Simple Nature - Light and Matter

Quantization of angular momentum

The angular momentum of a particle due to its motion through

space is quantized in units of~.

self-check I

What is the angular momentum of the wavefunction shown at the be-

ginning of the section? .Answer, p.

1063

Degeneracy

Comparing the oversimplified figure a/1 with the more realistic

depiction in a/2 using complex numbers, we see that there is a

direction of rotation, which was drawn as counterclockwise in the

figure. As in figures u/2 and u/3 on p. 914, we could have drawn

the clockwise version by putting the rainbow colors in the opposite

order, i.e., by letting the phase spin in the opposite direction in the

complex plane. This feature was hidden in a/1, where in order to

get a depiction using real numbers, we had to use a standing wave.

A standing wave, however, can be constructed as a superposition

of two traveling waves, so the issue was still there, just hidden.

We really havetwo quantum-mechanical states here, regardless of

whether we use standing waves or traveling waves. If we use standing

waves, they are of the form sin 8θ and cos 8θ, while in terms of

the traveling waves we havee^8 iθ ande−^8 iθ. By Euler’s formula

(sec. 10.5.6, p. 627), either traveling wave can be expressed as a

superposition of the two standing waves, and vice versa. (Physically,

there are not four different states here but two. The situation is a bit

like choosing a Cartesian coordinate system in the plane, where we

could choose one coordinate system (x,y), or some other coordinates

system (x′,y′) rotated with respect to the first one; but this does

not mean there are four coordinates needed to describe a plane.)

These two states are simplified models of states in an atom, so

it’s worth thinking about how we could tell, for a real atom, whether

the electron had angular momentum in one direction or the other.

One technique would be to look at absorption or emission spectra of

thin gases, as in example 13 on p. 898. But this only distinguishes

states according to their energies, and since these two states have the

same kinetic energy, that would not necessarily help. In quantum

mechanics, when we have more than one state with the same energy,

they are said to bedegenerate. In our example, the degeneracy of the

`= 8 state is 2. This degeneracy arises from the symmetry of space,

which does not distinguish one direction from another. Degeneracies

often, but not always, arise from symmetries. (Cf. p. 927.)

If we wanted to distinguish these two degenerate states obser-

vationally, one way to do it would be to “lift” the degeneracy by

920 Chapter 13 Quantum Physics