bei48482_FM

The third term of Eq. (6.6) is a function of azimuth angle only, whereas the other

terms are functions of rand only.

Let us rearrange Eq. (6.6) to read

r

2

 sin 

  E (6.7)

This equation can be correct only if both sides of it are equal to the same constant,since they

are functions of differentvariables. As we shall see, it is convenient to call this constant

ml^2. The differential equation for the function is therefore

m

2

l (6.8)

Next we substitute ml^2 for the right-hand side of Eq. (6.7), divide the entire equa-

tion by sin^2 , and rearrange the various terms, which yields

r

2

  E sin  (6.9)

Again we have an equation in which different variables appear on each side, requiring

that both sides be equal to the same constant. This constant is called l(l1), once

more for reasons that will be apparent later. The equations for the functions and R

are therefore

 sin l(l1) (6.10)

r

2

  El(l1) (6.11)

Equations (6.8), (6.10), and (6.11) are usually written

Equation for  ml^2  0 (6.12)

sin l(l1) ^ ^0 (6.13)

r

2

 E R^0 (6.14)

Each of these is an ordinary differential equation for a single function of a single vari-

able. Only the equation for Rdepends on the potential energy U(r).

l(l  1)



r^2

e^2



4  0 r

2 m



^2

dR



dr

d



dr

1



r^2

Equation

for R

ml^2



sin^2 

d



d

d



d

1



sin

Equation

for 

d^2



d^2

e^2



4  0 r

2 mr^2



^2

dR



dr

d



dr

1



R

d



d

d



d

1



sin

m^2 l



sin^2 

d



d

d



d

1



sin

m^2 l



sin^2 

e^2



4  0 r

2 mr^2



^2

dR



dr

d



dr

1



R

d^2



d^2

1



d^2



d^2

1



e^2



4  0 r

2 mr^2 sin^2 



^2

d



d

d



d

sin



dR



dr

d



dr

sin^2 



R

204 Chapter Six

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