Mathematical Physics Vol 1

Chapter 7. Partial differential equations

428

Multiplying the last relation by r 2 we obtain two members, one is the function of r only, and the other of θ only 1 R d d r r 2 d R d r + k 2 r 2 + 1 Θ sin θ d d θ sin θ d Θ d θ − m 2 sin 2 θ = 0 . This equation is now decomposed on two ordinary differential equations 1 Θ sin θ d d θ sin θ d Θ d θ − m 2 sin 2 θ = const . = − l ( l + 1 ) (7.436) 1 R d d r r 2 d R d r + k 2 r 2 = const . = l ( l + 1 ) . (7.437) Te first equation (7.436), after substitution cos θ = z , comes down to Legendre equation (5.21) ( 1 − z 2 ) d 2 Θ d z 2 − 2 z d Θ d z + l ( l + 1 ) − m 2 1 − z 2 Θ = 0 , the solution of which has the form Θ = C 5 P m l ( z )+ C 6 Q m l ( z ) . (7.438) The second equation (7.437), after introducing a new function u = R √ r , becomes d 2 u d r 2 + 1 r d u d r + " k 2 − ( l + 1 / 2 ) 2 r 2 # u = 0 , which is the Bessel equation (5.45), whose solution is of the form u = R √ r = C 7 J l + 1 / 2 ( kr )+ C 8 Y l + 1 / 2 ( kr ) . (7.439) Based on (7.438), (7.435) and (7.439) we can now write the solution of the initial equation(7.424) ψ =( C 1 sin ω t + C 2 cos ω t ) · ( C 3 sin m ϕ + C 4 cos m ϕ ) · · [ C 5 P m l ( cos θ )+ C 6 Q m l ( cos θ )] · 1 √ r C 7 J l + 1 / 2 ( kr )+ C 8 Y l + 1 / 2 ( kr ) . (7.440) The constants C i , ( i = 1 , 2 ,..., 8) are determined from initial and boundary conditions.