BioPHYSICAL chemistry

CHAPTER 12 THE HYDROGEN ATOM 241

Now let us use the separation of variables again. Define:

Y(θ,φ) =Θ(θ)Φ(φ) (db12.14)

After substituting this into the equation db 12.13 multiply by:

(db12.15)

This yields:

(db12.16)

These two equations can now be separated into:

(db12.17)

and

(db12.18)

The solutions to the equation for φshould be familiar to you already, as the equation is the
same as for the particle in a box. For this case, let us use exponential terms:

Φ(φ) =Aeimlφ (db12.19)

You can check that this is correct by substitution. Note that we needed to include ibecause
of the negative sign in the equation. The normalization constant is determined using the
condition that integration over every value of φshould yield a value of 1:

(db12.20)

Thus, the constant is:

A= (db12.21)

1

2 π

1

0

2

2

0

2

== =∫∫ΦΦ*d() () − d

π

φ

π

φφφAe eimllimφ φAA^2

0

2

φπ^2 2d

π

∫ =

sin sin

()

θ ()sin

θ

θ

θ

θ

θ

d

d

d

d

⎡ Θ

⎣

⎢

⎢

⎤

⎦

⎥

⎥

++⎣⎡ll 1 2 −−m^2 l⎦⎤Θθ()= 0

1 2

2

2

2

2

2

Φ

ΦΦΦ

()

() () ()

φφ

φ

φ

φφ

d

d

or

d

d

=−mmll=−

1 2

Φ()^2 Φ() Θ

sin

()

sin

φ

δ

δφ

φ

θ

θ

δ

δθ

θ

δ

δθ

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟+ ΘΘ() ( )sinθθ

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟++

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

ll 102 =

sin

() ()

(^2) θ
ΘΦθφ