1. No category

Chemistry 460
Spring 2015
Dr. Jean M. Standard
March 25, 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 (in addition, 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).
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Uhlenbeck and Goudsmit Postulate Electron Spin, 1925
In 1925, Uhlenbeck and Goudsmit (shown in Figure 3 with their advisor Kramers) 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 a
magnetic field. The electron spin and the related magnetic moment are quantized such that there are only two
possible discrete values for each. 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. Calculations predict that 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 experimentally 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.
Figure 3. Uhlenbeck (left), Kramers (center), Goudsmit (right) (from
http://en.wikipedia.org/wiki/George_Uhlenbeck).
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 4) developed a relativistic version of quantum
mechanics.
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Figure 4. 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 the results of the Stern-Gerlach experiment and also with Uhlenbeck and
Goudsmit’s postulate.
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 for electron spin 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 )
and ms corresponds to the magnetic
€ spin quantum number (analogous to the magnetic quantum number mℓ ). 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 =
€
1
2
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
€
Sˆ 2α = 43 ! 2α
Sˆ 2 β = 43 ! 2 β
Sˆzα =
€
1
2
!α
€
Sˆz β = − 12 !β .
= − 12 , and is
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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
2
2
x
x
y
y
z
z
y
z
= 0
i!Sˆx ;
[Sˆ , Sˆ ]
z
x
= i!Sˆy .
Raising and Lowering
Operators for Electron Spin
€
Raising and lowering operators can be defined for electron spin. These operators are Sˆ+ and Sˆ− and they are
defined as
Sˆ+ = Sˆx + iSˆy
Sˆ− = Sˆx − iSˆy .
€
€
The raising operator works on the spin eigenfunctions in the following way,
€
€
€
Sˆ+ β
Sˆ+ α
= ! α
= 0.
The raising operator raises the spin down eigenfunction β (with ms = −1 / 2 ) to the spin up eigenfunction α (with
ms = 1 / 2 ). The raising operator kills off the spin up eigenfunction α because it cannot be raised any higher
€
( ms = 1 / 2 is as high as the quantum number can go).
€
€
€
The lowering operator works on the spin eigenfunctions in€the following way,
Sˆ− β
Sˆ− α
= 0
= ! β .
The lowering operator kills off the spin down eigenfunction β because it cannot be lowered any further ( ms = −1 / 2
is as low as the quantum number can go). The lowering operator lowers the spin up eigenfunction α (with
€
ms = 1 / 2 ) to the spin down eigenfunction β (with ms = −1 / 2 ).
€
€
€
€
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