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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,
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