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>.