BioPHYSICAL chemistry

12.5

12.6

12.7 Principal, n=1, 2, 3,...; angular momentum, l=0, 1, 2,...,

n−1; magnetic ml=l, l−1, l−2,... , −l.

12.8

12.9 n, Quantization of energy; l, quantization of total angular momen-

tum; ml: quantization of the zcomponent of angular momentum.

12.10

12.11

12.12

12.13

1

22

1

4

2

0

3

0

2

3

0

πa

r

a

era

a

⎛ − /

⎝

⎜

⎜

⎞

⎠

⎟

⎟

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

−

00

2

4 2

∞

∫ ()πrrd

1

224

2

(^32)
0
3
0

Z 2

a

r

a

e

i

ra

π

− /

⎛

⎝

⎜

⎜

⎞

⎠

⎟

⎟

∇

⎛

⎝

⎜⎜

⎞

⎠

∫

− 0 Z ⎟⎟

⎟ −

⎛

⎝

⎜

⎜

⎞

⎠

⎟

⎟

(^1) −
224

2

(^32)
0
3
0
Z (^202)
a
r
a
errra
π
/ dd dθ φφ

=−

⎛

⎝

⎜

⎜

⎞

⎠

⎟

⎟

∞

−

∫

(^1) /
8

2

0

3

3

0 0

2

0

a

r

r

a

errad

ψψτ

π

*drr

a

r

a

=−era/

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟ −

∞

∫

1

16

1

2

2

0

3

0

2

0 0

0

∞∞

∫^4

πrr^2 d

λ

()(. )

==.

×

− =

9

8

9

829 11 10

2251122

ZRh cm

××= 10 −^8 cm 1 22. Å

ΔEZhcR

hc

=− h −

⎛

⎝

⎜⎜

⎞

⎠

(^222) ⎟⎟=

1

3

1

1 λ

λ

(. )

==.

×

− =×−

9

8

9

811 10

51 103 10^5

Rh cm

cmm

ΔEhcR

hc

=− h −

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟=

1

3

1

(^221) λ

0

8

15

0

3

2

0

(^00)
=−/ ⎛
⎝

⎜⎜

⎞

⎠

− ⎟⎟ ==

a

er

r

a

ra or ra ..29 10× − (^10) m

04 = ()^2

d

d

*

r

πψψr

==×. −

3

2

(^0) 79 10 11
a
m

4

44

0

3

3

0

2

0

32

0

a re^0 r a xe x ae^0

ra x

∞

− −

∞

−

∫∫==

/ dd 22

32

(^20)

3

4

3

4

3

8

x⎛−− − −xxx

⎝

⎜⎜

⎞

⎠

⎟⎟

∞

ψψτ

π

*drr π d

a

= errra/

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟ −

∞ ∞

∫∫

1

4

0

3

2

0 0

(^02) ==
∞
−
∫
(^4) /
0
3
3
0
(^20)
a
reradr

ANSWERS TO PROBLEMS 467

9781405124362_5_end.qxd 4/29/08 9:17 Page 467