Physical Chemistry Third Edition

758 17 The Electronic States of Atoms. I. The Hydrogen Atom

representation of the spin coordinates, we cannot write any mathematical formulas for

the functions or for the operators, but assign their properties to conform to the standard

pattern of angular momentum eigenvalues fors 1 /2. The spin functionsαandβ

are defined to be eigenfunctions of̂S^2 , the operator for the square of the spin angular

momentum:

̂S^2 αh ̄^2 (1/2)(3/2)αh ̄^23

4

α (17.7-8)

̂S^2 βh ̄^2 (1/2)(3/2)βh ̄^23

4

β (17.7-9)

The spin functions are also defined to be eigenfunctions of̂Sz, the operator for thez

component of the spin angular momentum:

̂Szαmshα ̄ +h ̄

2

α (17.7-10)

̂Szβmshβ ̄ −h ̄

2

β (17.7-11)

The spin functions are defined to be normalized and orthogonal to each other. Since we

do not have an explicit representation of any independent variables, we simply write

the equations

∫

α(1)∗α(1)ds(1)

∫

β(1)∗β(1)ds(1)1 (by definition) (17.7-12)

and

∫

β(1)∗α(1)ds(1)

∫

α(1)∗β(1)ds(1)0 (by definition) (17.7-13)

We interpret the probability density for one electron as follows. If an electron in a

hydrogen atom occupies the spin orbitalψnlm(r 1 ,θ 1 ,φ 1 )α(1), then we assert that the

probability of finding it in the volume elementr 12 sin(θ 1 )dr 1 dθ 1 dφ 1 d^3 r 1 with spin

up is|ψnlm(r 1 ,θ 1 ,φ 1 )|^2 d^3 r 1 and the probability of finding it with spin down is zero. If

an electron in a hydrogen atom occupies the spin orbitalψnlm(r 1 ,θ 1 ,φ 1 )β(1), then we

assert that the probability of finding it in the volume elementd^3 r 1 with spin down is

|ψnlm(r 1 ,θ 1 ,φ 1 )|^2 d^3 r 1 and the probability of finding it with spin up is zero.

Inclusion of the intrinsic angular momentum modifies the Schrödinger theory of the

electron so that it agrees adequately with experiment for most purposes. Further modi-

fications can be made to include additional aspects of relativistic quantum mechanics,

such as differences between the energies of “spin up” and “spin down” states for states

of nonzero orbital angular momentum. We will not discuss thespin–orbit coupling

that produces this effect. It is numerically unimportant for atoms in the first part of the

periodic table, but it is important in heavy atoms.^7

We are now able to apply the concept of a complete set of commuting observables

to the hydrogen atom. As explained in Chapter 16, measurement of a complete set of

(^7) Pilar,op. cit., p. 301ff (note 1); K. Balasubramanian,J. Phys. Chem., 93 , 6585 (1989).