Physical Chemistry Third Edition

730 17 The Electronic States of Atoms. I. The Hydrogen Atom

Equation (17.2-1) is a partial differential equation with three independent variables.

Comparison with Eq. (16.3-24) shows that the operator for the square of the angular

momentum is contained in the equation, so we can write the equation in the form:

− ̄

h^2

2 μr^2

∂

∂r

(

r^2

∂ψ

∂r

)

+

1

2 μr^2

̂L^2 ψ+V(r)ψErelψ (17.2-2)

The potential energy in Eq. (17.2-2) is expressed in terms of only one of the three
coordinates and the operator in the first term contains only the variabler, which suggests
x that we should try a separation of variables.

z

y

r 5 (x^21 y^21 z^2 )1/2

 5 cos^21 (zr (

 5 tan^21 (xy(

Figure 17.3 Spherical Polar Coor-
dinates. The First Separation of Variables

We separaterfromθandφby assuming the trial solution

ψ(r,θ,φ)R(r)Y(θ,φ) (17.2-3)

where we callR(r) theradial factorandY(θ,φ) theangular factor. ThêL^2 operator

contains onlyθandφ, so the radial factorR(r) is treated as a constant when̂L^2 operates.

The angular factorY(θ,φ) is treated as a constant when differentiation with respect to

ris carried out. Substitution of the trial solution into Eq. (17.2-2) gives

−

h ̄^2

2 μr^2

[

Y

d

dr

(

r^2

dR

dr

)

+

R

hr ̄^2

̂L^2 Y

]

+(V−Erel)RY 0 (17.2-4)

We divide this equation byRYand also multiply by 2μr^2 /h ̄. This separatesrfrom the

other variables, giving the equation

−

1

R

d

dr

(

r^2

dR

dr

)

+

2 μr^2

h ̄^2

(V−Erel)+

1

h ̄^2

1

Y

̂L^2 Y 0 (17.2-5)

The final term on the left-hand side of Eq. (17.2-5) does not depend onrand the other

terms do not depend onθorφ. Sincercan be held fixed whileθandφvary, the last

term must be a constant function ofθandφ, which we set equal to the constantK.

Multiplication byh ̄^2 Ygives the equation

̂L^2 Yh ̄^2 KY (17.2-6)

The angular factorYis an eigenfunction of̂L^2 with eigenvalueh ̄^2 K. We will determine

what the eigenvalues are later. Writing out the expression for̂L^2 we obtain

−h ̄^2

[

1

sin(θ)

∂

∂θ

(

sin(θ)

∂Y

∂θ

)

+

1

sin^2 (θ)

∂^2 Y

∂φ^2

]

h ̄^2 KY (17.2-7)

The Second Separation of Variables

To separate the variablesθandφwe assume the trial solution

Y(θ,φ)Θ(θ)Φ(φ) (17.2-8)