Chemistry 362
Fall 2015
Dr. Jean M. Standard
October 26, 2015
Electron Spin
Stern-Gerlach Experiment, 1921
In 1921, an experiment was performed by Otto Stern and Walther Gerlach in which a beam of silver atoms was
allowed to pass through an inhomogeneous magnetic field, as illustrated in Figure 1.
Figure 1. Apparatus for Stern-Gerlach experiment (from http://hyperphysics.phy-astr.gsu.edu/hbase/spin.html)
Stern and Gerlach used silver atoms in their experiment because the atoms have electron configuration [Kr]4d105s1
and therefore possess one unpaired electron (and the electron is located in an s-type atomic orbital so that there is no
associated orbital angular momentum). It is the behavior of this unpaired electron that the Stern-Gerlach experiment
probed. When the inhomogeneous magnetic field was on, the beam of silver atoms split into two parts, one
deflected up and the other deflected down, as shown in Figure 2.
Figure 2. Images of photographic plates from original Stern-Gerlach experiment for zero magnetic field (left) and
non-zero field (right); (from http://plato.stanford.edu/entries/physics-experiment/figure13.html).
2
Uhlenbeck and Goudsmit Postulate Electron Spin, 1925
In 1925, Uhlenbeck and Goudsmit proposed an explanation for the Stern-Gerlach experiment. They postulated that
the electron possesses an intrinsic angular momentum, referred to as "spin". This intrinsic angular momentum gives
rise to a magnetic moment in the electron that interacts with magnetic fields. The electron spin and the related
magnetic moment are quantized such that there are only two possible discrete values. This quantization of the
magnetic moment is what leads to the deflection of the beam of silver atoms either up or down in the
inhomogeneous magnetic field.
It is important to note that the term "electron spin" is a bit of a misnomer; the electron cannot be thought of as a tiny
ball of charge spinning about its axis. The surface of the electron would have to be rotating at greater than the speed
of light in order to produce a magnetic moment the size of that measured for the electron. Electron spin is an
intrinsic property of the particle, like charge or mass; however, in the case of spin, the property is purely quantum
mechanical in nature.
Dirac Equation, 1928
While Uhlenbeck and Goudsmit’s postulate of the existence of an intrinsic angular momentum for the electron
provided an explanation of the Stern-Gerlach experiment, there was no solid theoretical framework for their
postulate at the time. However, in 1928 Paul Dirac (Figure 3) developed a relativistic version of quantum
mechanics.
Figure 3. P. A. M. Dirac (from http://discover.positron.edu.au/popups/paul-dirac).
Using his relativistic equation, Dirac was able to derive the existence of electron spin and show that the properties of
electron spin were in accord with Uhlenbeck and Goudsmit’s postulate and the results of the Stern-Gerlach
experiment.
3
Electron Spin as an Intrinsic Angular Momentum
Because spin corresponds to an intrinsic angular momentum of the electron, it behaves in very much the same way
as orbital angular momentum. That is, electron spin is a vector quantity that possesses similar eigenvalue equations,
commutators, and measurement relations. For example, there are two eigenvalue equations that have the form
Sˆ 2 ψ spin = ! 2 s( s + 1) ψ spin
Sˆz ψ spin = ms! ψ spin .
€
Here, s corresponds to the spin quantum number (analogous to the orbital angular momentum quantum number ℓ )
m
and ms corresponds to the magnetic
€ spin quantum number (analogous to the magnetic quantum number ℓ ). The
major difference between the electron spin quantum numbers and the orbital angular momentum quantum numbers
is that s is a half integer instead of an integer,
€
€
s = 12
ms = ± s = ± 12 .
Because there are only two possible values of the magnetic spin quantum number corresponding to the two
ψ
deflections observed in the Stern-Gerlach
€ experiment, there are two electron spin eigenfunctions, spin . One spin
1
1
eigenfunction corresponds to the "spin up" state, with quantum numbers s = 2 , ms = 2 , and is usually labeled α.
1
The other spin eigenfunction corresponds to the "spin down" state, with quantum numbers s = 2 , m s
usually labeled β. The eigenvalue equations for the spin up and spin down states may be€written as
€
2
3 2
ˆ
S α = 4!α
Sˆ 2 β = 43 ! 2 β
Sˆzα =
1
2
= − 12 , and is
€
Sˆz β = − 12 !β .
!α
The spin up and spin down eigenfunctions are normalized and orthogonal,
€
∫ α α dσ
∫ α β dσ
*
=
*
=
∫ β β dσ
∫ β α dσ
*
= 1
*
= 0.
Here, the volume element dσ is used to denote integration over the electron spin variable.
€
The commutator relationships for the spin operators also are similar to those for the orbital angular momentum.
Each component€operator commutes with the spin squared; however, the components do not commute among
themselves.
[Sˆ , Sˆ ] = [Sˆ , Sˆ ] = [Sˆ , Sˆ ]
[Sˆ , Sˆ ] = i!Sˆ ; [Sˆ , Sˆ ] =
2
x
€
x
y
2
2
y
z
y
z
z
= 0
i!Sˆx ;
[Sˆ , Sˆ ]
z
x
= i!Sˆy .