Physical Chemistry Third Edition

17.2 The Relative Schrödinger Equation. Angular Momentum 733

TheΘFunction

After replacement of the last term on the left-hand side of Eq. (17.2-9) by−m^2 and

multiplication byΘ, we obtain

sin(θ)

d

dθ

(

sin(θ)

dΘ

dθ

)

−m^2 Θ+Ksin^2 (θ)Θ 0 (17.2-22)

This equation can be transformed into a famous equation called theassociated Legendre

equationby the change of variables:

ycos(θ), P(y)Θ(θ) (17.2-23)

We do not discuss the associated Legendre equation, but give its solutions in Appendix F.

The solutions are calledassociated Legendre functionsand are derivatives of polyno-

mials known asLegendre polynomials.

The equation is named for Adrien-Marie
Legendre, 1752–1833, a famous French
mathematician who solved the equation. For a solution of the associated Legendre equation to exist that obeys the relevant
boundary conditions, it turns out that the constantKmust be equal tol(l+1) wherel
is a positive integer at least as large as the magnitude ofm. There is one solution for
each set of values of the two quantum numberslandm:

Θ(θ)Θlm(θ) (17.2-24)

Since the equation containsm^2 , the solutions are the same for a given value ofmand

its negative:

Θlm(θ)Θl,−m(θ) (17.2-25)

where we insert a comma to avoid confusing two subscripts having valuesland−m

with a single subscript having a valuel−m.

If we choose the complexΦfunctions, theYfunctions are calledspherical harmonic

functions.

YYlm(θ,φ)Θlm(θ)Φm(φ) (17.2-26)

Table 17.1 gives the normalized spherical harmonic functions forl0,l1, and

l2. Additional functions can be derived from formulas in Appendix F.

Angular Momentum Values

The spherical harmonic functions,Ylm(θ,φ), are eigenfunctions of̂L^2 with eigenvalue

h ̄^2 K, as in Eq. (17.2-6).

̂L^2 Ylm̂L^2 ΘlmΦmh ̄^2 l(l+1)ΘlmΦm (l0, 1, 2,...) (17.2-27)

The square of the angular momentum can take on any of the values

L^2 0, 2h ̄^2 ,6h ̄^2 ,12h ̄^2 ,20h ̄^2 ,...,l(l+1)h ̄^2 ,... (17.2-28)

The magnitude of the angular momentum can take on any of the values

L|L|0,

√

2 h ̄,

√

6 h ̄,

√

12 h ̄,...,R

(√

l(l+1)

)

h ̄,... (17.2-29)