Page 140 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

5.1 Orbital angular momentum 131

If a force F acts on the particle, then the torque T on the particle is de®ned as
dp
T ˆ r 3 F ˆ r 3 (5:4)
dt
where Newton's second law that the force equals the rate of change of linear
momentum, F ˆ dp=dt, has been introduced. If we take the time derivative of
equation (5.1), we obtain

dL dr dp dp
ˆ 3 p ‡ r 3 ˆ r 3 (5:5)
dt dt dt dt
since
dr dr dr
3 p ˆ 3 m ˆ 0
dt dt dt
Combining equations (5.4) and (5.5), we ®nd that
dL
T ˆ (5:6)
dt
If there is no force acting on the particle, the torque is zero. Consequently, the
rate of change of the angular momentum is zero and the angular momentum is
conserved.
The quantum-mechanical operators for the components of the orbital angular
momentum are obtained by replacing p x , p y , p z in the classical expressions
(5.2) by their corresponding quantum operators,

" @ @
^ y ÿ z (5:7a)
L x ˆ y^ p z ÿ z^ p y ˆ
i @z @ y

" @ @
^
L y ˆ z^ p x ÿ x^ p z ˆ z ÿ x (5:7b)
i @x @z

" @ @
^
L z ˆ x^ p y ÿ y^ p x ˆ x ÿ y (5:7c)
i @ y @x
Since y commutes with ^ p z and z commutes with ^ p y , there is no ambiguity
^
regarding the order of y and ^ p z and of z and ^ p y in constructing L x . Similar
^
^
remarks apply to L y and L z . The quantum-mechanical operator for L is
^ ^ ^ ^
L ˆ iL x ‡ jL y ‡ kL z (5:8)
2
and for L is
^ 2 ^ : ^ ^ 2 ^ 2 ^ 2 (5:9)
L ˆ L L ˆ L ‡ L ‡ L
x y z
^
^
^
The operators L x , L y , L z can easily be shown to be hermitian with respect to a
^
^ 2
set of functions of x, y, z that vanish at 1. As a consequence, L and L are
also hermitian.