Stern Gerlach Experiment:Descriptions and Developments

Li QinxunUniversity of Science and Technology of China

June 8, 2020

Abstract

The Stern-Gerlach Experiment (SGE) performed in 1922 is a seminal benchmark experiment of quantumphysics providing evidence for several fundamental properties of quantum systems. To explain the resultof SGE, several descriptions were developed. The most popular description is always shown in textbooks.In this article, we introduce this textbook description and use it as an example discuss some concepts. Toshow the drawback of textbook description, we discuss SGE with more details and introduce Potel’s fullquantum mechanical description. Then we use the parameters of Stern and Gerlach’s original apparatus toestimate whether the textbook description effective here, which has never been discussed before. After givingcorrections to textbook description, we turn to introduce some developments and applications of SGE, suchas longitude SGE.

I. Introduction

Stern-Gerlach Experiment(SGE) , carried out in1922, is considered as a canonical experiment thatled us on the way to quantum mechanics[1]. In thefamous experiment, Stern and Gerlach demonstrateda device to measure the possible values of the mag-netic moment for silver atoms. They sent a beamof these atoms through an inhomogeneous magneticfield, and found a non-trivial result from the screen.Rather than a continuous band as the classical pre-diction, the deflection pattern on the screen showstwo distinct bands[2]. According to the electrody-namics, this means that the magnetic dipole moment,and thus, the angular momentum of the silver atomshave only two possible values.

There are several descriptions of the result. Actu-ally, Stern and Gerlach’s idea for SGE was to exam-ine Bohr atomic model, or, more exactly, the quan-tization of orbit angular momentum of electrons[3].Though there seems to be a beautiful agreement be-tween Bohr theory and SGE, it proved to be an in-teresting coincidence[1]. Modern quantum mechanics

claims the splitting of atomic beam in SGE was dueto spin angular momentum, for the orbit momentumof ground state atom is zero[4].

In the quantum framework, we can use some ap-proximations to simplify the problem and then solveit[5]. In some popular textbooks, a semi-classical de-scription is applied to introduce SGE, which we willdiscuss later in Section II. This textbook description,in which SGE is an ideal measurement, provides anapproach to teach the basic concepts of quantum me-chanics, as Sakurai[2] did. As for the shortcomes ofthis description, we will introduce some delicate de-scriptions in Section III.

Besides its fundamental role in teaching, SGE isalso the focus of research efforts, because of its plen-tiful applications. Many beam-splitting of atomicbeam in the resonant and nonresonant experiments,or Stern-Gerlach-like experiments, have been devel-oped for different particles and different fields since1922[1]. Starting from SGE and its key concept ofspace quantization, some useful apparatus have beendesigned for various applications, including nuclearmagnetic resonance, the laser and atomic clocks[1].

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Introduction to quantum mechanics

Figure 1: A memorial plaque honoring Otto Sternand Walther Gerlach[1]

In Section IV, we will introduce some further devel-opments of SGE.

II. Textbook Description to SGE

As Figure.2 shows, the SGE apparatus contained fourcomponents, an oven to heat the atoms, a collima-tor to produce a narrow atom beam, a pair of polepieces to produce a non-uniform magnetic field, and ascreen. The beams, generated from effusion of metal-lic vapour in the 1000◦C oven, went through the in-homogeneous deflecting magnetic field, and finallyreached the screen. The magnetic field consists ofconstant components of intensity B0 and a gradientof intensity b[4]. That is to say:

B = −B1xi+ 0j + (B0 +B1x)k (1)

The directions of three axes are shown in Figure.3. According to classical electrodynamics, a neutralparticle in the field feels a deflecting force which isdecided by the field intensity B and its magnetic mo-ment µ[6]:

Figure 2: The Stern–Gerlach apparatus.[2]

Figure 3: The directions of three axes. [7]

F = ∇(µ · B) (2)

II.1. An Ideal MeasurementIn our textbook description, we treat the neutralatoms as some semi-classical magnetic needles, whichhave certain locations, certain momentum and quan-tized angular momentum[5]. Or, you can say theatoms here are mass points with quantized angularmomentum.

To describe µ, we should find the angular momen-tum. For a silver atom made up of a nucleus and47 electrons, the inner 46 electrons form a spher-ically symmetrical electron cloud, thus distributingzero angular momentum[7]. So we can infer that theangular momentum of an atom are solely due to the47th electron outside the closed shells, if you ignorethe nuclear spin. Moreover, the 47th electron is on

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Introduction to quantum mechanics

Figure 4: SGE as a measurement to spin projectionin the z-direction[7]

the 5s state, which means n = 5, l = 0. Thus, its or-bit angular momentum is also zero. What we concernhere is just their spins, or intrinsic angular momen-tum:

µ = γJ = γ(L+ S) = γS (3)The proportionality constant here, γ, is called the

gyromagnetic ratio. The deflecting force is:

F = ∇(µ · B) = −γB1Sxi+ γB1Sz k (4)We assume that bx ≪ B0, bz ≪ B0, so that the

evolution of state can be described by Larmor pre-cession. So the Hamiltonian can be simplified to be:

H = −γB0Sz (5)The eigenvector of this Hamiltonian is simply those

of Sz, so [H,Sz] = 0, which means expectation valueof Sz conserves. The Hamiltonian in matrix form is:

H = −γB0Sz = −γB0h

2

(1 00 −1

)(6)

So the stationary state of this system is:

|+⟩ =(10

), |−⟩ =

(01

)(7)

The initial state can be expressed as:

|ψ(0)⟩ = a |+⟩+ b |−⟩ =(ab

)=

(cosαsinα

)(8)

So the time-dependent state is:

|ψ(t)⟩ =[cosα e−iγB0t/2

sinα e−iγB0t/2

](9)

Then we consider the expectation value of Sx:

⟨ϕ(t)|Sx |ψ(t)⟩ =h

2⟨ϕ(t)|

(0 11 0

)|ψ(t)⟩

=h

2sin 2α cos(γB0t)

Then we can tell the expectation value of Bx os-cillates rapidly surround the average value 0. Asan approximation, we can ignore the force in the x-direction and get:

F = γB1Sz (10)

For textbook description, you can also ignore thefield intensity on x-direction at the beginning, whichhelps you realize why the z-axis component of spinconserves. The only reason we need the x-axis com-ponent is that our field must follow the rules ofMaxwell: ∇ ·B = 0.

Now we can derive that SGE device is a measure-ment to the spin projection along the magnetic field.The atoms with different Sz deflect in different dis-tance after traveling the field, and this forms the SGdeflection spectrum. Figure.5 shows a typical resultof SGE, two separate bands. As we know, electronis a spin-1/2 particle, so Sz has two possible values:+ 1

2 h or − 12 h. Delicate measurement gives:

µ = ± eh

2me= ±1

2γh (11)

as the quantum mechanics predicts.In conclusion, the textbook description assumes 3

points:

1. Classical Particle Trajectory: The movement ofatoms can be described in Newton’s mechanicsframework.

2. Larmor precession: The evolution of state in spinspace is mainly influenced by B0.

3. Macroscopic Electromagnetic Field: The gradi-ent part deflects atoms on a classical way.

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Introduction to quantum mechanics

Figure 5: Gerlach’s postcard, dated 8 February 1922, to Niels Bohr. It shows a photo-graph of the beamsplitting, with the message, in translation: “Attached [is] the experimental proof of directional quantization.We congratulate [you] on the confirmation of your theory.”

[1]

Figure 6: The first set of sequential SGE[7]

II.2. Sequential SGE

Now, we use Stern-Gerlach experiment as thearchetype of the measurement of a quantum mechan-ical property. When we try to measure different me-chanical quantity in sequence, for example, Sz andSx, some unbelievable things happen. Albert said,“Here’s an unsettling story about something that canhappen to electrons. The story is true.” [8]

In a sense, the SGE device is an apparatus forpreparation of certain quantum state[2]. We canblock Sz = − h

2 component(Sz− component in short),

and in this way our SGE apparatus outputs a pureSz+ beam with a mixed Sz input.

Then, let the Sz+ component travel through an-other SGE apparatus measuring Sz(SGz in short),and we will find that there is only one output, theSz+ beam[2]. This means our measurements with Sz

are repeatable, thus “Sz+ state” makes sense[8].If we turn the orientation of the dipole magnets,

the orientation of the inhomogeneous field would alsoturn. For an SGE device with a field in the x-direction, discussions in Section II.1 reveal that it’san ideal measurement to Sx(SGx). Sequential SGxshow a similar repeatable result.

We now consider whether two different measure-ments interact. In this turn, two apparatus are setup, first an SGz with a block on Sz−, and then anSGx. The outputs are as same as a single SGx,both Sx+ and Sx− components exist. This result isstill easy to understand in classical way, which claimsthe atoms with Sz = + h

2 consists of equal amount ofSx = + h

2 and Sx = − h2 .

However, the third apparatus shows an extremelyquantum behavior. Let the Sx+ component from the

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Introduction to quantum mechanics

Figure 7: The second set of sequential SGE[7]

second apparatus go through another SGz. The clas-sical theory (i.e. Bohr model) predicts there is onlySz+ component, for previous apparatus select bothSz and Sx quality so that the input of the third ap-paratus is purely Sz+, Sx+ atoms. In contrast, theresult is indeed similar to a single SGz without previ-ous apparatus, outputting both Sz+ and Sz−. This“unsettling”[8] phenomenon reveals almost the mostimportant feature of quantum mechanics, noncom-mutation and uncertainty principle[9, 10].

So, what happened to those Sz+, Sx+ atoms? Infact, quantum mechanics does not allow the descrip-tion of “atoms with Sz = + h

2 and Sx = + h2 ”. We use

an Hermite operator to represent an observable, andfor an eigenstate the eigenvalue of the operator is theobserved value. In the first set of sequential devices,the first apparatus produces atoms in Sz+ state, andwhen they travel through the second SGz:

Sz |Sz; +⟩ = +h

2|Sz; +⟩ (12)

The result is completely decided, Sz = + h2 .

For other states, or what we called superpositionstate, we can use the linear combinations of eigen-state to describe. For example,

|Sz; +⟩ = 1√2|Sx; +⟩+ 1√

2|Sx;−⟩ (13)

And the most important and mysterious axiom ofquantum measurement, Born-Lüders rule tells us, forobservable A and pure state

|ψ⟩ =n∑

i=1

ψi |αi⟩ , ψi = ⟨αi|ψ⟩ (14)

where |αi⟩ is the eigenvector of operator A and eigen-value αi, the observed value must be one of the eigen-values αi. And the probability to get αi is:

pi = |ψi|2 = |⟨αi|ψ⟩|2 (15)

Therefore, the second set of devices is simply ob-servable Sx acting on state |Sz; +⟩. According toEq.13, the probability to get value + h

2 and state|Sx; +⟩ is p+ =

∣∣1/√2∣∣2 = 1/2, while that of − h

2

and |Sx;−⟩ is p− =∣∣1/√2

∣∣2 = 1/2. These explainwhy there are two beams with almost equal intensity.

As for the third set, this situation is actually thesame as the second set, if you ignore the differentbeam intensities. The influence of previous SGz isjust to cut off the beam intensity, for the outputstate of SGx is always Sx+ and the blocked Sx−whether the previous SGz exists or not. So the out-put of the set is just 50 percent of |Sz; +⟩ and 50 per-cent of |Sz; +⟩, consistent with the experiment phe-nomenons.

Up to now, we have reviewed the familiar textbookdescription of sequential SGE and the basic conceptsof quantum measurement[2]. In next section, we willdiscuss SGE with more details.

III. Discussions With More Details

While the textbook description is quite simple in con-cept and useful to illustrate the Principles, the lackof details makes it too rough to be useful in practicalmeasurements and applications. In fact, for example,the sequential SGE is never able to realize in practice,for the spin projection indeed changes during travel-ing the field[11, 5]. In this section, we first introducea method in a quantum mechanics framework, thenapply some approximations to describe the practicalsituation and give some corrections to textbook de-scription.

III.1. Quantum Mechanic DescriptionNow we try to work out a description in the frame-work of quantum mechanics. Consider an atom being

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Introduction to quantum mechanics

Figure 8: The third set of sequential SGE[7]

sent into the field, the Hamiltonian is:

H =P 2X + P 2

Y + P 2Z

2M− γB · S (16)

where M is the mass of an atom. Because of themovement along y-axis, we introduce τ = L/VY =LM/hKY as the traveling time, or interaction time.In this section, we use the capital letters X, Y, Z, Tto represent magnitudes with dimensions. Low casex, y, z, t correspond to dimensionless quantities:

x =X

σ(17)

y =Y

σ(18)

z =Z

σ(19)

t =T

τ(20)

h =Hτ

h(21)

The initial wave packet of the atom can be writtento a Gaussian wave function with a momentum iny-direction[4]:

⟨XY Z,m|Ψ(T = 0),m0⟩

=Ce−X2+Y 2+Z2

2σ2 eiPY Y

h δ(m−m0)

=C exp

{−X

2 + Z2

2σ2

}exp

{− Y 2

2σ2+ ikY Y

}δ(m−m0)

where C is the normalization constant, KY is thewave number in Y-direction. m , m0 here representsthe spin projection in Z-direction.

Notice that the Y part of wave function can beseparated, and it just means the longitude movement,which is trivial and unimportant. What we reallycare about is the X −Z component, and now we candeal with it solely. The Hamiltonian of X − Z hereis:

H =P 2X + P 2

Z

2M− γB · S (22)

Using some dimensionless quantities, we can rewriteit:

h =A

2(p2x + p2z)− S[Sz(z + z0)− Sxx] (23)

The definitions of these dimensionless parametersare:

A =hτ

Mσ2(24)

S =γB1τσ

h(25)

z0 =B0

σB1(26)

These parameters describe the feature of the appa-ratus and subsequently decide whether a descriptionworks out or not.

With this Hamiltonian, we can write down the di-mensionless time-dependent Schrödinger equation:

h |Φ(t);m0⟩ = i∂

∂t|Φ(t);m⟩ (27)

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Introduction to quantum mechanics

where Φ(t) is the x-z wave function in coordinatespace. The initial Φ can be derived from separatingvariables:

⟨x, z;m|Φ(t = 0);m0⟩ =N exp

{−1

2(x2 + z2)

}δ(m−m0)

So far, we have accomplished the quantum me-chanical formulation of Stern Gerlach system. In asense, the full quantum mechanics description is al-ready done. There are two ways to continue our way,using numerical methods to find the exact solution,or applying some approximations to get an analyticalsolution.

To get the exact solution, in our spin-1/2 system,where m = ±h/2, it’s convenient to expand the wavefunction into two components:

⟨xz;m = h/2|Φ(t);m0⟩ = α(x, z, t)eitSz0/2 (28)⟨xz;m = −h/2|Φ(t);m0⟩ = β(x, z, t)e−itSz0/2 (29)

so that the Schrödinger equation for x-z plane can bewritten in a symmetry form:

H[α(x, z, t)β(x, z, t)

]= i

∂

∂t

[α(x, z, t)β(x, z, t)

](30)

H =

[A2 (p

2x + p2z)− S

2 zS2 x

S2 x

A2 (p

2x + p2z) +

S2 z

](31)

When z0 is large. The Hamiltonian in Equation.31consists of simply x, z, , px, pz. This form leads usto the method that we apply in dealing with the har-monic oscillator model. Use the harmonic oscillatorfunctions ϕn(x) and ϕm(z) as a set of base to expandα and β:

α(x, z, t) =∑nm

anm(t)ϕn(x)ϕm(z)

β(x, z, t) =∑nm

bnm(t)ϕn(x)ϕm(z)(32)

And the creation operators and destruction opera-

tors are:

ax =1√2(x+ ipx)

a†x =1√2(x− ipx)

az =1√2(z + ipz)

a†z =1√2(z + ipz)

(33)

To get the solution, we need to apply these opera-tors on Equation 30, then put Equation.31, Equa-tion.32 and Equation.33 into Equation 30. Whenz0 ≫ 1, the numerical solution of Equation 30was derived by Garraeay and Stenholm in 1999[12].For more general situation, Potel, Barranco et al.solved it in 2005 with a fourth order Runge-Kuttamethod[11].

In addition, a correction should be applied on this“exact solution” when you turn to experiments. Itshould be noticed that a beam of particles is usu-ally not given by a pure state, but rather by a mix-ture of small quantum wave packets. It is reasonalto assume that the distribution of this mixture is an-other Gaussian with standard deviation σm. To dealwith this problem, we can replace σ with a synthesisof two Gaussian σt =

√σ2 + σ2

m. For instance, wecan define a new parameter z′0 = B0/σtB1 to replacez0 = B0/σB1

III.2. Approximations and CorrectionsIn this part, I’d like to turn to some approximationsand talk about the physics meanings.

At the beginning, let’s review the dimensionlessconstant in the Hamiltonian. The adiabaticity pa-rameter A is the ratio of the interaction time τ di-vided by the natural time of expansion of the Gaus-sian packet tex =Mσ2/h. The separation parameterS is the ratio of the momentum change induced bythe magnetic field gradient γB1τ to the momentumwidth of the Gaussian packet h/σ, which shows therelative influence of magnetic field on the momentum.The inhomogeneity parameter z0 determines the rel-ative change of the magnetic field in the range of the

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Introduction to quantum mechanics

Gaussian. Comparing with z0, z′0 takes the width ofthe beam into account.

Using these parameters, the assumptions in SectionII.1 can be discussed in detail.

For textbook description, except the last assump-tion that can not be described in the quantum me-chanics, we can summarize that the conditions to ap-ply textbook description are S′ ≫ 1 and z′0 ≫ 1. Thisis not difficult to understand. The classical trajectoryassumption requires the field strongly influence themomentum of atoms, which refers to γB1τ ≫ h/σ.The Larmor precession assumption and our neglec-tion to velocity dispersion in textbook descriptionlead to a large z′0.

Going back to Stern and Gerlach’s original devicein 1922, we can see how the textbook descriptionworks here. The field had 0.1T of B0 and 0.1T/mof B1[1]. As for the dispersion, σ for each atom canbe estimated as de Broglie wave length[13]:

σ ≃ λ =h

p=

h

MVheat≃ h√

kBMT(34)

where M is the mass of silver atom, M = 107amu =1.77 × 10−25kg[3]. T , the temperature of the oven,was about 1300K[1]. Put these numbers into theequation and we have σ ≃ 10−12m.

To estimate σm, we use the half of beam width orthe thickness of collimator, 3× 10−5m[1]. Obviously,σt is much larger than σ, therefore σt ≃ σm. Thedimensionless parameter z′0 is:

z′0 =B0

σtB1≃ B0

σmB1≃ 105. (35)

As we expected, z′0 is much larger than 1.Similarly, we can also calculate S for Stern and

Gerlach’s apparatus. The length of the field is L =3.5 × 10−2m. The speed of atoms along Y-axis isapproximately Vheat ≃

√kBT/M = 3 × 102m/s. So

the traversing time τ is about L/Vheat = 10−4s.S canbe estimated as S = eB1τσ/mh ≃ 1023. So we cantell S ≫ 1 here.

While textbook description did work in Stern andGerlach’s original apparatus in 1922, it should be no-ticed that the beam size, or the width of the collima-tor, has a determinant influence in the magnitude of

z′0. Therefore, SGE is no easy to perform. In fact,it took more than a year for Stern and Gerlach toaccomplish the apparatus.

When the beam size could not be restrictedwell, which often happens dealing with neutronsor other low-mass particles, z′0 grows to be sosmall that the textbook description fail to ex-plain the phenomenons. Cruz-Barrios and Gómez-Camacho developed a semiclassical description forthis situation(z0 ≫ 1, S ≫ 1) based on CoherentInternal States[5].

To be concise, we only talk about the correctionsfrom this semiclassical description. When z′0 ≫ 1,some additional effects occur[11]:

1. There is a focusing effect, so that the particlesdeviating in the direction in which the field de-creases tend to focus, while those going in thedirection of increasing fieldtend to de-focus.

2. There are some particles with a given spin pro-jection which deviate as those with a differentspin projection. So, the Stern-Gerlach setup isnot, even in theory, a “completely reliable” mea-suring apparatus.

3. There are some particles, with a definite spinprojection along the quantization axis, whichchange the spin projection as they go throughthe magnet. So, the Stern-Gerlach setup is notan “ideal” measurement apparatus, as successivemeasurements will not give exactly the same re-sults. In other words, sequential SGE woulddifferent phenomenons from discussions in Sec-tion.II.

In this semi-classical description, the SGE can al-ter the spin projection, which leads to a non-idealmeasurement. Moreover, the position of particle isnot always correlated with the spin projection, whichmakes the measurement based on space quantizationless reliable.

Potel, Barranco et al. discussed more approxi-mations, including adiabatic approximation, pseudo-adiabatic approximation and symmetrized approxi-mation. Their physical meanings and their deviationsfrom the numerical result of exact solution was alsoshown in Ref [11].

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Introduction to quantum mechanics

IV. Developments of SGE

Stern-Gerlach experiment shows an excellent poten-tial in measurement and preparation. Since 1922,lots of Stern-Gerlach-like experiments have been de-veloped. In this section, we will first focus on SGEfor electron, then turn to introduce some other Stern–Gerlach-like experiments.

Longitude SGEThe spin of the electron plays a central role in theexplanation of atomic spectra[14]. However, Stern-Gerlach experiment does not work with beams of elec-trons because of the combined effects of the Lorentzforce and the Heisenberg uncertainty principle[15].The Lorentz precession of electron charge destroysthe effort of magnetic moment to achieve spacequantization. Besides, according to our discussionin Section III.2, electron is so light that textbookdescription and space quantization method do notwork. Brillouin, however, designed a longitude SGEto measure the spin magnetic moment of electrondirectly[16]. Based on Bohr’s inference, Pauli ana-lyzed Brillouin’s proposal and made a more generalargument that no device based on the concept ofclassical particle trajectories and macroscopic mag-netic fields could be used to separate an electronbeam by spin or to measure the electron’s magneticmoment[17]. Because of Pauli’s inference, Brillouin’sidea had lied idle for years. In the 1980s and 1990s,some theory physicists paid their attention to revisitBohr and Pauli’s argument, and some mistakes werefound[18, 19].

Along Brillouin’s way, Dehmelt and his colleaguesdid a beautiful job on direct measurement to electronspin[20]. In 1986[21], they described an ideal deli-cate apparatus, where electrons’ trajectories are par-allel to the magnetic field to turn the on-axis Lorentzprecession to off-axis one. A Penning trap was usedto make electrons’ trajectories periodic and stable.We called the particles bounded in the Penning trapGeonuims, for they are bounded on the earth. Afterfixing the naughty electron on the operating table, aweak magnetic bottle, named Larrence magnetic bot-tle, was adopted to cause an axial oscillation. The

Figure 9: Electron in Penning trap, the Geoniumatom. In the simple mono-electron oscillator modeshown, the electron moves only parallel to the mag-netic field B and along the symmetry axis of the elec-trode structure. Each time it gets too close to oneof the negatively charged cap electrodes, it is turnedaround and a rf oscillatory motion results.[20]

frequency of this oscillation is spin dependent:

ωz(↑)− ωz(↓) = δ ≃ 2πHz (36)

Then the quantization of electron spin can be ob-served as the discrete frequency of oscillation. Itshould be mentioned that this experiment can be re-peated on the same particle once and once again, forit doesn’t destructive the mixture in space. If youlike, this experiment can be even done continuously.

In 1986, they realized this experiment and mea-sured the magnetic moment of electron, includingthe anomaly part, which is the most importantsupporting evidences to quantum field thoery[22].Their result at the time of submission, g =2.002319304400(80), is the most accurately deter-mined parameter of any elementary charged parti-cle which in addition can be directly compared withtheory value 2.002319304402. This unbelievable ac-curation shows the strength of field theory and thepotential of SGE.

Optical SGE Ref.[23] reports the observation ofdeflection and splitting of atomic beam in the res-onant laser field. This optical Stern-Gerlach effect

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Introduction to quantum mechanics

Figure 10: Schematic diagram of the optical SGEsetup[23].

was first predicted by Kazantsev in 1975[24, 25]. TheHamiltonian here is:

H =p2

2m− d · E(x, t) (37)

where d is the electric dipole moment of the atom.Instead of the gradient of magnetic field, a gradi-ent of optical field cause a light pressure on atomsthat splits atomic beams according to their dipolemoment.

Computer-Simulated SGE Based on textbookdescription, Schroeder and Moore build a computer-simulated system of SGE[26]. This system, calledspin, is designed for teaching. The user can designand run experiments involving successive spin mea-surements, illustrating incompatible observables, in-terference, and time evolution. Figure 4 is drawnusing this program. This computer-simulated lab-oratory combined with spin-based textbook gives anew way to start the journey of learning quantummechanics[7].

V. Conclusion

The story of Stern-Gerlach experiment is quite won-derful. It was designed to examine Bohr’s modeland get a nice agreement, but this agreement wasa lucky coincidence indeed[1]. It was the first exper-iment to imply the spin of electron, though the early

description focused on orbit angular momentum[3].The popular textbook description treats the problemby many classical approximations, such as classicaltrajectory, but some textbook authors called it “theleast classical system”[2].

The SGE shows different meanings for differenttheory. In the age of Bohr’s model, it was the directevidence of angular momentum quantization. Fornon-relativistic quantum mechanics, it illustrated thebasic concept of quantum measurement and Pauli’sspin theory, while spin itself is a relativistic effect[13].For quantum field theory, the measurement of elec-tron’s magnetic moment by SGE serves as one of themost accurate evidence.

Since 1922, our realization to Stern–Gerlach exper-iment has been enriched and deepened through sev-eral descriptions and developments. All these worksshow the great significance of SGE.

References

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[2] Jun John Sakurai, Jim Napolitano, et al. Mod-ern quantum mechanics. Pearson Harlow, 2014.

[3] Friedel Weinert. Wrong theory—right experi-ment: The significance of the stern-gerlach ex-periments. Studies in History and Philosophy ofScience Part B: Studies in History and Philoso-phy of Modern Physics, 26(1):75–86, 1995.

[4] David J Griffiths and Darrell F Schroeter. In-troduction to quantum mechanics. CambridgeUniversity Press, 2018.

[5] S Cruz-Barrios and J Gomez-Camacho. Semi-classical description of stern-gerlach experi-ments. Physical Review A, 63(1):12101, 2000.

[6] Daniel E Platt. A modern analysis of thestern–gerlach experiment. American Journal ofPhysics, 60(4):306–308, 1992.

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Introduction to quantum mechanics

[7] David H McIntyre. Spin and quantum measure-ment. Quantum, 1:4, 2002.

[8] David Z Albert. Quantum Mechanics and Expe-rience. Harvard University Press, 1994.

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[11] G Potel, F Barranco, S Cruz-Barrios, andJ Gómez-Camacho. Quantum mechanical de-scription of stern-gerlach experiments. PhysicalReview A, 71(5):52106, 2005.

[12] B. M. Garraway and S. Stenholm. Observing thespin of a free electron. Phys. Rev. A, 60:63–79,Jul 1999.

[13] Claude Cohen-Tannoudji, Bernard Diu, andFrank Laloe. Quantum Mechanics (1 vol. set).Wiley-Interscience, 2006.

[14] Marlan O Scully, Willis E Lamb Jr, and AsimBarut. On the theory of the stern-gerlach ap-paratus. Foundations of Physics, 17(6):575–583,1987.

[15] Nevill Francis Mott. The scattering of fast elec-trons by atomic nuclei. Proceedings of the RoyalSociety of London. Series A, Containing Pa-pers of a Mathematical and Physical Character,124(794):425–442, 1929.

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