BioPHYSICAL chemistry

CHAPTER 12 THE HYDROGEN ATOM 243

This will be true if l=0 and ml=0. Another possible solution is:

Θ(θ) =Bcosθ (db12.28)

Substitution gives:

sinθ(− 2 Bsinθcosθ) +[l(l+1)sin^2 θ−ml^2 ]Θ(θ) = 0 (db12.29)

This is true if l=1 and ml=0. In general the solutions are polynomials of trigonometric
functions.

Radial solution

The radial equation was found previously (eqn db 12.11) to be:

(db12.30)

Multiplying this by R(r)/ryields:

(db12.31)

These are called Laguerre functions and are eigenfunctions with a series of solutions. To
determine the general functional form of the solutions, let Π(r) =rR(r). Then:

(db12.32)

Consider the case where l=0 then, after multiplying by − 2 m/Z, the equation reduces to:

(db12.33)

To solve this equation, try a solution with exponentials with a constant, α:

Π(r) =re−αr (db12.34)

(db12.35)

(db12.36)

d

d

2

22

2

0

2

4

0

Π

Π

()

()

r

r

me

r

++Er

⎛

⎝

⎜

⎜

⎞

⎠

⎟

Z πε ⎟ =

−+

+

−

⎡

⎣

ZZ^22

2

2

2

2

(^20)

1

mr 24

r

ll

mr

e

r

d

d

Π()

()

πε

⎢⎢

⎢

⎤

⎦

⎥

⎥ΠΠ()rEr= ()

−

+

−

+

Z^22

2

2

0

2

242 m

rR r

r

e

r

rR r

m

l

d

d

[()]

[()]

πε

(( ) [()] [()]l

r

+= 1 rR r rR r E

1

2

Z

−

+

−

−+

Z^22 Z

2

2

0

2

24 m

r

Rr

rR r

r

er

rE

()

d[()]

d πε

2

2

10

m

ll()+=

d

d

2

2

2

r

Π()rreee=− −αα( −−ααrr+ )−α−αr=αrre−−ααrr− 2 αe

d

dr

Π()rr e e=−( )α −−ααrr+