UH Manoa , Sept. 17, 2015

The Stern-Gerlach

Experimentand

the Discovery(?) ofthe Electron Spin

Sandip PakvasaUniversity of Hawaii

HonoluluHawaii

In many textbooks, we learn aboutthe Stern-Gerlach experiment as a classic

way to measure spin and illustrate fundamental quantum mechanical behaviour 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 “Utliscwur”: ”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!

In Frankfurt, he carried out the measurement of velocities of

molecules emitted from a heated wire and confirmed the Maxell-Boltzmann

distribution, in an ingenious experiment.

The molecular beam technique was invented by L. Dunoyer(1911) but

exploited thoroughly by Stern.

He spent the war years(1914-18) in the army and worked as a meteorologist

on the eastern front.

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, l-1,……..-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 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…”.

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!)

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

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.

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 earlyexample 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. Herschbachdurimg a reenactment (2002)

Plaque honoring Stern and Gerlach in Frankfurt

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. Afterthe 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)

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 rBut only c/2 for rC !

Classical electron radius=r= e2/mc2

Compton radius of electron = rC= h/mc= r /α

where α= 1/137 is fine structure constant.

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, ghad 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 byscattering 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 Stern-Gerlach. 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….

.

So a discovery(Stern-Gerlach, 1922) became an invention(Goudsmit-

Uhlenbeck,1925)This is opposite of the usual/normal

order: inventionfollowed by discovery

e.g. Quarks invented in 1963(Gell-Mann-Zweig), discovered in 1968-

9(SLAC-MIT)Hence the confusion……

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!

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

became the generally accepted dogma…..

Stern-Gerlach for free electrons?

Brillouin’s Longitudinal SGE1927-8

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!

Circuit diagram

Classical Motion

“Geonium”

CSGE – Schematic

CSGE ResultsTransitions between Landau levels, and between

magnetic moment levels

(Single electron)

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!

Dehmelt and co. measured the electron magnetic moment with fantastic accuracy:

g/2= 1.001159652188(4)

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

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…….

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!

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 para-

hydrogen, 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.

In 1932-3, Stern decided to measure the magnetic moment of the proton,

known also to have spin ½.

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!

When he had announced his intention to do this measurement,

he wasberated 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

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)

The Number of times Stern was nominated for Nobel

Prize:81He has been justly called:

“The Founding Father of experimental atomic physics”

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

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…

It can be claimed that descendents of the Stern-Gerlach experiment are legion…..including

nuclear magnetic resonance, optical pumping, atomic clocks, anomalous moments, and many

other practical applications ……

(Actually, he was trained as a theorist, had many close theorist friends, including Pauli and

would talk to him when he had problems!)

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!”)

References General reviews:1. B. Friedrich and D. Hercshbach," How a Bad Cigar Helped

Reorient Atomic Physics", Phys. Today, 56, 53(2003).2. J. P. Toennies et al., Otto Stern (1888-1969): The Founding

Father of Experimental Atomic Physics, arXiv:1109.4864. The rest:3. N. Bohr, On the Constitution of atoms and molecules, I, II

and II, Phil. Mag. 26, 1, 476, and 857(1913).4. A. Sommerfeld, Ann. der Physik, 51, 1(1916).5. A. Sommerfeld, Ann. der Physik, 491(1916).6. P. Debye, Physik. Zeits. 17, 507(1916).7. N. Bohr, "On the quantum theory of line spectra, Part II,

Mathematisk-Fyske Meddelsoer, Det Kgl. Danske Videnskabernes Selskab:Shrifter8, 4.1, 37(1918).

8. A. Sommerfeld, Atombau und Spektrallinien, Braunschweig:Vieweg (1919).

7. 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, 1775(1988).

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)..