Лекции по квантовой механике. Розман Г.А. - 77 стр.

                        d 2Φ
                             + m 2Φ = 0                          (19.10)
                        dϕ 2
è
                 1 d           dP           m2
                         sin θ     +  λ −
                                          P = 0 .           (19.11)
               sin θ dθ        dθ         sin 2 θ
     Ðàññìîòðèì óðàâíåíèå (19.10). Óáåäèìñÿ, ÷òî åãî ðåøåíèå èìååò âèä
Φ (ϕ ) = C ⋅ exp(imϕ ) . Äëÿ ýòîãî ïîäñòàâèì åãî â óðàâíåíèå (19.10):

           d 2Φ     d  dΦ  d
           dϕ 2
                =           =
                   dϕ  dϕ  dϕ
                                          (         )
                                        Cime imϕ = C ⋅ im
                                                          dϕ
                                                            d imϕ
                                                               e =

                = C ⋅ im ⋅ im ⋅ e imϕ = −Cm 2 ⋅ e imϕ = −m 2 Φ.
      Ñëåäîâàòåëüíî, óðàâíåíèå âûðîæäàåòñÿ â òîæäåñòâî: 0=0 .Ëåãêî
óáåäèòñÿ, ÷òî ôóíêöèÿ         Φ (ϕ ) îáëàäàåò ïåðèîäè÷íîñòüþ ñ ïåðèîäîì
2π : Φ (ϕ ) = Φ (ϕ + 2π ). Äëÿ ýòîãî äîñòàòî÷íî âîñïîëüçîâàòüñÿ ôîðìó-
ëîé Ýéëåðà exp(im ⋅ 2π ) = cos(2πm ) + i sin (2πm ). Ïîñòîÿííûé ìíîæèòåëü
m îáÿçàí áûòü öåëûì ÷èñëîì. Ó÷èòûâàÿ, ÷òî óðàâíåíèå (19.10) ÿâëÿåòñÿ
äèôôåðåíöèàëüíûì óðàâíåíèåì âòîðîãî ïîðÿäêà, ìû ïîëó÷èì ïîëíîå ðå-
øåíèå ýòîãî óðàâíåíèÿ, åñëè ïîòðåáóåì, ÷òîáû ïîñòîÿííàÿ m ïðèíèìàëà
çíà÷åíèÿ 0, ± 1,±2.... Ïîñòîÿííóþ Ñ ìîæíî îïðåäåëèòü èç óñëîâèÿ íîðìè-
               2π

               ∫Φ       dϕ = 1 èëè
                    2

               o
               2π               2π                      2π
ðîâêè:
               ∫ Φ ⋅ Φ dϕ = ∫ Ce ⋅ Ce dϕ = C ∫ dϕ = C ⋅ 2π = 1
                  •             imϕ  − imϕ  2        2

               0                 0                      0



îòêóäà
                                     1
                               C=       .                        (19.12)
                                     2π
         Èòàê, ðåøåíèåì óðàâíåíèÿ (19.11) ÿâëÿåòñÿ ôóíêöèÿ

                               Φ (ϕ ) = 2π exp(imϕ ).
                                              1
                                                                 (19.13)


                                              77