Approximate wavefunction Total z-component spin
He(1s 1 )(1s 2 ) 1
He(1s 1 )(1s 2 ) 0
He(1s 1 )(1s 2 ) 0
He(1s 1 )(1s 2 ) 1At this point, experimental evidence can be introduced. (The necessity of
comparing the predictions of theory with experiment should not be forgot-
ten.) Angular momenta of charged particles can be differentiated by magnetic
fields, so there is a way to experimentally determine whether or not atoms have
an overall angular momentum. Since spin is a form of angular momentum, it
should not be surprising that magnetic fields can be used to determine the
overall spin in an atom. Experiments show that ground-state helium atoms
have zero z-component spin. This means that of the four approximate wave-
functions listed above, the first and last are not acceptable because they do not
agree with experimentally determined facts. Only the middle two, (1s 1 )(1s 2 )
and (1s 1 )(1s 2 ), can be considered for helium.
Which wavefunction of the two is acceptable, or are they both? One can sug-
gest that both wavefunctions are acceptable and that the helium atom is dou-
bly degenerate. This turns out to be an unacceptable statement because, in
part, it implies that an experimenter can determine without doubt that elec-
tron 1 has a certain spin wavefunction and that electron 2 has the other spin
wavefunction. Unfortunately, we cannot tell one electron from another. They
are indistinguishable.
This indistinguishability suggests that the best way to describe the electronic
wavefunction of helium is not by each wavefunction individually, but by a
combination of the possible wavefunctions. Such combinations are usually
considered as sums and/or differences. Given nwavefunctions, one can math-
ematically determine ndifferent combinations that are linearly independent.
So, for the two “acceptable” wavefunctions of He, two possible combinations
can be constructed to account for the fact that electrons are indistinguishable.
These two combinations are the sum and the difference of the two individual
spin orbitals:He,1
1
2
[(1s 1 )(1s 2 ) (1s 1 )(1s 2 )]He,2
1
2
[(1s 1 )(1s 2 ) (1s 1 )(1s 2 )]The term 1/ 2 is a renormalization factor, taking into account the combi-
nation of two normalized wavefunctions. These combinations have the proper
form for possible wavefunctions of the helium atom.
Are both acceptable, or only one of the two? At this point we rely on a pos-
tulate proposed by Wolfgang Pauli in 1925, which was based on the study of
atomic spectra and the increasing understanding of the necessity of quantum
numbers. Since electrons are indistinguishable, one particular electron in he-
lium can be either electron 1 or 2. We can’t say for certain which. But because
the electron has a spin of^12 , it has certain properties that affect its wavefunction
(the details of which cannot be considered here). If electron 1 were exchanged
with electron 2, Pauli postulated, the complete wavefunction must change sign.
Mathematically, this is written as
(1, 2) (2, 1)378 CHAPTER 12 Atoms and Molecules