BioPHYSICAL chemistry

12.19 A wavefunction can have a node where the value is exactly equal

to zero. For the 2s orbital this occurs at:

12.20Φ(φ+ 2 π) =Φ(φ)

Aeimlφ=Aeiml(φ+^2 π)=Aeimlφeiml^2 π

1 =eiml^2 π=cos(2πml) +isin(2πml)

with ml=0, ±1, ±2, ±3,....

12.21

12.22

−Bsinθ(2sinθcosθ) + 2 Bcosθsin^2 θ= 0

0 = 0

12.23

The second term is zero since:

This leaves the first term and:

E

me

=−

4

2

0

32 πεZ^2

α

πε

=

me

Z^2

2

(^40)
re
mE
e
−−rre

• 
  

⎛

⎝

⎜⎜

⎞

⎠

αααα⎟⎟+−+

πε

2

2

2

0

2

2

4

2

Z

mm

Z^2

0

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟=

αα

πε

2 αα

2

2

0

2

2

4

re e

me

r

−−rr−+ +Er

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

Z

ee−αr= 0

d

d

2

2

2

r

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

d

dr

Π()rr e e=−( )α −−ααrr+

−+=BBsinθ [sin ] cos sin

θ

θθθ

d

d

(^2220)
sinθ [sin ( sin )] cos sin
θ
θθ θθ
d
d

−+BB 202 =

sinθ sin ( cos ) [ ( )

θ

θ

θ

θ

d

d

d

d

B

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

++11 1 ssin^2 θθ−= 00 ]( cos )B

d

d

d

d

2

2

2

φ

φ

φ

Φ()==( ( )Aim e φφ) Aim e( )

l

im

l

llim =−m ()

l

(^2) Φφ
d
d
d
φ d
φ
φ
Φ()==(Aeimφφ) A im e( )
l
llim
ψ
π
ρ ρ
200
3
0
3
(^14)
224

2

2

=−/ 0

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟ − =

Z

a

e wheenρ= 4

ANSWERS TO PROBLEMS 469

9781405124362_5_end.qxd 4/29/08 9:17 Page 469