Physical Chemistry , 1st ed.

differentiate between orbitalangular momentum and spinangular momen-

tum. Both observables are angular momenta, but they arise from different

properties of the electron: one from its motion about a nucleus, the other from

its very existence.

The spin angular momentum of an electron can have only certain specific

values. Spin is quantized.Like the zcomponent of orbital angular momentum,

mshas 2s1 possible values. In the case of the electron,s^12 , so the only

possible values ofmsare ^12 and ^12 . The specification of an electron’s spin

therefore represents two other quantum numbers that can be used to label the

state of that electron. In practice, however, it is convenient to not specify s,

since it is always ^12 for electrons. This gives us a total of four quantum num-

bers: the principal quantum number n, the orbital angular momentum quan-

tum number , the orbital angular momentum zcomponent m, and the spin

angular momentum (zcomponent) ms. These are the only four quantum num-

bers needed to specify the complete state of an electron.

Example 12.2

List all possible combinations of all four quantum numbers for an electron

in the 2porbital of a hydrogen atom.

Solution

In tabular form, the possible combinations are

Symbol Possible values

n 2

 1

m 1 0 1

ms ^12 or ^12  ^12 or ^12  ^12 or ^12 

There are a total of six possible combinations of the four quantum numbers

for this case.

Although not considered until now, the msof the electron in a hydrogen

atom is either ^12 or ^12 . A fascinating astronomical consequence of spin is the

fact that an electron in hydrogen has a slightly different energy depending on

the relative spin orientations of the electron and the proton in the nucleus. (A

proton also has a characteristic spin quantum number of^12 .) If an electron in

a hydrogen atom changes its spin, there is a concurrent energy change that is

equivalent to light having a frequency of 1420.40575 MHz, or a wavenumber

of about 21 cm^1 , as shown in Figure 12.2. Because of the pervasiveness of hy-

drogen in space, this “21-cm^1 radiation” is important for radio astronomers

who are studying the structure of the universe.

Finally, since spin is part of the properties of an electron, its observable val-

ues should be determined from the electron’s wavefunction. That is, there

should be a spin wavefunction part of the overall . A discussion of the exact

form of the spin part of a wavefunction is beyond our scope here. However,

since there is only one possible observable value of the total spin (s^12 ) and

only two possible values of the zcomponent of the spin (ms^12 or ^12 ), it is

typical to represent the spin part of the wavefunction by the Greek letters and

, depending on whether the quantum number msis ^12 or ^12 , respectively.

12.2 Spin 373

400Å 500Å 600Å 700Å

H

2468 cm^1

21 cm^1
Figure 12.2 A very high resolution spectrum
of the hydrogen atom shows a tiny splitting due
to the spin on the electron. This splitting is caused
by the electron spin interacting with the nuclear
spin of the hydrogen atom’s nucleus (a proton).