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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
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