The Stern-Gerlach
Experiment
and
the Discovery(?) of
the Electron Spin
Sandip Pakvasa
University of Hawaii
Honolulu
Hawaii
UH Manoa , Sept. 17, 2015
In many textbooks, we learn about
the Stern-Gerlach experiment as a classic way to
measure spin and illustrate fundamental quantum
mechanical behaviour e.g. entanglement etc. and
why it is so important in the discussion of the
measurement problem usually early in the first
chapter
e.g.
Bohm(1952)(pioneer)
Merzbacher(1962)
Feynman(1965)
Messiah(1966)
Gottfried(1966)
Baym(1969)
Sakurai(1985)
Abers(2004)
Weinberg(2013)
etc
We also learn that the Stern-Gerlach
experiment
Was performed and published in 1921-2!
A few pages later,
the same textbooks ascribe the “Discovery” of
electron spin to the theoretical proposal by Goudsmit
and Uhlenbeck in 1925!
How come the discovery of electron spin was not
ascribed to Stern-Gerlach? And the due credit not given
to them?
I started worrying about this anomaly only after having
been teaching Quantum Mechanics for over 40 years
and never having questioned it(well, almost never)!
In the process I learnt a lot about the original
experiment, the troubles with its interpretation,
the astoundingly creative career of Otto Stern and
other tidbits which I will share with you today.
(courtsey Xerxes Tata)
Otto Stern was born in Sohrau,Germany in 1888,
and received his Ph. D. in 1912. He joined Einstein in
Prague as his first “post-doc”, and followed Einstein
to Zurich. He became close friend and companion of
Max von Laue (and later many other theorists).
They shared profound misgivings about the atomic
model of Bohr published in 1913. While hiking on a
mountain near Zurich, they took what Pauli later
called “Utlischwur”: ”If this nonsense of Bohr should
prove to be right in the end, we will quit physics”!
Needless to say, Bohr was right AND they did NOT
quit physics!
He spent the war years(1914-18) in the army
and worked as a meteorologist
on the eastern front.
In Frankfurt, he carried out the
measurement of velocities of molecules
emitted from a heated wire and confirmed
the Maxwell-Boltzmann distribution, in an
ingenious experiment.
The molecular beam technique was invented by L.
Dunoyer(1911) but
exploited thoroughly by Stern.
Stern ended up in Frankfurt with Max Born as his mentor.
Finally, in early 1921, Stern thought he had come up with an
idea to disprove the Bohr “orbits”!
In the meantime the Bohr model had been embellished by
Sommerfeld and others to include eliptical orbits, orbital
angular momentum etc. And Debye and Sommerfeld had
described the orbits with magnetic moments orientable in
magnetic fields. Since orbits can only be in certain planes
w.r.t. the magnetic field direction, this was called space
quantisation.
Stern thought he could test this idea experimentally.
In august 1921, he submitted a paper to ‘Zeitschrift fur
physik’ entitled “ A way towards the experimental examination
of spatial quantisation in a magnetic field”
Of course, he was hoping to prove it wrong!
With Sommerfeld’s generalisation,Bohr’s orbits
acquired orbital angular momentum and new
quantum numbers: l and ml .
For a given n, l takes values from 0 to n -1; and
for a given l, ml takes values from l to –l, total
of 2l +1 values. The z-component of orbital
angular momentum takes the values ml = +l, l1,……..-l.
So for l=1, it can be +1,0 or -1. Since the
direction of Lz with respect to an applied
magnetic field is thus quantised, (in this case to
0, 90 or 180 deg.), this was referred to as space
quantisation. This was considered a crucial
prediction of the Bohr-Sommerfeld theory.
Although in Debye-Sommerfeld proposal, for l=1,
there should be three possible values of ml = +1, 0
and -1, and hence three orientations of the orbit; in
1918 in a series of papers on atomic spectra, Bohr
said that he was troubled by the case ml=0, because
it corresponds to the orbit being parallel to the
magnetic field direction and is hence unstable!
(essentially,” I don’t like it!” kind of an argument).
Sommerfeld in his influential textbook repeated
Bohr’s claim.
This is the reason why, when Stern wrote the paper
with his proposed experiment, following this
lead(from eminent theorists), he also predicted that
they would see the beam split in two!
In the paper he already mentions that he and Gerlach
have started to work on the proposed experiment.
He had realised how difficult the experiment was and found a collaborator:
Walter Gerlach who was already in Frankfurt, and a superb experimenter. By
early 1921, they had already started on the design and execution of the
experiment.
They were not encouraged by theorists,
even sympathetic ones like Born! As Born recalled:”In fact it took me quite a
while before I took this idea seriouslyl! I thought that space quantisation was a
kind of symbolic expression for something which you did not understand. But to
take it literally like Stern did, this was his own idea…I tried to persuade Stern
that there was no sense to it, but then he told me that it was worth a try.”
Eventually Born came around and became an enthusiastic supporter of the
experiment.
Although, at one time when they did not see any effect, Debye said, ”surely
you did not really believe that the orientation of orbits will be physically real…”.
A note on Stern’s style of working:
He always had a cigar in one hand, and
he left actual work with hands to others,
as he did not trust his own manual
dexterity!
If there was an imminent crash, he would
raise both his hands and stay away, as he
said that it is better to let things fall
where they may, rather than trying to
interfere.
He described the beneficial effects
of a large wooden hammer that
he kept in his lab and used to
threaten apparatus if it did not
behave!(apparently it worked!)
The experiment turned out to be even more difficult than
anticipated. It took more than a year to complete. There were
also financial difficulties. A series of public lectures by Born and
others were arranged with a charge for admission were arranged
to help defray the cost of the experiment! A friend suggested to
Born to write to a Henry Goldman(in New York) who had family
roots in Frankfurt. He received a charming response with a
cheque for a few hundred dollars , which helped them tide
over…. (Henry Goldman had started the Woolworth chain and
founded Goldman Sachs.) In the meantime, Born had moved to
Gottingen and Stern had moved to Rostock. Gerlach had to travel
to Rostock to show the results to Stern…On one such visit, they
reviewed the results, and in view of disappointing results, decided
to quit! However, Gerlach had to spend the night in Rostock due
to a railway strike, and decided to try one more time when he got
back to Frankfurt!
W. Gerlach and O.Stern, Z. f. Physik.
9,353(1922):
actual values: L=3.5 cm, B=0.1T, gradient of
10T/cm, resultant Δz = 0.22mm. They saw two
lines, and the value of Δz corresponds to a
value of
magnetic moment of exactly one Bohr magneton
as expected! The shift was larger than Stern’s
estimate due to their using a higher field
gradient.
They had confirmed the Bohr-Sommerfeld
theory! Gerlach sent a congratulatory postcard
to Bohr!
Aftermath: It was generally acclaimed as a triumph of the “old” quantum
theory (of course the “new” quantum theory or quantum mechanics or
electron spin were yet to be discovered) and to confirm the reality of
Bohr orbits, and converted the non-believers(including Stern). No one
questioned the two instead of three lines….nor raised any other
questions!(not even Pauli, nor Heisenberg……..etc).
There was an interesting reaction from Einstein and Ehrenfest. In a paper
written a few weeks after the publication of the S-G result, they raised a
question.
They did a semi-classical calculation of the time it would take the atom to
change from one polarity to the other in the magnetic field by releasing
the energy difference as radiation, and found a time of several hundred
years.
The puzzle was that the amount of time spent by the atoms in the magnet
was much smaller, about
10-4 s! They considered various possible explanations….
Although this was BEFORE Quantum Mechanics or the concept of a wave
function; obviously it was an early
example or premonition(?) of wave function collapse!
Incidentally, the fact of there being two, rather than three lines
(expected in the Debye-Sommerfeld ) was not raised!
The fact that the values for the magentic moment agreed with
that expected for an orbit with l =1, and
ml = +1 and -1 was accepted without question.
The value for μ deduced from the observed splitting was exactly
one Bohr magneton, namely
eh/2mc.
As it turned out, the presence of two lines was due to two
possibilities
for the spin and the coincidence of the magnetic moment being
the same was due to the factor of 2 cancelling out with the
factor of ½!
So although the Stern-Gerlach experiment was indeed the first
observation of electron spin, it was credited as having
confirmed space quantisation and Bohr-Sommerfeld model of
the atom and NOT the electron spin!
B. Friedrich and D. Herschbach
durimg a reenactment (2002)
Plaque honoring Stern and Gerlach in
Invention of Electron Spin:
By 1924-5 the anomalous Zeeman effect, and generally the detailed
spectra including the fine structure had become very confusing. There
were proposals for ½ integral quantum numbers with empirical formulas
such as the Lande g-factor, which was essentially data fitting.
As early as 1921, Arthur Compton had raised the notion of a magnetic
electron, which had spin. But the first serious proposal of electron spin
was due to Ralph Kronig in 1925. He was just 21, and after finishing his
Ph. D. at Columbia in NY, was joining Pauli as his assistant in Tubingen.
When he arrived, he had the idea for electron spin with angular
momentum of s=h/2 , which could take two values along any given
direction, but with an associated magnetic moment
μ = (ge/2mc)s, where g = 2, rather than 1 as it is in the case of orbital
magnetic moment, to explain the anomalous Zeeman effect. Pauli was
very negative and angry and rejected his proposal, and he got a similar
reception from Heisenberg, Bohr etc.; Kronig did not publish his idea and
put his notes away. After
the acceptance of Goudsmit-Uhlenbeck publication of the same idea, he
published a summary of the objections!
(μL = (e/2mc)L, μS = g(e/2mc)S)
Classical electron radius=r= e2/mc2
Compton radius of electron = rC=
h/mc= r /α
where α= 1/137 is fine structure
constant.
To get the angular momentum =mvr
to be h/2, the velocity at the
periphery has to be
= 1/2(c/α) = (137/2)c for r
But only c/2 for rC !
A few months later, Goudsmit and Uhlenbeck, students of
Ehrenfest in Leiden had the same idea as Kronig! But in
contrast to Pauli, Ehrenfest was very supportive, and said,
write it up and publish it! Both proposals of Kronig and of
Goudsmit-Uhlenbeck suffered from a problem of “factor of
two” as pointed out by Heisenberg.
Namely although the factor g had to be 2 to obtain the
correct splitting in the anomaolous Zeeman effect, g
had to be 1 in the calculation of fine structure splitting to
get the right value. When Ehrenfest took Goudsmit and
Uhlenbeck to the train station to meet Einstein and told him
about their work and the factor of two problem, Einstein
said without a moment’ s thought that it must be a
relativistic effect! As Thomas showed in a few days!
One of Pauli’s objections was that a spinning sphere with
the spin of (h/2) and a classical charge radius would have
speeds larger than c at the periphery! But if the radius is
Compton, it is not a problem….

When did it become clear that the spin “invented”
by Kronig, Goudsmit and Uhlenbeck in 1925 had already
been observed in 1922 by Gerlach and Stern?:
In 1927, two experiments were done by young graduate
students in Urbana, Illinois and in Aberdeen, Scotland!
R.G.J. Fraser measured the shape of hydrogen atom by
scattering and found it to be spherically symmetric; T. E. Phipps
and J. B. Taylor did a Stern-Gerlach experiment with hydrogen
atoms and confirmed that they behave just like Silver atoms and
split into two beams. They concluded that the atoms were in the
l=0, ground state of the Schrodinger wave function Ψ(1,0,0) and
hence the Stern-Gerlach effect was entirely due to the spin of
the single electron and had nothing to do with the so-called
“space-quantization” or electron orbits in the atom. By then it
was clear that silver and many other atoms with a closed shell
and one electron in the outer shell had zero orbital angular
momentum! This was the first clear statement that Stern-Gerlach
had indeed observed the electron spin!…
By mid ‘30’s most textbooks (e. g.
Slater, 1935, etc.) explained the
initial confusion and the final
clarification giving full credit to SternGerlach. But eventually there ceased
to be any discussion of this detailed
history in most textbooks
except by philosophers and historians
of science who continue to argue to this
day…….
about ontology and epistemology….
One issue raised by Bohm in his 1952 book and later by
Wigner(1963) was whether one could take the two split
beams and do an “inverse”
S-G experiment to restore the original beam?
This was answered by Schwinger, Scully and Englert in
a series of papers in 1988-9..
“Is Spin Coherence like Humpty-Dumpty?”
The answer is yes and no!
Humpty-Dumpty sat on a wall,
Humpty-Dumpty had a great fall;
All the king’s horses and all the king’s men
couldn’t put Humpty-Dumpty together again.
The riddle has a solution : egg!
In the case of spin, the egg can be put together but
the egg-shell has some cracks!
They found that in order to restore the original spin state
to 1 part in 100(i.e. to 1 %) one needed accuracy of 1
part in 105!
Stern-Gerlach for free
electrons?
In the Solvay Conference in 1927, Bohr and
Pauli argued that for free electrons, a S-G
experiment to measure the magnetic moment
was impossible, and furthermore,
for free electrons the concept was
meaningless! The argument was published by
Mott in 1928.
This was reproduced in textbooks such as
Mott, Mott and Massey,and modern ones like
Baym, Gottfried etc… and is generally
accepted as dogma…..
Brillouin’s Longitudinal SGE
1927-8
June 23, 2010
TU Chemnitz
Page 30
Brillouin’s Longitudinal SGE (2)
Velocity along z changes as δB/δz changes until eventually
coming to rest
(α = insertion angle)
very difficult, has never been tried!
Enter Dehmelt!
In the 1970’s, culminating in 1984, Dehmelt and his
team in Seattle,
started experiments with a Penning trap which
seemed to do the impossible as defined by Bohr and
Pauli!
Penning trap is a device which traps charged
particles using a combination of inhomogeneous
electric quadrupole field and an axial magnetic dipole
fields.
Dehmelt calls his way of measuring magnetic
moment CSGE for Continuous Stern-Gerlach
Experiment as opposed to the original one which is
dubbed TSGE for Transient Stern-Gerlach
Experiment!
“Geonium”
Circuit diagram
Classical Motion
CSGE – Schematic
CSGE Results
Transitions between Landau levels, and between magnetic
moment levels
(Single electron)
Dehmelt and co. measured the electron magnetic
moment with fantastic accuracy:
g/2= 1.001159652188(4)
To be compared to the QED
prediction(e.g. Kinoshita) of:
g/2= 1.0011159652133(29)
Leading to Nobel Prize for Dehmelt (1989)
and to admission by old-timers like Peierls
who were around in the ‘20’s that “the
electron is free in the sense intended by
Bohr. This was one of the cases where
Bohr was wrong.”
Although some philosophers of science may still
not be convinced!
CSGE
Longitudinal, like Brillouin proposal
 New detection scheme – frequency
instead of observing changes in
classical particle trajectories
 Greatly increased detection sensitivity
 Essentially free individual electron
whose spin relaxation time is practically
infinite

There is still some interest in trying to
construct an experiment more like
original SGE for free electrons or a la
Brillouin proposal and attempts persist!
Some recent attempts : Batelaan et al.,
Phys. Rev. Lett. 79,45(1997).
G.H.Rutherford and R. Grobe, J. Phys.
A31, 9331(1998).
B. M. Garraway and S. Stenholm, Phys.
Rev. A60, 63(1999)
Turning back to Stern and his
career…….
In 1932-3, Stern decided to measure the
magnetic moment of the proton, known
also to have spin ½.
He had to use clever tricks, as proton is also
charged and would have Lorentz force. He
chose neutral H2 molecules which have two
protons each.(In hydrogen atom, the 2000
times larger electron moment overwhelms that
of the proton!) The trick was to choose parahydrogen, in which the protons spins are
aligned so their moments add up, the electrons
in ground state have total spin zero and
orbital angular momentum also zero, The small
correction due to non zero temperature and
rotational motion is inferred from the study of
ortho-hydrogen. So one measures twice the
magnetic moment of the proton.
When he had announced his intention to
do this measurement, he was
berated by theorists(including Pauli) for
wasting his time since one “knew” the
answer as it was given by Dirac equation,
namely one nuclear magneton= eh/2Mc
As it turned out, he was vindicated when he
found a value almost 3 times bigger than
the expectation!
Later on in the mid-fifties this large
anomalous value led to some question about
the existence of anti-proton since proton
apparently did not obey Dirac equation!
Otto Stern(1888-1969)
– Note: Stern received the Nobel
Prize in 1943 "for his contribution
to the development of the
molecular ray method and his
discovery of the magnetic moment
of the proton". He did the
following:
– 1. Tested for the first time the
Maxwell-Boltzmann
distribution(speed as a function of
the temperature) (1920)
– 2. “discovered” electron spin albeit
unknowingly!(1922)
– 3. experimentally tested using
diffraction the de Broglie
relationship for many atoms and
molecules for the first time(1926)
– 4. measured the magnetic
moment of the proton (1933)
Students, Assistants and “post-docs” of
Stern who went on to win Nobel Prizes:
Isidore Rabi, 1944,
Felix Bloch, 1952
Polykarp Kusch, 1955,
Emilio Segre, 1959
Norman Ramsey, 1989
The Number of times Stern was
nominated for Nobel Prize:81
He has been justly called:
“The Founding Father of experimental
atomic physics”
It can be claimed that descendents of the SternGerlach experiment are legion…..including
nuclear magnetic resonance, optical pumping, atomic
clocks, anomalous moments, and many other practical
applications ……
Subsequent time-line:
In 1934, after the Nazi takeover, Stern moved
to USA at the Carnegie Inst. Tech. after
ensuring that all in his lab had found secure
positions. At Carnegie with an ill-equipped
lab he couldn’t perform any interesting expts.
He retired to Berkeley in 1945, and stayed
there until his death in1969
As for Gerlach, he worked on radiometric
pressure, and material science. In 1944, he
became head of German Nuclear Research
Program and was detained at the Farm Hall
along with other German Physicists in 1945…
My takeaway from Stern’s career?
It is OK to have theorists as friends, even be
familiar with what they are talking about, but
don’t take them too seriously, certainly don’t pay
(too much) attention to their advice, and above
all; any experiment that can be done is worth
doing! (quote from Gerlach: ‘there is no such
thing as an experiment that is too dumb!”)
(Actually, he was trained as a theorist, had many close
theorist friends, including Pauli and would talk to him
when he had problems!)
References
General reviews:
1.
2.
1.
2.
3.
4.
5.
6.
B. Friedrich and D. Hercshbach," How a Bad Cigar Helped Reorient
Atomic Physics", Phys. Today, 56, 53(2003).
J. P. Toennies et al.,” Otto Stern (1888-1969): The Founding
Father of Experimental Atomic Physics”, arXiv:1109.4864.
The rest:
N. Bohr, “On the Constitution of atoms and molecules”, I, II and
III, Phil. Mag. 26, 1, 476, and 857(1913).
A. Sommerfeld, Ann. der Physik, 51, 1(1916).
A. Sommerfeld, Ann. der Physik, 491(1916).
P. Debye, Physik. Zeits. 17, 507(1916).
N. Bohr, "On the quantum theory of line spectra, Part II”,
Mathematisk-Fyske Meddelsoer, Det Kgl. Danske Videnskabernes
Selskab:Shrifter8, 4.1, 37(1918).
A. Sommerfeld, “Atombau und Spektrallinien”,
Braunschweig:Vieweg (1919).
O. Stern, Zeit. fur Physik, 7, 249(1921).
8. W. Gerlach and O. Stern, "Der Experimentalle Nachweis der
Richtungsquantelung im Magnetfeld", Z. fur Physik.,9, 349,
353(1922).
9. A. Einstein and P. Ehrenfest, Zeit. fur Physik, 11 31(1922).
10. A. Lande, Zeit. fur Physik, 5, 231, (1921), 7, 398(19210; 11,
353(1922).
11. A. H. Compton, "the Magnetic Electron", J. Franklin Inst.,
192, 294(1921).
12. S. Goudsmit and G. Uhlenbeck, Nature, 117, 264(1926).
13. L.H. Thomas, Nature, 117, 5141926).
14. R. L. Kronig, Nature, 117, 555(1926).
15. T. E. Phipps and J. R. Taylor, Phys. Rev. 29, 309 (1927); R.
G. J. Frazer, Proc. Roy. Soc., A114, 212(1927).
16. M. O. Scully, J. Schwinger and B. G. Englert, "Is spin
coherence like Humpty- Dumpty?" Phys. Rev. A40,
7.
17. N. Bohr, Collected Works, Vol. 6, J. Kalckar, ed. (N. Holland,
Amsterdam, 1985), pp. 305-350; W. Pauli, Proc. of the 6th
Solvay Conference, (Gauthier-Villars, Paris, 1932), p.217,
reprinted on p. 332 of above.
18. N. F. Mott, Proc. Roy. Soc. A124, 425(1929).
19. N. F. Mott and H. S. W. Massey, “The Theory of Atomic
Collisions”, Oxford, 1933 (1st Ed.).
20. K. Gottfried and T. M. Yan, “Quantum Mechanics”, Springer
(2004), G. Baym, “Quantum Mechanics”, Benjamin (1969).
21. L. Brillouin, Proc. Acad. Nat. Sci. USA, 14, 155(1928).
22. H. Dehmelt, Proc. Natl. Acad. Sc. 83, 2291(1983).
23. O. Stern, Z. Physik. 2, 49(1920), 2, 417(1920).
24. I. Estermann, R. Frisch and O. Stern, Z. Phys. 73, 348(1931).
25. I. Estermann and O. Stern, 85, 17(1933)..