Page 167 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics
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158                          The hydrogen atom
                                                        R ˆ iX ‡ jY ‡ kZ
                             By de®nition, the center of mass is related to r 1 and r 2 by

                                                             m 1 r 1 ‡ m 2 r 2
                                                         R ˆ                                    (6:2)
                                                                  M
                             where M ˆ m 1 ‡ m 2 is the total mass of the system. We may express r 1 and r 2
                             in terms of R and r using equations (6.1) and (6.2)
                                                                    m 2
                                                          r 1 ˆ R ÿ   r
                                                                    M
                                                                                                (6:3)
                                                                    m 1
                                                          r 2 ˆ R ‡   r
                                                                    M
                               If we restrict our interest to systems for which the potential energy V is a
                             function only of the relative position vector r, then the classical Hamiltonian
                             function H is given by
                                                              2      2
                                                          jp 1 j  jp 2 j
                                                     H ˆ       ‡      ‡ V(r)                    (6:4)
                                                          2m 1   2m 2
                             where the momenta p 1 and p 2 for the two particles are
                                                           dr 1             dr 2
                                                   p 1 ˆ m 1  ,     p 2 ˆ m 2
                                                           dt                dt
                             These momenta may be expressed in terms of the time derivatives of R and r
                             by substitution of equation (6.3)

                                                               dR    m 2 dr
                                                      p 1 ˆ m 1    ÿ
                                                                dt   M dt
                                                                                                (6:5)

                                                               dR    m 1 dr
                                                      p 2 ˆ m 2    ‡
                                                                dt   M dt
                             Substitution of equation (6.5) into (6.4) yields
                                                                2         2
                                                           dR         dr
                                                      1
                                                  H ˆ M          ‡ ì       ‡ V(r)               (6:6)
                                                                  1
                                                      2    dt     2    dt
                             where the cross terms have canceled out and we have de®ned the reduced mass
                             ì by
                                                            m 1 m 2   m 1 m 2
                                                       ì           ˆ                            (6:7)
                                                           m 1 ‡ m 2    M
                               The momenta p R and p r , corresponding to the center of mass position R and
                             the relative position variable r, respectively, may be de®ned as
                                                            dR              dr
                                                    p R   M    ,     p r   ì
                                                            dt              dt
                             In terms of these momenta, the classical Hamiltonian becomes
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