Physical Chemistry Third Edition

18.2 The Indistinguishability of Electrons and the Pauli Exclusion Principle 767

If the wave function is a product of two orbitals as in Eq. (18.1-6) the probability

density for two particles is the product of two one-particle probability densities:

|Ψ(1, 2)|^2 |ψ 1 (1)|^2 |ψ 2 (2)|^2 (18.2-2)

Although we have obtained a wave function that satisfies our zero-order Schrödinger

equation and the appropriate boundary conditions, it must be modified to conform to

a new condition, which is required to obtain adequate agreement with experiment:

Identical particles are inherently indistinguishable from each other.This condition is

plausible because of the uncertainty principle, which makes exact trajectories impos-

sible to specify. If two identical particles approach each other closely it might not be

possible to tell which is which after the encounter. Figure 18.2 shows two encounters

that could be distinguished from each other if classical mechanics were valid, but which

might not be distinguished according to quantum mechanics, because of the combined

uncertainties of position and momentum.

2

2

1

121

12

Figure 18.2 Two Encounters of

Classical Particles.

Our theory must not include anything that would allow us to distinguish one particle

from another of the same kind. In a helium atom the probability of finding electron 1

at location 1 and finding electron 2 at location 2 must equal the probability of finding

electron 1 at location 2 and finding electron 2 at location 1. Any difference in these two

probabilities could give an illusory means of distinguishing the particles. We say that

the probability densityΨ*Ψmust besymmetricwith respect to interchange of the two

particles’ coordinates:

Ψ(1, 2)∗Ψ(1, 2)Ψ(2, 1)∗Ψ(2, 1) (18.2-3)

The labels in this equation mean that in the function on the right-hand sider 2 ,θ 2 , and

φ 2 are placed in the formula for the wave function wherer 1 ,θ 1 , andφ 1 were in the

formula on the left-hand side, and vice versa.

There are two ways to satisfy Eq. (18.2-3). Either the wave function can be

symmetricwith respect to interchange of the particles:

Ψ(1, 2)Ψ(2, 1) (symmetric wave function) (18.2-4)

or the wave function can beantisymmetricwith respect to interchange of the particles:

Ψ(1, 2)−Ψ(2, 1) (antisymmetric wave function) (18.2-5)

Particles that require symmetric wave functions as in Eq. (18.2-4) are calledbosons,

and particles that require antisymmetric wave functions as in Eq. (18.2-5) are called

fermions. Electrons are experimentally found to be fermions, as are protons and neu-

trons. Photons are bosons, as are nuclei containing an even number of nucleons (protons

and neutrons). All fermions have spin quantum numbers that are equal to half-integers,

whereas bosons have spin quantum numbers equal to integers.

Bosons are named for Satyendra Nath
Bose, 1894–1974, an Indian physicist
who gained early recognition for
deriving Planck’s black-body radiation
by assuming that photons are bosons.
Fermions are named for Enrico Fermi,
1901–1954, an Italian-American
physicist who was well known for his
work on nuclear fission and who
received the 1938 Nobel Prize in
physics for his work on neutron
bombardment.

The simplest way to make a two-electron orbital wave function satisfy the antisym-

metrization requirement is to add a second term that is the negative of the first term

with the coordinate labels interchanged, giving

Ψ(1, 2)C[ψ 1 (1)ψ 2 (2)−ψ 2 (1)ψ 1 (2)] (18.2-6)