Principles of Quantum Mechanics   as Applied to Chemistry and Chemical Physics

Principles of Quantum Mechanics as Applied to Chemistry and Chemical Physics


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the same interpretation to the motion of an electron and regard the spin
as the angular momentum associated with the electron revolving on its axis.
However, as Dirac\u2019s relativistic quantum theory shows, the spin angular
momentum is an intrinsic property of the electron, not a property arising from
any kind of motion. The electron is a structureless point particle, incapable of
\u2018spinning\u2019 on an axis. In this regard, the term \u2018spin\u2019 in quantum mechanics can
be misleading, but its use is well-established and universal.
Prior to Dirac\u2019s relativistic quantum theory, W. Pauli (1927) showed how spin
could be incorporated into non-relativistic quantum mechanics. Since the
subject of relativistic quantum mechanics is beyond the scope of this book, we
present in this chapter Pauli\u2019s modification of the wave-function description so
7.1 Electron spin 195
as to include spin. His treatment is equivalent to Dirac\u2019s relativistic theory in
the limit of small electron velocities (v=c! 0).
7.2 Spin angular momentum
The postulates of quantum mechanics discussed in Section 3.7 are incomplete.
In order to explain certain experimental observations, Uhlenbeck and Goudsmit
introduced the concept of spin angular momentum for the electron. This
concept is not contained in our previous set of postulates; an additional
postulate is needed. Further, there is no reason why the property of spin should
be confined to the electron. As it turns out, other particles possess an intrinsic
angular momentum as well. Accordingly, we now add a sixth postulate to the
previous list of quantum principles.
6. A particle possesses an intrinsic angular momentum S and an associated magnetic
moment Ms. This spin angular momentum is represented by a hermitian operator S^
which obeys the relation S^ 3 S^ ˆ i"S^. Each type of particle has a fixed spin
quantum number or spin s from the set of values s ˆ 0, 1
2
, 1, 3
2
, 2, . . . The spin s for
the electron, the proton, or the neutron has a value 1
2
. The spin magnetic moment for
the electron is given by Ms ˆ ÿeS=me.
As noted in the previous section, spin is a purely quantum-mechanical concept;
there is no classical-mechanical analog.
The spin magnetic moment Ms of an electron is proportional to the spin
angular momentum S,
Ms ˆ ÿ gse
2me
S ˆ ÿ gsìB
"
S (7:1)
where gs is the electron spin gyromagnetic ratio and the Bohr magneton ìB is
defined in equation (5.82). The experimental value of gs is 2.002 319 304 and
the value predicted by Dirac\u2019s relativistic quantum theory is exactly 2. The
discrepancy is removed when the theory of quantum electrodynamics is
applied. We adopt the value gs ˆ 2 here. A comparison of equations (5.81) and
(7.1) shows that the proportionality constant between magnetic moment and
angular momentum is twice as large in the case of spin. Thus, the spin
gyromagnetic ratio for the electron is twice the orbital gyromagnetic ratio. The
spin gyromagnetic ratios for the proton and the neutron differ from that of the
electron.
The hermitian spin operator S^ associated with the spin angular momentum S
has components S^x, S^ y, S^z, so that
196 Spin
S^ ˆ iS^x ‡ jS^ y ‡ kS^z
S^2 ˆ S^2x ‡ S^2y ‡ S^2z
These components obey the commutation relations
[S^x, S^ y] ˆ i"S^z, [S^ y, S^z] ˆ i"S^x, [S^z, S^x] ˆ i"S^ y (7:2)
or, equivalently
S^ 3 S^ ˆ i"S^ (7:3)
Thus, the quantum-mechanical treatment of generalized angular momentum
presented in Section 5.2 may be applied to spin angular momentum. The spin
operator S^ is identified with the operator J^ and its components S^x, S^ y, S^z with
J^x, J^ y, J^z. Equations (5.26) when applied to spin angular momentum are
S^2jsmsi ˆ s(s‡ 1)"2jsmsi, s ˆ 0, 12, 1, 32, 2, . . . (7:4)
S^zjsmsi ˆ ms"jsmsi, ms ˆ ÿsÿ s‡ 1, . . . , sÿ 1, s (7:5)
where the quantum numbers j and m are now denoted by s and ms. The
simultaneous eigenfunctions jsmsi of the hermitian operators S^2 and S^z are
orthonormal
hs9m9sjsmsi ˆ äss9äms m9s (7:6)
The raising and lowering operators for spin angular momentum as defined by
equations (5.18) are
S^‡ \ufffd S^x ‡ iS^ y (7:7a)
S^ÿ \ufffd S^x ÿ iS^ y (7:7b)
and equations (5.27) take the form
S^‡jsmsi ˆ

(sÿ ms)(s‡ ms ‡ 1)
p
"js, ms ‡ 1i (7:8a)
S^ÿjsmsi ˆ

(s‡ ms)(sÿ ms ‡ 1)
p
"js, ms ÿ 1i (7:8b)
In general, the spin quantum numbers s and ms can have integer and half-
integer values. Although the corresponding orbital angular-momentum quan-
tum numbers l and m are restricted to integer values, there is no reason for
such a restriction on s and ms.
Every type of particle has a specific unique value of s, which is called the
spin of that particle. The particle may be elementary, such as an electron, or
composite but behaving as an elementary particle, such as an atomic nucleus.
All 4He nuclei, for example, have spin 0; all electrons, protons, and neutrons
have spin 1
2
; all photons and deuterons (2H nuclei) have spin 1; etc. Particles
with spins 0, 1, 2, . . . are called bosons and those with spins 1
2
, 3
2
, . . . are
fermions. A many particle system of bosons behaves differently from a many
7.2 Spin angular momentum 197
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 ms to designate the state. Thus,
for a particle in a specific spin state, the state function is denoted by
Ø(r, ms, t), where ms has only the (2s‡ 1) possible values ÿs", (ÿs‡ 1)",
. . . , (sÿ 1)", s". While the variables r and t have a continuous range of
values, the spin variable ms has a finite number of discrete values.
For a particle that is not in a specific 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 jsmsi,
Ø(r, ó , t) ˆ
Xs
msˆÿs
Ø(r, ms, t)jsmsi (7:9)
If Ø(r, ó , t) is normalized, then we have
hØjØi ˆ
Xs
msˆÿs
…
jØ(r, ms, t)j2 dr ˆ 1
where the orthonormal relations (7.6) have been used. The quantity
jØ(r, ms, t)j2 is the probability density for finding the particle at r at time t
with the z-component of its spin equal to ms". The integral
„ jØ(r, ms, t)j2 dr
is the probability that at time t the particle has the value ms" 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
case s ˆ 1
2
is the most important for studying chemical systems. For s ˆ 1
2
there
are only two eigenfunctions, j1
2
, 1
2
i and j1
2
, ÿ1
2
i. For convenience, the state s ˆ 1
2
,
ms ˆ 12 is often called spin up and the ket j12, 12i is written as j"i or as jÆi.
Likewise, the state s ˆ 1
2
, ms ˆ ÿ12 is called spin down with the ket j12, ÿ12i
often expressed as j#i or jâi. Equation (7.6) gives
hÆjÆi ˆ hâjâi ˆ 1, hÆjâi ˆ 0 (7:10)
The most general spin state j÷i for a particle with s ˆ 1
2
is a linear com-
bination of jÆi and jâi
198 Spin
j÷i ˆ cÆjÆi ‡ câjâi (7:11)
where cÆ and câ are complex constants. If the ket j÷i is normalized, then
equation (7.10) gives
jcÆj2 ‡ jcâj2 ˆ 1
The ket j÷i may also be expressed as a column matrix, known as a spinor
j÷i ˆ cÆ
câ
\ufffd \ufffd
ˆ cÆ 10
\ufffd \ufffd
‡ câ 01
\ufffd \ufffd
(7:12)
where the eigenfunctions jÆi and jâi in spinor