Question

Quantum mechanics
An atom has even parity if for the non-radial part of the wave function: Y_lm(Theta,Phi)=Y_lm((pi-theta),(pi-phi)) and an odd parity if Y_lm(Theta,Phi)=-Y_lm((pi-theta),(pi-phi)), where in both cases, Y_lm(Theta,Phi)=THETA_lm(theta)*PHI(phi). This requires that THETA_lm((pi-theta))=(-1)^l-m*THETA_lm(theta), where:

THETA(theta) = P_l^m(Psi) with Psi = Cos(theta)

P_l^m(x) = (-1)^m - (1-x^2)^(m/2)*[d^mP_l(x)/dx^m]

and rodregez formula P_l(x) = (1/2^l*l!) * [d^l(x^2-1)^l/dx^l]


Using these equations, show that THETA_lm (pi-theta)=(-1)^(l-m)*THETA_lm (theta).