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1 This is the publishe version of a paper publishe in Entropy. Citation for the original publishe paper (version of recor): Wennerström, H., Westlun, P-O. (2017) A Quantum Description of the Stern Gerlach Experiment. Entropy, 19(5): Access to the publishe version may require subscription. N.B. When citing this work, cite the original publishe paper. Permanent link to this version:

2 entropy Article A Quantum Description of the Stern Gerlach Experiment Håkan Wennerström 1, * an Per-Olof Westlun 2, * 1 Division of Physical Chemistry, Chemical Center, P.O. Box 124, University of Lun, SE Lun, Sween 2 Department of Chemistry, Biological an Theoretical Chemistry, Umeå University, Umeå, Sween * Corresponence: hakan.wennerstrom@fkem1.lu.se (H.W.); per-olof.westlunl@umu.se (P-O.W.); Tel.: (H.W.); (P-O.W.) Acaemic Eitors: Mariela Portesi, Alejanro Hnilo an Feerico Holik Receive: 16 February 2017; Accepte: 20 April 2017; Publishe: 25 April 2017 Abstract: A etaile analysis of the classic Stern Gerlach experiment is presente. An analytical simple solution is presente for the quantum escription of the translational an spin ynamics of a silver atom in a magnetic fiel with a graient along a single z-irection. This escription is then use to obtain an approximate quantum escription of the more realistic case with a magnetic fiel graient also in a secon y-irection. An explicit relation is erive for how an initial off center eviation in the y-irection affects the final result observe at the etector. This shows that the mouth shape pattern at the etector observe in the original Stern Gerlach experiment is a generic consequence of the graient in the y-irection. This is followe by a iscussion of the spin ynamics uring the entry of the silver atom into the magnet. An analytical relation is erive for a simplifie case of a fiel only along the z-irection. A central question for the conceptual unerstaning of the Stern Gerlach experiment has been how an initially unpolarize spin ens up in a polarize state at the etector. It is argue that this can be unerstoo with the use of the aiabatic approximation. When the atoms first experience the magnetic fiel outsie the magnet, there is in general a change in the spin state, which transforms from a egenerate eigenstate in the absence of a fiel into one of two possible non-egenerate states in the fiel. If the irection of the fiel changes uring the passage through the evice, there is a corresponing aiabatic change of the spin state. It is shown that an application of the aiabatic approximation in this way is consistent with the previously erive exact relations. Keywors: Stern Gerlach experiment; quantum escription an interpretation; aiabatic approximation; spin ynamics; spin ensity matrix, relaxation 1. Introuction In 1922, Stern an Gerlach publishe a paper [1] reporting experimental finings on how silver atoms coul be eflecte when travelling through a magnet with a fiel graient in the irection of the main component of the fiel. They observe that the silver atoms followe one of two paths corresponing to a eflection of equal size but of opposite sign. Figure 1 shows a schematic view of the experiment. The result was surprising to the authors, but they immeiately unerstoo that they ha observe another manifestation of the quantum worl that was in the process of being unravele at that time. Stern even sent a postcar to Bohr the ay after the observation to congratulate him for a successful theoretical preiction! It woul soon turn out, however, that Bohr s analysis in t provie the correct explanation of the observation. A fascinating historical account of the Stern Gerlach (SG) experiment an the scientific iscussion it initiate was recently publishe by Schmit-Böcking et al. [2]. It was later realize that the observation of a splitting Entropy 2017, 19, 186; oi: /e

3 Entropy 2017, 19, of 13 into two paths was ue to an interaction between the spin of the unpaire electron in the groun s state of the silver atom. It was also conclue that the measure eflection was consistent with a g-factor of 2 for the electron spin. The calculation was base on treating the interaction spin magnetic fiel in a quantum formalism, while treating the ynamics of the atom using classical mechanics. Figure 1. A schematic picture of the Stern Gerlach experimental set up. The report by Stern an Gerlach was soon followe by other investigations using the same concept. Stuies were mae on alkali an hyrogen atoms [3,4] an the experimental results followe the establishe expectations supporting the original interpretations. At this stage, the SG experiment was scientifically establishe an was no longer a part of the research front. It was, however, one of the rare experiments that were consiere to be funamental enough to take a role in forming a conceptual unerstaning. Consequently, a large fraction of textbooks of quantum mechanics makes use of the experiment to illustrate the spin concept in quantum theory [5 17]. In these textbook accounts, one typically keeps the classical escription of the translational motion an remarks that the funamental quantum effect is that the silver atoms behave as if their spin takes one of two possible values. Some authors point out [13,14] that there is a question of how the initially unpolarize spins transform into aopting one of two possible states. A further complication aresse by D. Bohm [5], for example, is that a graient in the fiel must, accoring to Maxwell s equations, have at least two vectorial components so one along the z-irection requires one also along the y-irection. Consiering its conceptual importance, it took a long time before the SG experiment was analyze using a more complete quantum escription. A. Bohm [17], Scully et al. [18,19] an Utz et al. [20] an Gomis an Perez [21] have shown that, using a Hamiltonian accounting for translation an a fiel in the z-irection, it is possible to erive the basic SG observation of a splitting into two separate signals at the etector without invoking further approximations. Even though the authors of these papers provie a formal escription of the experiment, they refrain from aressing some of the interpretation problems. How is the polarization of the spin create? For a spin polarize in the x, y-plane, there is no force generate on the atom. There is, however, still a force transmitte by the graient. What happens with the spin on the entry to the magnetic fiel? What is the effect of the graient in the perpenicular irection? In the present paper, we aim at getting more insight into these questions by combining a more realistic escription of the magnetic fiel with an ambition to push the quantum treatment of the problem as far as is feasible. The final escription is evelope in three steps. In Section 2, we first present a formally exact solution to the problem of a spin one half particle traveling initially in the x-irection in a magnet with a fiel B = (0, 0, B 0 + βz). This problem has been

4 Entropy 2017, 19, of 13 solve previously [18 22], but we present an alternative formally simple erivation. It has the ual avantage of proviing both a more obvious connection to the semi-classical text-book escription an a basis for a generalization incluing also the effect of the fiel graient in the y-irection. In Section 3, the latter case is analyze consiering the extra term in the Hamiltonian as a perturbation. The thir step is in Section 4 where we consier the ynamics of the particle an its spin associate with the entrance of the particle into the magnet. The magnetic fiel has a complex space epenence in this region, an we consier explicitly only the situation when a particle enters the magnet with z = 0 strictly. In Section 5, the formal results are then iscusse in terms of possible interpretations. The observations of the original SG experiment are quantitatively accounte for, incluing the role of the sprea in the initial position an momentum in the y-irection. Section 6 conclues that the formal results are consistent with the valiity of the aiabatic approximation applie to spin ynamics from the point where the spin enters the magnetic fiel, to its exit from the magnet. Section 7 gives a iscussion of alternative escriptions base on spin relaxation an ecoherence concepts an final conclusions are summarize in Section Silver Atom insie a Magnet with Fiel Exclusively in the z-direction Discussions of the SG experiment are typically base on the simplifying assumption that the magnetic fiel, B, is strictly along the z-axes, with a main component an a graient β B z z, which is a measure of the strength of the graient: The Hamiltonian of the silver atom in the fiel is then B z = B 0 + βz. (1) H = p2 2m + s z + γ e β hzs z. (2) Here, ω 0 γ e B z, where the electron gyromagnetic ratio for the 5 s electron is γ e = (s 1 T 1 ), an s z is the imensionless spin angular momentum operator. In the following erivations, we simplify the notation using γ hγ e β. This is vali insie the magnet 0 < x < L, where L is the length of the magnet. In orer to obtain manageable bounary conitions, it is assume that, at t = 0, the silver atoms enter the magnet with a momentum, p x, solely along the x-irection. In reality, there has to be a istribution of momenta also in the other irections. In the original experiment, two slits were use to keep p y (0) an p z (0) small, while simultaneously having y(0) an z(0) as well-efine as possible. The Hamiltonian Equation (2) an the initial conitions specifies a quantum mechanical problem. Below, we present a solution to this problem that iffers formally, but not funamentally, from previously reporte solutions [17 22]. The starting point is the general equation of motion of an observable Ô involving the commutator of Ô an H h Ô = i [H, Ô]. (3) t Since the spin operator s z in the z-irection commutes with H it follows that t s z = 0, (4) an s z is a constant of the motion. The other central operator in the problem is the momentum p z escribing the translational motion in the z-irection. One has t p z = iγ [zs z, p z ] = γ s z. (5)

5 Entropy 2017, 19, of 13 Since s z is a constant of the motion it follows that the erivative t p z = γ s z 0 (6) is time inepenent. This escribes the time evolution of the mean value of p z. However, to have a more complete escription, we nee information on how the istribution of p z evelops. Thus, consier t p2 z = iγ [zs z, p 2 z] = 2γ s z p z. (7) It follows that the evolution of p 2 z is etermine by the correlation between s z an p z, which coul appear ifficult to calculate. However, t s zp z = iγ [zs z, s z p z ] = γ s 2 z = γ/4. (8) This operator has a constant time erivative an Inserting this result into Equation (7) yiels, since p z 0 = 0, s z p z = s z p z 0 γt/4. (9) p 2 z t = p 2 z 0 + γ 2 t 2 /4. (10) Note that this result is inepenent of the initial state of the spin system. Equations (6) an (8) give explicit expressions for how the mean an the root mean square values of p z change uring the passage through the magnet. In Equation (6), there is a epenence on the initial state of the spin. Consier first a spin that is polarize in the z-irection so that s z = 1/2 an use p z (0) = 0, an then p z t = γt/2; p 2 z t = γ 2 t 2 /4. (11) Thus, if there is no initial sprea in the istribution of p z, the motion through the magnet oesn t give rise to such a sprea since p z 2 t = p 2 z t. (12) Note that the expression for p z in Equation (11) is ientical to the one obtaine in a semi-classical escription. For the case when s z 0 = 0 an there is no spin polarization in the z-irection, Equation (6) shows that the average momentum p z remains at zero if it is initially zero. For p 2 z, on the other han, Equation (10) still applies an the mean square average of p z evolves as for the fully polarize case. Experimentally, one never observes eviations in the z-irection that excee those for fully polarize spins. On both experimental an theoretical grouns, one has the relation p z γt/2. Combine with the two conitions this can only be realize by a istribution p 2 z t = γ 2 t 2 /4, p z 0 = 0, (13) f (p z ) = 1 2 δ(p z + γt/2) δ(p z γt/2). (14) Equation (14) implies that a spin system, initially with no net polarization, behaves with a probability 1 2 as oes a system with s z = 1/2 an with the same probability as a system with s z = 1/2. This conclusion is consistent with the textbook account of the SG experiment. There is, however, a remaining question of how to interpret these probabilities. For a pure spin state, s z = 0 implies that the spin is polarize in the x, y-plane. The magnetic fiel in the z-irection gives rise to an in plane Larmor

6 Entropy 2017, 19, of 13 precession an there is formally no net force on the silver atom. The formalism still gives that p 2 z is time epenent accoring to Equation (10). This issue will be iscusse more in Section 5. To obtain a better basis for the iscussion of this conceptual problem an to get a more complete escription of the SG experiment, we consier two circumstances that have been ignore by the use of the Hamiltonian of Equation (2). 3. Silver Atom insie a Magnet Incluing Also the y-component of the Fiel A virtue of the formalism use in the previous section is that it can provie a basis for a feasible treatment of a more general case. To make the magnetic fiel of Equation (1) consistent with Maxwell s equations, it is necessary to a a graient also in the y-irection. Then, there is an aitional term in the Hamiltonian H 1 = γs y y. (15) With this term in the Hamiltonian, the spin s z is no longer a constant of the motion an h t s z = γ y s x. (16) Equation (5) for the time erivative of the momentum p z remains the same as well as Equation (7) for the evolution of p 2 z. We no longer get a close set of equations, however, since, instea of Equation (8), one has t s zp z = γ/4 + γ h y p zs x. (17) To make progress, it seems necessary to introuce approximations into the calculations utilizing the fact that eviations of y from zero are small making H 1 a weak perturbation. Consier the case where initially p y = 0. There is a force on the atom in the y-irection, but it is much smaller than the force in the z-irection, an it is assume that one can ignore the change in the y-coorinate uring passage through the magnet. Then, one can replace y by y(0) in the equations. The time epenence of p z is no longer linear an, to fin the correction, one can take the time erivative of Equation (5) The time variation in s x is 2 t 2 p z = γ t s z = γ2 h y s x γ2 y 0 h s x. (18) t s x = ω 0 s y + γ h (zs y + ys z ). (19) By assumption, we have /γ >> z, y. If one then neglects the last term in Equation (19), one is back to a normal Larmor precession in a magnetic fiel B 0 an Using this relation, Equation (18) can be integrate to yiel s x s x 0 cos(ω 0 t). (20) p z t = γ s z 0 + ( γ ) 2 y 0 s x. (21) There is, thus, in aition to the leaing term proportional to t, also an oscillating correction term in the expression for p z (t). We can now evaluate the correction term in Equation (17) as approximately yp z s x y 0 { ( γt s z 0 + ( γ ) 2 y 0 s x )s x } = y 0 γt s z 0 s x + y 2 0 ( γ ) 2 s 2 x y 2 0 ( γ ) 2 /4. (22)

7 Entropy 2017, 19, of 13 Here, the last equality follows from the circumstance that s x t oscillates rapily so it averages to zero over a short time. Using Equation (22), the erivative in Equation (17) t s zp z = γ 4 [1 ( y 0γ ) 2 ] (23) is time inepenent as in the absence of a y-component of the fiel, but with a somewhat smaller value. After integrating the equation using the initial conition s z p z = 0, one has t p2 z = γ2 2 [1 ( y 0γ ) 2 ]t. (24) Thus, the presence of a graient in the y-irection has only a limite influence on the ynamics of the silver atoms. This conclusion was reache by more qualitative arguments by Bohm [5] an Le Bellac [14]. However, as not shown previously, a eviation in the initial y-value from zero has the effect of reucing p 2 z an thus the eviation in the z-coorinate of the atoms uring the passage. Integration of Equation (24) results in our final expression for the ynamics of the silver atoms in a realistic magnetic fiel p 2 z t = p 2 z 0 + γ 2 /4[1 ( y 0γ ) 2 ]t 2. (25) This result is, as for the corresponing Equation (10), inepenent of the initial spin polarization. In Section 2, it was conclue that Equation (10) was only compatible with a elta function istribution of p z. Is a similar argument vali for Equation (25)? For a position with y = 0, the magnetic fiel is not along the z-irection. If one assumes that there initially is a perfect polarization of the spin along the z-irection s z = ±1/2, there will be a rapi precession an s z will have a component that averages to zero over a short time. The net force on the atom in the z-irection is then reuce by a factor cos(θ), where θ is the angle between the fiel an the z-axes. Similarly, if one assumes that the spin is polarize along the irection of the fiel, the net force in the z-irection is also reuce by a factor cos(θ). For ω 0 >> γz, γy, it follows that Thus, the maximum/minimum value of p z is cos(θ) = ( y 0γ ) 2. (26) p z t = ± γt 2 [1 1 2 ( y 0γ ) 2 ]. (27) Comparing Equations (25) an (27), it follows that, to the leaing orer, p 2 z 1/2 = p z max,min t. (28) Thus, also for the case y = 0, the atoms move as if they follow one of two trajectories where the force is maximal. This behavior is inepenent of the initial conition for the spin. 4. Spin Dynamics on Entering the SG Magnet In the previous sections, the escription was focuse on the events once the silver atoms ha entere the magnet. The initial conitions were chosen at t = 0 an x = 0. A somewhat bewilering formal result was that the behavior insie the magnet was partly inepenent of the initial conition for the spin state. This concerns in particular how the square of the momentum in the z-irection evelope. In most accounts of the SG experiment, one avois an explicit iscussion of the behavior of the system prior to the entry into the magnet. It is remarke that there is a narrow zone of the same orer as the gap with where the magnetic fiel goes from essentially zero to the value insie

8 Entropy 2017, 19, of 13 the magnet. The magnetic fiel outsie the magnet isn t controlle explicitly. The experimental result can, however, be accounte for solely on the basis of the events insie the magnet, an this suggests that what happens outsie the magnet is irrelevant. Below, we scrutinize this somewhat optimistic assumption. The fact that the outcome of the experiment seems to be inepenent of the etaile properties of the entrance magnetic fiel suggests that one can use a moel magnetic fiel in the iscussion of entrance effects. The fiel insie the magnet is known from Equations (1) an (15). Outsie the magnet, the fiel can be estimate (see Appenix A) by integrating the ipolar contribution from a magnet with uniform gap D. Then, for x < 0, B z (x, z) B 0 π {arctan( z + D ) arctan( z D )}, x x B x (x, z) B 0 2π ln{ x2 + (z + D) 2 x 2 + (z D) 2 }, B y 0. (29) Due to the presence of the graient in the z, y plane insie the magnet, there is not a perfect fit between the expressions for the fiel aroun x = 0, but we ignore this complication. It is a substantial challenge to solve the ynamic quantum equations for a silver atom moving across such a fiel. Even though the transition zone is narrow, the resience time of the silver atoms is still much larger than the inverse of the varying Larmor frequency. There is no obvious basis for assuming that the spin state is, in general, conserve uring the entrance into the magnet. To obtain some information on what can happen to the spins in the entrance zone, consier the iealize case where silver atoms enter along z = 0. In this limit, Equation (29) shows that there is only a fiel in the z-irection. It varies strongly with the position x an there is a coupling between the s z component of the spin an the translational motion along the x-irection. In a semi-classical approximation, the Hamiltonian is H = hω(x)s z, (30) where the varying Larmor frequency is given by ω(x) B z (x, 0)γ e. If the spin is initially polarize along the the z-irection in the spin up state, Equation (30) implies that the spin energy is hω(x)/2. It is increasing in time as x increases an, by conservation of energy, there has to be a corresponing loss of kinetic energy of the atoms. With a spin in the opposite state, the spin energy ecreases, resulting in an increase of the kinetic energy of the atoms. For the case of an initial polarization of the spin in the x, y-plane so that s z = 0, it follows from the Hamiltonian of Equation (30) that the average spin energy H is zero throughout the entrance phase. Analogous with the analysis of the properties of the momentum p z in the previous section, it is revealing to consier the mean square energy of the spin system an H 2 = h 2 ω(x) 2 s 2 z = [ hω(x)/2] 2. (31) This value is inepenent of the character of the spin state. For a pure spin up, or spin own, this is fully consistent with the value for H. All other initial spin states behave as if they were either spin up or spin own with probabilities given by the iagonal elements of the spin ensity matrix. Note that, by making the semi-classical approximation by treating the translational motion classically, the spin state, for example, represente by a ensity matrix, nee not correspon to a pure state. 5. Quantitative Interpretation of the SG Observations In Sections 2 4, we have erive a number of exact an approximate relations vali for the SG experiment, but without explicitly aressing the question concerning the pattern observe on the etector. Equation (11) in Section 2 shows how the momentum in the z-irection evolves in time for a spin system that is initially spin polarize along the z-irection. This epenence is erive from a quantum formalism. To fin a preicte pattern on the etector place at the exit of the magnet,

9 Entropy 2017, 19, of 13 a relation between position an momentum is neee. This, in turn, involves an interpretation step. By inserting the quantum result for the momentum into a classical trajectory calculation, one obtains the preiction that the silver atom with mass m is eflecte to z max,min = t 0 P max,min m (t )t = ± 1 t γt m 0 2 t = ±γt 2 /(4m) (32) in full accorance with observations an with the stanar escription. It thus appears that the position-momentum relation for the atoms can be seen as fully classical in this particular case. For a system that is initially unpolarize in the z-irection, the mean of p z is zero, but, accoring to Equation (14), this reflects the fact that p z is istribute into two values corresponing to the ones foun for the two fully polarize states. There are two main possibilities for interpreting this situation. One possibility (see, for example, Utz et al. [20]) is to see the particle state as a superposition of the two space/spin combinations an there is then a reuction of the wavefunction on the etector. The other possibility (see, for example, Schmit-Böcking et al. [2] is to, analogously with the case for an initially polarize situation, consier two separate classical trajectories where each particular atom follows one of these trajectories with equal probability. Schmit-Böcking et al. argue in etail in favor of this interpretation. The main result of Section 3 was the conclusion that an initial eviation of the position in the y-irection, from zero, results in a reuction of the increase in momentum uring the passage through the magnet. If one also in this case applies a classical relation between momentum an position, one has at the etector in position y z f inal = ± γ 4m t2 [1 1 2 ( y 0γ ) 2 ]. (33) Thus, as the y 0 -value increases, the graient in the y-irection consequently ecreases the gap between the two final z-positions. This conclusion explains semi-quantitatively the observe mouth-like pattern on the etector in the original experiment. In reality, we expect that there is also a contribution from an initial sprea in the momentum p y aroun its average value of zero. First, when this effect is also consiere, it is possible to etermine the initial eviations in y an p y from their mean value of zero base on the observe pattern on the etector. It emerges from the calculations, however, that the observe mouth-like pattern on the etector (see Figure 1) is a generic consequence of the necessary presence of a magnetic fiel graient also in the y-irection [23]. In Section 2, it was conclue that the evolution of the momentum in the z-irection square p 2 z has a time evolution that is inepenent of the initial spin state. This poses the question whether the system behaves is if it was in either a spin up or a spin own state or if the spin actually is in both of these states. The calculations in Section 4 gave further insight into this question. It was conclue that the same ilemma appears also for the conitions in the entry zone just outsie the magnet. It was further conclue that there is a covariation between spin coorinate an the momentum in the x-irection. In this situation, there is also a choice between seeing the state of a particular silver atom as a superposition of two spin/momentum combinations or consiering the atom as aopting one of two possible states with equal probability. 6. Applying the Aiabatic Approximation A virtue of consiering the ynamics of the silver atoms from a position clearly outsie the SG magnet to the point of exit is that it provies a basis for consiering all events from preparation to etection. Above it was foun, from the point of entrance into the magnetic fiel to the exit at the etector, the spin of the silver atoms has the property of behaving as if it was polarize along the z-irection. This property was foun to apply irrespective of the initial state of the spin before entering the magnetic fiel. It was eucte that this was the case, however, at an early stage base on the outcome of the experiment. The first to formulate the conceptual problem were

10 Entropy 2017, 19, of 13 Einstein an Ehrenfest [24], in a paper from 1922 [25]. In aition, Stern expresse concerns, even late in life [2] that such a property is har to unerstan conceptually. Similar remarks also appear in textbooks. A main conclusion from Section 4 is that, in general, the state of the spin evolves alreay when entering the magnetic fiel. Even though one can account for the experimental outcome by consiering the ynamics for position x = 0 an on using an empirical rule for the spin state, this leaves out a conceptually important aspect. In Section 4, the entry of the silver atoms into the fiel was escribe using a time-epenent magnetic fiel, which implies a semi-classical approximation. For such a case, one can apply the aiabatic approximation, which states that a system in an eigenstate will remain in an eigenstate if the time variation of the Hamiltonian is slow enough. Except at the very entrance to the fiel, the Larmor frequency is large relative to the time variation of the Hamiltonian as measure by p x md 105 s 1. The criterion for applying the aiabatic approximation thus seems to be fulfille, except at positions clearly outsie the magnet. What are the consequences of applying the aiabatic approximation for the SG experiment? Consier first the case of an atom with a spin in the s z = 1/2 state before entering the magnet. Assume also, as in Section 4, that the atoms come in along the x-axis with y, z = 0. The spin will then interact with a fiel along the z-axes. Since it is initially in an eigenstate of the spin Hamiltonian, it will remain so. It will then follow a trajectory an reach the etector at z = γt 2 /4m. This behavior is consistent with the calculations in Sections 2 an 4 as well as with the conventional analysis of the SG experiment. If we introuce now the complication that the y-coorinate is slightly off center so that y 0 = 0, but we still have the spin polarize along the z-irection. If we assume a fiel as in Equation (15) an aopt the aiabatic approximation, the spin enters the magnet unchange. There a y-component to the fiel appears an, accoring to the aiabatic approximation, the polarization changes slightly to follow the irection of the fiel. Then, the force in the z-irection on the atom is reuce leaing to a smaller eviation in z at the exit. This eviation is quantitatively consistent with the fining in Section 3. A thir case is when the spin outsie the magnet is polarize in the x-irection with s x = 1/2. Since the fiel is zero, it is in a ( egenerate) spin eigenstate, but, on entering the fiel, this is no longer the case. There is a change from a egenerate state into a situation with two nonegenerate states. The aiabatic approximation oesn t provie a rule for which of the two possible states will be populate. Lacking such an explicit rule one can apply a symmetry argument using the fact that there is no inherent preference for either state connecting it to a possible final state. The natural assumption is then that they are populate with equal probability. The preiction is then that the silver atoms reach the etector at +z or z with equal probability of 1 2. In aition, for this case, the preiction is consistent with observations using two SG magnet in series, but with the fiel irection rotate 90. In the original SG experiment, there was no active preparation of the spin state of the silver atoms leaving the oven. The initial spin state can then be represente by a spin ensity matrix of the form σ = [ but with a unknown reference irection Ω. To apply the aiabatic approximation, one first transforms the spin ensity matrix to a coorinate system along the magnetic fiel at the entrance σ = [ aa ba ] Ω ] ab bb B (34). (35) An equation was erive for evolution of the momentum p z an its relation to the expectation value of s z. If one now assumes that also uring the entry phase the atoms experience a fiel strictly

11 Entropy 2017, 19, of 13 along the z-axis, consistency with the application of the aiabatic approximation is achieve by reucing the ensity matrix of Equation (35) to a iagonal form: σ = [ aa 0 0 bb ] B. (36) This no longer refers to a pure spin state an it can be interprete as if the spin up state is aopte with probability aa an the spin own state with probability bb = 1 aa. For the ensemble of silver atoms, all Ω occurs with equal probability. Thus, on average, spin up an spin own occur with equal probability an one shoul have z = ± z max with equal probability at the etector, which is consistent with the observations. The main conclusion of the present section is that one can account for the observations in the original SG experiment an closely analogous ones by explicitly consiering the spin ynamics uring the full passage from oven to etector by aopting the aiabatic approximation. The problem of fining a selection rule for connecting an initially oubly egenerate spin state with one of two nonegenerate states insie the fiel was solve by requiring consistency with exact equations erive for special conitions of the fiel. A secon implication of the use of the aiabatic approximation is that the silver atom follows one of two possible trajectories uring the passage through the etector. There is no nee to invoke a superposition of the two alternatives. 7. The Relation to Other Interpretations of the SG Experiment In a previous paper [23], we have analyze the SG experiment in terms of the motion of the silver atoms insie the magnet by consiering the effects of the spin relaxation. It was conclue that, by assuming a very rapi T 2 -relaxation, it was possible to account for the experimental finings. This inclues the qualitative observation that a eviation in the initial y-coorinate, y 0, gives rise to a ecrease in the z-coorinate at the etector resulting in a mouth-like pattern. The effect of the T 2 relaxation is to reuce a spin ensity matrix of the form in Equation (35) to the iagonal one in Equation (36). The thermal fluctuations of the magnetic fiel were consiere to be the main cause of this relaxation. The effect to reuce the ensity matrix to a iagonal form is, however, the same as achieve by applying the aiabatic approximation. It is the conclusion of the present paper that the coupling between the translational motion an the spin ynamics is more essential for unerstaning the SG experiment than the coupling spin thermally fluctuating fiel. A complete escription woul require consierations of both aspects. In [21], Gomis an Perez iscuss the reuction of the spin ensity matrix from the form in Equation (35) to the one of Equation (36) in terms of a ecoherence process [26]. They use a Caleria Legett master equation to escribe the coupling between the spin an a thermal bath. Although formally ifferent, this is conceptually closely analogous to the approach of [23], where the coupling to the thermal bath was escribe using the relaxation concept. In their recent thorough account of the SG experiment an its historical context, Schmit- Böcking et al. [2] also iscuss how to escribe the motion of the silver atoms in the magnet. They conclue base on a careful argumentation the atom ynamics can, with a goo approximation, be escribe in terms of classical trajectories. In Section 2, we have provie aitional support to this conclusion, by showing within an exact quantum escription that the evolution of the momentum p z is consistent with the semi-classical picture. Scully et al. [19], Bohm [17], Utz et al. [20], Gomis et al. [21] an Rey et al. [22] all treat the case consiere in Section 2 within a full quantum escription of spin an translation, but not of the magnetic fiel. The specific formalisms iffer between these authors, but the main results are the same an they are in accorance with the formal results of Section 2. Utz et al. also iscuss specifically how the polarization of the spin occurs. They fin, starting from t = 0 an x = 0, in the notation use in Section 2, that an initial spin polarize in the xy-plane is rapily polarize into a state with a spin ensity matrix as in Equation (36), where the off-iagonal elements have ecaye to zero. They make

12 Entropy 2017, 19, of 13 an interpretation that each silver atom travels as a wave packet with nonzero amplitue aroun the two positions of the classical trajectory. At the etector, there is then a reuction of the wave packet. The ifficulties associate with this view are iscusse in the paper by Schmit-Böcking et al. [2]. One important application of the SG experiment has been in iscussions of coincience effects involving the stuy of an entangle spin system. For two atoms in an initial combine spin singlet state that are entering two separate SG magnets, one expects a correlation between the measure eviations at the two etectors. Such a case was analyze by Bell who arrive at a conclusion, now referre to as Bell s theorem [27]. One basis for the argument leaing to the theorem is an assumption that the singlet character of the spin state is preserve uring the passage through the magnet. For the special case of two magnets with the same irection of the fiel, this implies that if a spin up is observe at one magnet, this implies that one woul observe spin own for the other atom. A consequence of the iscussion in Section 4 above is that the spin state evolves uring the entry to the magnet. Is the singlet character preserve uring this evolution? The characteristic feature of the singlet state is that it is antisymmetric with respect to particle exchange. This symmetry is preserve only if the two atoms experience ientical fiels uring the entry of the magnet. This can t be the case in general. There is thus no conservation rule ensuring the preservation of the singlet character. It can still be approximately vali, but a theorem base on such an approximation appears a bit unsatisfactory. By aopting the view expresse in Section 5, that the SG experiment can be unerstoo by applying the aiabatic approximation, a basis for an analysis of the coincience measurement is provie. It was conclue that an atom with an initially polarize spin in the irection of the fiel woul pass the whole SG evice with an unchange spin state. For the other extreme with a spin polarize perpenicular to the irection of the fiel, there is a change of the spin state into one of two possibilities with equal probability. Lacking an explicit escription of this ynamic event, the most natural way is to consier it as a stochastic event. There is no basis for assuming that there is a correlation between two such stochastic events in the two magnets of a coincience measurement. There is no conservation rule that woul ensure such a correlation. When the arguments for these two limiting cases are applie to an initial singlet system analyze in two SG evices of the same orientation, the preicte correlation is 1/3 rather than 1 as assume by Bell. We have previously publishe a more etaile analysis of this argument elsewhere [28]. 8. Conclusions We have iscusse above four questions concerning the SG experiment. First, an most significantly, is the problem of how the initially unpolarize spin can behave as polarize insie the magnet an leave it in a specifie spin state. A secon question (relate to the first) concerns the possibility of obtaining a fully quantum escription of the ynamics insie the magnet instea of the conventional semi-classical argument, which requires a separation of the spin ynamics from the translation of the silver atom. A thir issue concerns what happens in the transition zone between the silver atom source an the entry to the magnet. Finally, a fourth question concerns the significance of the presence of a y-component of the magnetic fiel. The answers to the four interpretation issues are: 1. The spins become polarize uring the entry to the magnet, where there can occur a transfer of energy an angular momentum between spin an translational motion. 2. When having only a z-component of the fiel, it is possible to obtain an exact analytical solution to the equation of motion. For realistic magnetic fiels, however, such a solution remains to be foun. 3. The fiel outsie the magnet has a complex structure an there are significant changes in the spin state occurring uring the entry to the magnet. 4. For trajectories eviating from y = 0, the force on the particles is along the fiel irection, an for larger y-values, the eviation of the trajectory in the z-irection becomes smaller. This explains the mouth-like shape foun on the etector in the original experiment.

13 Entropy 2017, 19, of 13 Acknowlegments: The authors thank H. Schmit-Böcking for sharing their paper about the historical backgroun to the Stern-Gerlach experiment prior to publication. We are grateful to M.H. Levitt for sening us their manuscript prior to publication an to G.W. Driver for linguistic corrections of the paper. This work was supporte by Sweish Research Council (VR). Author Contributions: Håkan Wennerström an Per-Olof Westlun esigne the project an Håkan Wennerström wrote Sections 2 4 while the remaining sections have contributions from both authors. Conflicts of Interest: The authors eclare no conflict of interest. Appenix A. Calculation of the Fiel outsie a Large Magnet with a Uniform Gap of With D For a magnet extening, inefinitely in the y-irection, inefinitely for x > 0 an inefinitely in the z-irection except for a gap at D/2 < z < D/2, the magnetic fiel in the free space is etermine by the magnetic ipolar ensity ρ M. Assume a uniform ρ M insie the magnet with ipole oriente in the z-irection. The magnetic fiel outsie the magnet is obtaine by integrating the contributions from the iniviual ipoles. The fiel in the y-irection is zero by symmetry, an, for the z-irection, one has B z (x, 0, z) = = D/2 I xy (z )z + I xy (z )z = D/2 D/2 I xy (z )z I xy (z )z = D/2 D/2 D/2 I xy (z )z, (A1) where I xy (z ) = 0 For the fiel in the x-irection, one has similarly ( 1 r 3 3z 2 r 5 ) y x ; r = (x x, y, z z ). D/2 B x (x, 0, z) = I xy(z )z, D/2 (A2) (A3) I xy(z ) = 0 3r x r z r 5 y x. (A4) A teious but straightforwar evaluation of the integrals result in the functional form of the fiel in Equation (29). The two components of the fiel are consistent with the Maxwell equations. References 1. Gerlach, W.; Stern, O. Der experimentelle Nachweis er Richtungsquantelung im Magnetfel. Z. Phys. 1922, 9, Schmit-Böcking, H.; Schmit, L.; Lüe, H.J.; Trageser, W.; Tempelton, A.; Sauer, T. The Stern Gerlach Experiment Revisite. Eur. Phys. J. H 2016, 41, Taylor, J.B. Magnetic Moments of the Alkali Metal Atoms. Phys. Rev. 1926, 28, Rabi, I.I. Zur Methoe er Ablenkung von Molekularstrahlen. Z. Phys. 1929, 54, Bohm, D. Quantum Theory; Renewe 1979; Dover Publications Inc.: Mineola, NY, USA, Messiah, A. Quantum Mechanik; Ban 1; Walter e Gruyter & Co.: Berlin/Heielberg, Germany, Blochinzev, D.I. Elementary Quantum Mechanics; Nauka: Moscow, Russia, (In Russian) 8. Park, D. Introuction to the Quantum Theory; Dover Publications Inc.: Mineola, NY, USA, Ballentine, L.E. Quantum Mechanics: A Moern Development; Worl Scientific Publishing Co. Pte. Lt.: Singapore, Merzbacher, E. Quantum Mechanics; John Wiley & Sons Inc.: New York, NY, USA, Schwabl, F. Quantum Mechanics; Springer: Berlin/Heielberg, Germany, Sakurai, J.J. Moern Quantum Mechanics; Aison-Wesley Pub. Comp. Inc.: Boston, MA, USA, Feynman, R.P.; Leighton, R.B.; San, M. Feynman Lecture on Physics; Aison-Wesley Pub. Comp. Inc.: Boston, MA, USA, 1983; Volume 3.

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