Principles - Methods of Molecular Quantum Mechanics - page 9

with components

! !

hy @

@ z z @

hz @

@ x x @

L x ¼ i L y ¼ i

; ;

@ y @ z

!

ð 1 : 37 Þ

hx @

@ y y @

L z ¼ i

@ x

In the theory of angular momentum, frequent use is made of the ladder
(or shift) operators:

L þ ¼ L x þ iL y ð step-up Þ; L ¼ L x iL y ð step-down Þ

ð 1 : 38 Þ

These are also called raising and lowering operators 4 respectively.
Angular momentum operators have the following commutation rela-
tions:

8
<

½ L x ; L y ¼ iL z ; ½ L y ; L z ¼ iL x ; ½ L z ; L x ¼ iL y

½ L z ; L þ ¼ L þ ; ½ L z ; L ¼ L
½ L

ð 1 : 39 Þ

:

2 2

; L k ¼½ L ; L ¼ 0 k ¼ x ; y ; z

The same commutation relations hold for the spin vector operator S
(Chapter 5).

1.2.15 Transformations in Coordinate Space

We now give the definitions of the main coordinate systems useful in
quantum chemistry calculations (Cartesian, spherical, spheroidal), the
relations between Cartesian and spherical or spheroidal coordinates, and
the expressions of the volume element dr and of the operators r and r

2 in
the new coordinate systems. We make reference to Figures 1.1 and 1.2.

(i) Cartesian coordinates (x,y,z):
x ; y ; z 2ð¥; ¥Þ ð 1 : 40 Þ

dr ¼ dx dy dz ð 1 : 41 Þ

4 Note that the ladder operators are non-Hermitian.