Principles - Methods of Molecular Quantum Mechanics - page 11

dr ¼ r 2 dr sin u d u d w ð 1 : 46 Þ

r¼ e r @

1

r @

1
r sin

@
@w

@ r þ e u

þ e w ð 1 : 47 Þ

@u u

þ

2

r 2 @

1

r 2 @
@ r

1
r 2

1
sin

@
@u

u @
@u

1
sin 2

@

2

r ¼ sin þ

@ r u @w

2

u

2

r L

= h 2
r 2

2

¼r ð 1 : 48 Þ

where e r , e u , and e w are unit vectors along r, u , and w .In

Equation (1.48):

¼ @

2
@ r 2 þ r 2 @

1

r 2 @
@ r

2

r @

2

r

r ¼

ð 1 : 49 Þ

@ r @ r

is the radial Laplacian and

2 3

8
<

!
þ

2

1
sin

@
@u

u @
@u

1
sin 2

@

2

L

¼ L L ¼ h 2

4 5

sin

u @w

2

u

0
@

1
A

ð 1 : 50 Þ

:

2 2

@

u @
@u

1
sin 2

@

¼ h 2

2 þ cot

þ

@w

2

@u

u

is the square of the angular momentum operator (1.36). For the
components of the angular momentum vector operator

L we

have

w @
@u

w @
@w

L x ¼ i

h sin cot u ð 1 :

51 Þ cos

w @
@u

w @
@w

L y ¼ i

h cos cot u sin ð 1 : 52 Þ

h @
@w

L z ¼ i

ð 1 : 53 Þ

@
@u

u @
@w

L þ ¼ h exp ð i

wÞ þ i cot ð 1 : 54 Þ

wÞ @
@u

u @
@w

L ¼ h exp ð i

þ i cot ð 1 : 55 Þ