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The discovery of quantized electron spin.
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Mark J. Moody
Honors Quantum Mechanics
4-26-2013
The History of Electron Spin
Contrary to common sense and semantics, electrons do not physically spin. Electron spin
is an intrinsic form of angular momentum carried by an electron, the name “spin” carrying over
from a classical analog reminiscent of a planet not only orbiting about a celestial body, but also
spinning on its axis. Over time, this conception proved incorrect, but the name remains. In
reality, no part of an electron is physically spinning and we have no entirely accurate classical
analog to describe the property of spin. Unlike physically spinning objects, electron spin can
take on half integer values, the direction of “spin” can be changed but its “speed” cannot, and the
magnetic induction necessary to explain the observed magnetic dipole moment of an electron
requires a different internal charge distribution than the measured, accepted value. Electron spin
is a purely quantum mechanical effect. Electrons have been shown through Young’s double slit
diffraction experiment and observed discreet atomic energy levels to exhibit wave like behavior,
and are best described as a wave function generally represented by Ψ nlm with an additional term
outside the wave function representative of spin since spin does not alter the wave function itself.
Everything we know about a particle can be derived from the particle’s wave function and spin.
Four quantized values known as quantum numbers characterize the complete and unique
quantum wave function of any electron: n, l, m, and s. The principal quantum number, n, is a
positive integer which determines the size of the electron orbital and the energy of the electron(s)
associated with that shell. Each electron shell can hold 2 n2 electrons. As n increases, orbital
size increases, allowing an electron to exist at a greater radial distance from the positively
charged nucleus. This separation of charge increases the potential energy between the electron
and nucleus and so determines the electron’s energy for that state. For single electron systems
such as is found in a hydrogen atom, an electron’s energy depends exclusively upon n. For multi
electron systems, the electron’s energy also depends upon l.
The azimuthal quantum number, l, is an integer ranging from zero to n−1 which defines
angular momentum and the three dimensional shape of the orbital. Similar to the above
described method for organizing orbitals into shells according to n, orbitals within a shell are
also grouped into subshells according to l. The letters s, p, d, and f are typically used to describe
the value for l, zero being s, one being p, two being d, three being f, and any integer four or
greater following alphabetically using g,h,i and so on. The letters s, p, d, and f originally came
from descriptions of atomic spectra, and indicate sharp, principal, diffuse, and fundamental,
respectively. An orbital with principal quantum number 2 and azimuthal quantum number 1
would be a 2 p orbital.
The magnetic quantum number, m, is an integer ranging from –l to l describing the spatial
orientation of the electron orbital, loosely indicating the direction of the angular momentum
vector. While m does not influence the electron’s energy, it affects the space quantization of the
electron’s probability cloud. The 2l+1 values for m restrict the portion of the total angular
momentum along a particular quantization axis in agreement with Heisenberg’s uncertainty
principle. For each subshell l, there are 2l+1 unique spatial orientations for the comprising
orbitals. As the simplest example, the z component of an electron’s angular momentum, with
respect to a standard Cartesian coordinate system, upon reduction, is given by L z=mℏ. [2]
The azimuthal quantum number yields the quantum mechanical analog to classical
angular momentum r × p where p=mv classically and p=−iℏ∇ quantum mechanically. For
both cases r is the radial distance from the center of orbit and p is momentum. In the first case,
m represents the mass of the orbiting object and v represents its velocity. In the quantum
mechanical case,i=√−1 , ℏ= h2 π
where h=6.626 e−34 m2 kgs
is Planck’s action quantum, and
∇= ∂∂ x
+ ∂∂ y
+ ∂∂ z
is an operator (shown in Cartesian coordinates) that acts upon the wave
function Ψ . This electron orbital angular momentum is very similar to the classical notion of
orbital angular momentum. While orbital angular momentum is derived from an electron’s
rotating trajectory about a nucleus, spin angular momentum is inferred from experiments such as
the Stern-Gerlach experiment, in which orbital angular momentum cannot account for all of the
observed angular momentum effects.
In 1922, Otto Stern and Walther Gerlach performed their now famous experiment on the
deflection of particles. The Stern-Gerlach experiment shows that electrons and atoms have
intrinsic quantum properties, and exhibits that measurements made on a quantum mechanical
system affect the system being measured. This means that a wave function initially existing as a
superposition of several eigenstates collapses to a single determinate eigenstate upon interaction
or measurement (indeed, measurement of any sort requires interaction). It is this quantum
property that relates the wave function of a physically realizable object with the classical
observables position and momentum.
The Stern-Gerlach experiment involves sending a beam of particles through an
inhomogeneous magnetic field and observing their deflection. In such a field, there is a force (in
addition to the standard torque) exerted on a magnetic dipole, the force being given by ¿∇ (μ ∙ B)
, where μ represents the magnetic dipole and B represents the magnetic vector field.
Were the dipole produced by physical spin, one would expect the distribution of their
spin angular momentum vectors to be continuous. Interestingly, Stern and Gerlach observed
discreet, consistent values of deflection, either up or down. This means that such particles have
quantized magnetic moments of two possible values.
This same experiment was performed for electrons, but since electrons are charged
particles, their motion through a magnetic field deflects their trajectory into a curved path
proportional to the electron’s velocity with an exerted Lorentz force F=qv × B where q
represents charge and v represents velocity. An appropriate electric field is used to cancel this
force so that measurements can be made regarding the previously defined force in a consistent
manner. Electrons were measured to have two possible spin values, ±ℏ2
, meaning that they are
spin 12
particles, though this interpretation was not discovered until later. [1]
In mid-September of 1925, George Uhlenbeck and Samuel Goudsmit discovered what
has become known as electron spin. Goudsmit believed that a great deal of auspicious chance
accompanied himself and Uhlenbeck in their discovery. While Goudsmit was a doctoral student
at Leiden University, he once needed to accompany his father on a business trip to Germany.
Goudsmit’s professor, Paul Ehrenfest preferred that Goudsmit not interrupt his studies and upon
learning Goudsmit’s destination explained: “Nearby is a university and there is a spectroscopist,
Paschen. You are interested in spectroscopy; go and have a look”. Paschen treated Goudsmit
more as a colleague than a student and showed Goudsmit the experimental setup he used for the
spectral line of ionized helium. At the time, Goudsmit didn’t understand the experiment in its
entirety, but noted a few interesting attributes of the helium spectral lines: fine structure, the
Sommerfield relativistic explanation, and a forbidden component that was clearly observable.
Many theoreticians simply excluded the forbidden component claiming it was due to inaccuracy
in experimentation.
Soon, Goudsmit’s interest in spectroscopy let him to discover a formula for the doublets
in the helium spectra, and he made the claim that his formula was identical to the formula used
by Sommerfeld for X-ray doublets. Doublets arise from the observation that certain spectral
lines of an excited, ionized gas, split into separate lines in the presence of a strong magnetic
field. This is also more generally known as the Zeeman effect. Upon telling Ehrenfest about his
formula, Ehrenfest saw that it lacked supporting evidence and derivation, but not wanting to
discourage Goudsmit, offered to have it published. Goudsmit’s main article on the topic was
published in Holland and in French, resulting in very few people reading the article.
Nevertheless, as a young graduate student, Goudsmit was quite proud of the publication. Two
and a half years later, Robert Millikan developed the same formula, supported by experimental
data confirming its accuracy.
Soon thereafter, Ralph Kronig traveled from America to Leiden University and
collaborated with Goudsmit in his spectroscopy work including the intensities of the Zeeman
effect. Quantum theory was in formulation, and quantum mechanics as a complete subject of
study did not yet exist. Goudsmit and Kronig made educated guesses and tested various
formulae, developing a feel for each test’s correctness. When Ehrenfest learned about
Goudsmit’s inexact testing methods, he sent Goudsmit to Amsterdam to serve as a part-time
assistant to Pieter Zeeman, who took a more formal approach in his spectroscopy testing.
During this time, Goudsmit recognized that the recent publication of the Pauli principle,
despite being one of the most important publications to date in physics, was cumbersome and
difficult to understand. Goudsmit felt the principle was more difficult to understand than
necessary, and wrote that Pauli’s principle becomes easier to digest if two new quantized
numbers are introduced to the system: ml and ms , with ms always being ±12
(Goudsmit
originally used 1 and 0, but the effect is the same and ±12
is the currently accepted value).
Using these quantized numbers with the Pauli principle significantly simplifies it. Goudsmit’s
change didn’t fundamentally alter the Pauli principle and is equivalent to a simplistic linear
transformation. Though the fundamentals of the Pauli principle weren’t altered, this new form
rendered understanding and teaching much more manageable. By the spring of 1925, the
introduction of ml and ms as quantum numbers supported that electrons possess quantized spins,
but this feature went unnoticed. Goudsmit wrote a letter to Kronig explaining his discovery of
the simplified Pauli principle, but Kronig seemed uninterested, writing back to Goudsmit about
several other matters, never mentioning Goudsmit’s discovery.
During this time, George Uhlenbeck interrupted his research to take a position as tutor for
the children of the Netherlands Ambassador in Rome. Ehrenfest introduced Uhlenbeck to
Goudsmit, telling Goudsmit that Uhlenbeck had learned classical physics in Italy, and
recommending the two should work together for Uhlenbeck to learn about atomic structure and
spectral analysis. Goudsmit suspected that Ehrenfest really intended the reverse—that Ehrenfest
wanted Goudsmit to learn some formal classical physics from Uhlenbeck.
While Uhlenbeck initially understood little regarding this new physics of atomic spectra,
he helped Goudsmit with an article he had promised to write for “Physica”. Since Uhlenbeck
was unfamiliar with the subject, he asked many questions that Goudsmit hadn’t previously
considered, leading to some significant results.
The first result was a new interpretation of the hydrogen spectrum, replacing and
updating the Sommerfeld hydrogen spectrum. Since Goudsmit had so familiarized himself with
atomic spectra intensity rules and such, he was able to essentially guess which formulae they had
to use. Goudsmit and Uhlenbeck used the classical expressions, with the exception of using half
integral quantum numbers rather than integral quantum numbers. This method precisely fit the
hydrogen spectrum, even predicting Paschen’s forbidden spectral line as though it would be
present, and indeed, it was. The forbidden spectral line was not in fact forbidden or brought
about by erroneous measurements, but a valid spectral line that Goudsmit and Uhlenbeck’s
formulae predicted. Goudsmit and Uhlenbeck had little idea how important their result would
become, but nevertheless published their results in Dutch in the “Physica” journal. They also
published a paper in “Zeitschrift fur Physik” on the quantum vectors L and S.
Some time later, Goudsmit introduced Uhlenbeck to the Pauli principle using his ml and
ms quantum numbers. Uhlenbeck recognized that this quantization implied a fourth degree of
freedom for the electron, meaning that it would be expected to spin or rotate. Uhlenbeck did not
know of Landé or Heisenberg’s work, since he was trained classically, and Goudsmit did not
know what a degree of freedom was. A degree of freedom is any of the minimum number of
coordinates needed to completely specify the motion of a mechanical system. For example,
Cartesian coordinates in three dimensions x, y, and z (mutually perpendicular) yield three
translational degrees of freedom: up/down, left/right, and forward/backward; and three rotational
degrees of freedom: tilting forward/backward, swiveling left/right, and pivoting side to side.
When Uhlenbeck made this remark, Goudsmit realized this conception fit precisely with the
interpretation of the hydrogen spectrum they had published just four weeks prior.
Electromagnetic induction indicates that electric charges and magnetic fields are
intrinsically related. A moving charge will induce a magnetic field radially about the charge’s
linear path and a constantly changing magnetic flux will induce an electromotive force on nearby
charged particles, resulting in a current when the collection of charged particles move through a
conducting wire. If we treat a system of uniformly distributed charges about a sphere, the
sphere’s rotation would induce a magnetic moment along the axis of rotation. This is the first
conception of electron spin. The electron was believed to be physically spinning, inducing a
magnetic moment. The magnetic moment is necessary, it fully describes and fits the Zeeman-
effect and it came out naturally along with the Landé formulae.
Goudsmit and Uhlenbeck eagerly informed Ehrenfest, and various other physicists how
their model fitted perfectly with the formalism. Goudsmit had an excellent feel for the formulae
of atomic spectra but did not understand all of the rigorous physics in the formal model.
Uhlenbeck, on the other hand, began to ponder about rotating charges. Uhlenbeck went to
Hendrik Lorentz regarding the matter and received the reply, “Yes, that is very difficult because
it causes the self energy of the electron to be wrong.”
Goudsmit and Uhlenbeck had just submitted an article to Ehrenfest, who in turn planned
to send it off for publication in “Naturwissenschaften”. Uhlenbeck then urgently traveled to
Ehrenfest following his discussion with Lorentz, and requested Ehrenfest not to send the paper to
be published. Uhlenbeck had come to realize the impossibility of having an electron that rotates
at the necessary high angular velocity (near the speed of light) to induce the observed magnetic
moment. Unfortunately, Ehrenfest had already submitted the paper for publication including the
now-known erroneous material, and the paper was published under Goudsmit and Uhlenbeck’s
names. When Goudsmit learned of this, Uhlenbeck’s consoled Goodsmit, “Well, that is a nice
idea, though it may be wrong. But you don’t yet have a reputation, so you have nothing to lose.”
Following publication of the paper, Heisenberg wrote a letter to Goudsmit and Uhlenbeck
referring to their courageous publication. Goudsmit did not find the submission courageous at
all, possibly because he had not yet made a public name for himself, or didn’t understand the
consequences of submitting erroneous information. In any case, Heisenberg’s response
incorporated a formula (most of which Goudsmit did not understand, considering the formula
was given without derivation) and a question about what Goudsmit and Uhlenbeck had done
with the factor two. Goudsmit had not the slightest idea of what factor Heisenberg was referring
to or what this implied.
Goudsmit and Uhlenbeck discovered electron spin because it fit very nicely into their
formalism, but they had never constructed a quantitative calculation regarding the size of the
spectral line splitting. If one were to accept the notion of electron spin up or spin down, they
were not sure if the discrepancy in energy would be yielded correctly. They had developed
convenient formulae fitting with previous knowledge, but had no idea if it were even possible to
derive said formulae.
Goudsmit believed the derivation process would be incredibly laborious and difficult, but
we now know that nearly any physics or chemistry student studies the spectral line of hydrogen
at some point and can perform the derivation quite easily. In an odd sense, it seems fortuitous
that they had not worked out the derivation, because the discrepancy of the factor of two would
have led them to believe their formulae to be incorrect and they may never have published their
paper. They believed all the formulae fit together perfectly, but this term seemed to ruin that
notion and would have prevented their publication.
Understandably, this series of events around publishing an error discouraged both
Goudsmit and Uhlenbeck. Fortunately, Lorentz was approaching the fiftieth anniversary of his
doctorate degree, and many prominent physicists joined him for a celebration in Leiden, very
close to where Goudsmit and Uhlenbeck worked. They all met at Ehrenfest’s home and
discussed a myriad of topics, including the problem of electron spin. Goudsmit recounts that he
learned a great deal during these meetings.
Neils Bohr took particular interest in Goudsmit and Uhlenbeck’s formulae and he
reasoned that the factor two could be trivial. Bohr invited Goudsmit to Copenhagen for research
opportunities. Goudsmit considered this a mistake, that Bohr should have invited Uhlenbeck
instead. Goudsmit did not seem to be making progress over six weeks time, so Bohr sent him
back to Hague. While in Copenhagen, Goudsmit met a young man named Thomas who had
become proficient in working with Einstein’s theory of relativity. Thomas took Goudsmit’s
forlmulae and worked out the factor of two that Heisenberg noticed. Thomas informed
Goudsmit that the factor of two represented the relativistic factor and that the formulae were
actually correct, just misunderstood.
Goudsmit told Bohr about the discovery, and Bohr recommended that he stop in
Hamburg on his way home to explain the justification to Pauli. Goudsmit made the attempt, but
since his understanding of the derivation and relativistic theory was inadequate, he failed to
explain it to Pauli. According to Goudsmit, Pauli seemed to have a predisposition for
disbelieving the theory of spin even before the justification, and Goudsmit’s attempt apparently
did nothing to alleviate this. Upon Goudsmit’s return to Leiden, Einstein was still there, and he
again made the attempt to explain his justification. According to Goudsmit, this went even
worse, presumably because Einstein developed the theory of relativity and knew the subject
intimately. Einstein would therefore be able to quickly point out when Goudsmit was misusing it
or misunderstanding it in his supposed justification. Goudsmit left the subject and its
justification alone for the time being, but later received a letter from Pauli explaining that
Thomas had shown him the work and that he was convinced.
Goudsmit met once more with Thomas, but the case was finally settled. The formulae fit
the atomic spectra, full derivations were provided, ml and ms were introduced, and a new
interpretation of the hydrogen spectrum was given. Thomas informed Goudsmit that over a year
prior, Kronig had presented a similar idea to Pauli regarding a spinning electron, but Pauli
ridiculed the work and it reached no further result. Even so, Kronig was not the first. Compton
speculated on the idea of a spinning electron. In neither Compton’s nor Kronig’s case were the
current interpretations of the hydrogen spectrum or the ml and ms quantum numbers available,
and their results they achieved were few and ultimately erroneous. [3]
The final result published by Goudsmit and Uhlenbeck contains a very fitting model to
describe previously misunderstood phenomena in spectroscopy. The circumstances surrounding
this final discovery are remarkable; yet nature often reveals its secrets in surprising ways to those
who seek understanding, and sometimes by using sources that bring mankind a necessary
humility in the face of all our seemingly vast knowledge.
Works Cited
[1]D.J. Griffiths, Introduction to Quantum Mechanics, (Prentice Hall, New Jersey, 2005), 2nd ed.
[2]McMurry, John, and Robert Fay. Chemistry. 6th. Upper Saddle River: Pearson Prentice Hall, 2012. 161-
162. Print.
[3]van der Waals, J.H.. "The Discovery of Electron Spin." . N.p.. Web. 26 Apr 2013.
<http://www.lorentz.leidenuniv.nl/history/spin/goudsmit.html>.