Page 139 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

5

Angular momentum

Angular momentum plays an important role in both classical and quantum
mechanics. In isolated classical systems the total angular momentum is a
constant of motion. In quantum systems the angular momentum is important in
studies of atomic, molecular, and nuclear structure and spectra and in studies
of spin in elementary particles and in magnetism.

5.1 Orbital angular momentum

We ®rst consider a particle of mass m moving according to the laws of classical
mechanics. The angular momentum L of the particle with respect to the origin
of the coordinate system is de®ned by the relation
L r 3 p (5:1)
where r is the position vector given by equation (2.60) and p is the linear
momentum given by equation (2.61). When expressed as a determinant, the
angular momentum L is

i j k

L ˆ x y z

p x p y p z
The components L x , L y , L z of the vector L are
L x ˆ yp z ÿ zp y

L y ˆ zp x ÿ xp z (5:2)
L z ˆ xp y ÿ yp x
The square of the magnitude of the vector L is given in terms of these
components by
2
2
2
:
L ˆ L L ˆ L ‡ L ‡ L 2 z (5:3)
y
x
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