Page 207 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

198 Spin

particle system of fermions. This quantum phenomenon is discussed in Chap-
ter 8.
The state of a particle with zero spin (s ˆ 0) may be represented by a state
function Ø(r, t) of the spatial coordinates r and the time t. However, the state
of a particle having spin s (s 6ˆ 0) must also depend on some spin variable. We
select for this spin variable the component of the spin angular momentum
along the z-axis and use the quantum number m s to designate the state. Thus,
for a particle in a speci®c spin state, the state function is denoted by
Ø(r, m s , t), where m s has only the (2s ‡ 1) possible values ÿs",(ÿs ‡ 1)",
... ,(s ÿ 1)", s". While the variables r and t have a continuous range of
values, the spin variable m s has a ®nite number of discrete values.
For a particle that is not in a speci®c spin state, we denote the spin variable
by ó. A general state function Ø(r, ó, t) for a particle with spin s may be
expanded in terms of the spin eigenfunctions jsm s i,
s
X
Ø(r, ó, t) ˆ Ø(r, m s , t)jsm s i (7:9)
m s ˆÿs
If Ø(r, ó, t) is normalized, then we have
s
X 2
hØjØiˆ jØ(r, m s , t)j dr ˆ 1
m s ˆÿs
where the orthonormal relations (7.6) have been used. The quantity
2
jØ(r, m s , t)j is the probability density for ®nding the particle at r at time t
„
2
with the z-component of its spin equal to m s ". The integral jØ(r, m s , t)j dr
is the probability that at time t the particle has the value m s " for the z-
component of its spin angular momentum.

7.3 Spin one-half
Since electrons, protons, and neutrons are the fundamental constituents of
atoms and molecules and all three elementary particles have spin one-half, the
1
1
case s ˆ is the most important for studying chemical systems. For s ˆ there
2 2
1
1 1
1
1
are only two eigenfunctions, j , i and j , ÿ i. For convenience, the state s ˆ ,
2
2
2
2 2
1 1 1
m s ˆ is often called spin up and the ket j , i is written as j"i or as jái.
2 2 2
1 1 1 1
Likewise, the state s ˆ , m s ˆÿ is called spin down with the ket j , ÿ i
2 2 2 2
often expressed as j#i or jâi. Equation (7.6) gives
hájáiˆhâjâiˆ 1, hájâiˆ 0 (7:10)
1
The most general spin state j÷i for a particle with s ˆ is a linear com-
2
bination of jái and jâi