
E ( ) =tz, ¦ a q ( ) ( zkt sin ), (A.19)
x n n n
n
the magnetic field is immediately obtained from Maxwell’s equation,
∇ × E = − ∂ B t ∂ , as,
§ q () t ε ·
H ( ) = ¦ a ¨ n 0 ¸ cos ( zk ), (A.20)
t
z,
y n ¨ ¸ n
n © k n ¹
and the total field energy (Hamiltonian), which is given by,
ε
H = 1 ³ ( E 2 + µ H 2 )dV , (A.21)
Field 0 x 0 y
2 V
becomes, upon substituting (A.19) and (A.20) into (A.21),
q
H = 1 ¦ (m ω 2 q 2 + m 2 )= 1 ¦ § ¨ m ω 2 q 2 + p 2 n ·
¸ , (A.22)
Field n n n n n ¨ n n n ¸
2 n 2 n © m n ¹
provided one makes the associations: m = a 2 Vε 2ω , and p = m q .
2
n n 0 n n n n
The fact that each term in (A.22) is identical to the energy of a harmonic
oscillator of frequency ω , implies that the field may be visualized as
n
consisting of (or being populated by) a number n of such oscillators
(photons), and the analysis given above follows directly. Accordingly, we
can write
E = = ¦ ω § 1 · ¸ . (A.23)
¨ +n
Field n
n © 2 ¹
Again, ideally for n=0, it is concluded that the electromagnetic vacuum
possesses infinite energy. Furthermore, it can be shown [183] that the
averages of the field and its magnitude squared are E = 0 and
x
2 2
E = 2 E ( +n 1 ) 2 , where E = = ω ε V has dimensions of
x x n n 0
electric field. Thus, even when there is no field present, n=0, the vacuum is
endowed with a non-zero root-mean-square deviation. These are called zero-
point vacuum fluctuations and are the essence of the Casimir effect [19].
A.5 Second Quantization [ 232], [233]
Systems like the monatomic linear chain and the electromagnetic field,
whose behavior can be described in terms of fictitious particles, such as