Physical Chemistry Third Edition

17.2 The Relative Schrödinger Equation. Angular Momentum 731

Substitution ofΘ(θ)Φ(φ) into Eq. (17.2-7) followed by division byΘ(θ)Φ(φ) and

multiplication by sin^2 (θ) gives

sin(θ)

Θ

d

dθ

(

sin(θ)

dΘ

dθ

)

+

1

Φ

d^2 Φ

dφ^2

−Ksin^2 (θ) (17.2-9)

The variablesθandφare now separated.

TheΦFunctions

The last term on the left-hand side of Eq. (17.2-9) depends only onφand the other

terms depend only onθ. The independent variableθcan be held fixed whileφranges,

so the last term must be a constant function ofφ. We denote the constant by−m^2. With

this choice,mwill turn out to be a real integer. Multiplication byΦgives the equation

d^2 Φ

dφ^2

−m^2 Φ (17.2-10)

Except for the symbols used Eq. (17.2-10) is the same as several equations already

encountered. Its general solution can be written in the form of Eq. (15.3-6)

ΦCcos(mφ)+Dsin(mφ) (17.2-11)

or in the form of Eq. (15.3-26)

ΦAeimφ+Be−imφ (17.2-12)

whereA,B,C, andDare constants.

We always assume that a wave function must be continuous at all locations. The

variableφranges from 0 to 2πwhen measured in radians. Sinceφ0 andφ 2 π

refer to the same location for given values ofrandθ, we must require that

Φ(0)Φ(2π) (17.2-13)

This condition is satisfied only ifmis real and equal to an integer.

Exercise 17.2

Show thatmis real and equal to an integer ifΦ(0)Φ(2π).

There are two versions of theΦfunction. The first version ofΦis an eigenfunction

of̂Lz, which is given in spherical polar coordinates by Eq. (16.3-20):

̂Lzh ̄

i

∂

∂φ

(17.2-14)

We operate on the version ofΦin Eq. (17.2-12) witĥLz:

̂LzΦh ̄

i

(

imAeimφ−imBe−imφ

)

(17.2-15)