Quantum mechanical description of Stern-Gerlach experiments

Quantum mechanical description of Stern-Gerlach experiments G. Potel,1F. Barranco,2S. Cruz-Barrios,3,1 and J. Gómez-Camacho1 1Departamento de Física Atómica, Molecular y Nuclear, Apartado Postal 1065, 41080 Sevilla, Spain 2Departamento de Física Aplicada 3, E.S.I. Isla de la Cartuja, Sevilla, Spain 3Departamento de Física Aplicada 1, E.U.P. Virgen de África, Sevilla, Spain 共Received 21 September 2004; published 27 May 2005兲 The motion of neutral particles with magnetic moments in an inhomogeneous magnetic field is described in a quantum mechanical framework. The validity of the semiclassical approximations which are generally used to describe these phenomena is discussed. Approximate expressions for the evolution operator are derived and compared to the exact calculations. Focusing and spin-flip phenomena are predicted. The reliability of SternGerlach experiments to measure spin projections is assessed in this framework. DOI: 10.1103/PhysRevA.71.052106 PACS number共s兲: 03.65.Sq, 03.65.Ta, 03.65.Wj, 03.65.Nk I. INTRODUCTION The Stern-Gerlach experiment consists in taking a beam of particles that have a neutral electric charge, but a finite magnetic moment, and passing them through an inhomogeneous magnetic field. The observed result is that the particles deflect differently depending on the spin projection along the magnetic field. So, by measuring the deflection, one can infer the value of the spin projection of the particles along the direction of the magnetic field. The description of this phenomenon is done with the following assumptions. 共i兲The spin projection along the zaxis, taken along the magnetic field at the center of the beam, is conserved. 共ii兲Particles with different spin projections along the z axis, as they go through the inhomogeneous magnetic field, suffer a force in the zdirection that is given by the product of the magnetic moment times the gradient of the field times the spin projection. This is what we will call the textbook description of the Stern-Gerlach experiment 关1–4兴. Thus, considering the particle position as a pointer and the spin projection as the quantum property to be measured, the Stern-Gerlach setup is associated with a measurement operator on the spin state which has as eigenvalues the spin projections along the zaxis. Under the textbook description, the Stern-Gerlach experiment corresponds to an “ideal” measurement, in the sense of von Neumann 关5兴, because the quantum state is not modified by the measurement process when it is an eigenstate of the measuring apparatus. Besides, it is “completely reliable,” in the sense discussed in 关6兴, because the position is completely correlated with the spin projection. However, when the experiment is investigated in more detail, the situation becomes more complicated. As the magnetic field has zero divergence, then it is not possible to have a gradient of the field only in one direction. This produces terms in the Hamiltonian that can change the spin of the incident particle. A detailed investigation of these effects was made in a recent publication 关7兴, making use of the concept of coherent internal states 关8兴in a semiclassical approach. In this approach, it is shown that the quantum mechanical wave function which describes the motion of a system with internal degrees of freedom can be approximated by a single trajectory only for certain internal states which are called coherent internal states. These internal states evolve in time according to an evolution operator which is determined by the interaction evaluated along the trajectory. The coherent internal states, in the case of the Stern-Gerlach experiments, are states with definite projection along the direction of the magnetic field. This direction may vary depending on the position of the particle, because the magnetic field is not homogeneous. The main result of 关7兴is that, indeed, when a beam of particles goes through a Stern-Gerlach magnet, the different spin projections deviate depending on the spin projection. However, when the size of the beam is not very small compared to the range of inhomogeneity of the magnetic field, additional effects occur. 共i兲There is a focusing effect, so that the particles deviating in the direction in which the field decreases tend to focus, while those going in the direction of increasing field tend to defocus. 共ii兲There are some particles with a given spin projection which deviate as those with a different spin projection. So the Stern-Gerlach setup is not, even in theory, a “completely reliable” measuring apparatus. 共iii兲There are some particles, with a definite spin projection along the quantization axis, which change the spin projection as they go through the magnet. So the Stern-Gerlach setup is not an “ideal” measurement apparatus, as successive measurements will not give exactly the same results. This is what we will call the semiclassical description of the Stern-Gerlach experiment. Note that if we associate the particle position after the magnet as a “pointer,” which gives the result of the measurement of the spin projection along the zaxis, then we conclude that, in the semiclassical description, the Stern-Gerlach experiment is not an ideal measurement, because it can alter the spin projection, or a completely reliable one, because the position is not always correlated with the spin projection. These conclusions were obtained in a semiclassical framework, in which the motion of the particles was described by classical trajectories which depended in the spin projection along the magnetic field that they encountered. Our motivation here is to see whether the same conclusions hold when the full quantum mechanical problem is considered. In Sec. II we formulate the time-dependent quantum PHYSICAL REVIEW A 71, 052106 共2005兲 1050-2947/2005/71共5兲/052106共10兲/$23.00 ©2005 The American Physical Society052106-1 mechanical problem of a wave packet going through a SternGerlach magnet and discuss the validity of the textbook and semiclassical approaches. In Sec. III we present the numerical solution of the quantum mechanical problem. In Sec. IV we investigate several analytic approximations to the problem, considering the validity of the concept of coherent internal states. In Sec. V we discuss the interpretation of SternGerlach experiments as measurements devices. Sec. VI is for a summary and conclusions. II. QUANTUM MECHANICAL FORMULATION We want to investigate the effect of an inhomogeneous magnetic field on the evolution of a quantum wave packet. The situation that we will consider is a magnetic field that has components in the Xand Zdirections, but not in the Y direction. This magnetic field has a length L, and it can be written as B ជ =共B0+B1Z兲u ជ z−B1Xu ជ x,0艋Y艋L.共1兲 We use the capital letters X,Y,Z,Tto represent magnitudes with dimensions. Lowercase x,y,z,tcorrespond to dimensionless quantities. We neglect border effects around Y=0 or Y=L. Note that this field fulfills ⵜB ជ =0 and also ⵜ⫻B ជ =0,as should be expected for a magnetic field in the region where there are no currents. These conditions were not fulfilled in the case discussed in textbooks such as 关1–3兴. The Hamiltonian which describes a nonrelativistic neutral particle which enters in this field is given by H=PX 2+PY 2+PZ 2 2M− ␮ B ជ ·I ជ ,共2兲 where ␮ is the magnetic moment and I ជ is the spin operator. We consider now a wave packet 兩⌿共T兲;m0典which enters into this field. Initially, the wave packet can be characterized in coordinate space as a Gaussian which is moving in the y direction, while the initial spin projection along the Zaxis is m0: 具XYZ,m兩⌿共T=0兲;m0典=Nexp 冉 −X2+Y2+Z2 2 ␴ 2 冊 ⫻exp共ikyY兲 ␦ 共m,m0兲.共3兲 Note that, neglecting the effects of the border, the Ycomponent of the wave function is not affected by the interaction. Border effects will be relevant when the transit time, which is the time that the particle takes to go from the situation in which the field vanishes 共B ជ =0兲to the situation where the field is stationary 共B ជ ⯝B0u ជ z兲, is short compared to the precession time of the spin in the magnetic field. An estimate of the transit time is tt=B0/共B1v兲. The precession time is tp =ប/共B0 ␮ 兲. It can be seen that, for realistic cases, the transit time, although short compared to the time that the beam spend within the magnet, is always larger than the precession time. So the spins of the particles of the beam have the time to adapt to the magnetic field in which they enter. The wave function is given by a wave packet that can be factorized into a Ycomponent and an 共X,Z兲component. The Ycomponent will evolve freely inside the magnet, because the Hamiltonian does not have any interaction term which depends on Y, once that the border effects 共occurring at Y =0 and Y=L兲are found to be negligible. Note that the wave packet will stay within the magnetic field during a time ␶ =L/vy, where vy=បky/M. Assuming that the size of the wave packet ␴ is very small compared to L and taking into account that the transit time ttis much smaller than T, we can consider that the magnetic field starts at T=0 and finishes at T= ␶ . So we focus on solving the two-dimensional time-dependent problem, which corresponds to calculating the time evolution between the time T=0 and T= ␶ in a Hamiltonian H=PX 2+PZ 2 2M− ␮ B ជ ·I ជ ,共4兲 considering that the initial wave function is 具XZ;m兩⌽共T=0兲;m0典=Nexp 冉 −X2+Z2 2 ␴ 2 冊 ␦ 共m,m0兲.共5兲 It is convenient to make use of dimensionless variables. So we define x=X/ ␴ ,z=Z/ ␴ ,t=T/ ␶ , and h=H ␶ /ប. Then, the equation of motion becomes h兩⌽共t兲;m0典=id dt兩⌽共t兲;m0典.共6兲 The dimensionless Hamiltonian can be written as h=h0+v, with h0=A 2共px 2+pz 2兲,v=−S关Iz共z+z0兲−Ixx兴,共7兲 where px=−id/dx,pz=−id/dz, and the dimensionless parameters A,S,z0are A=ប ␶ M ␴ 2,S= ␮ B1 ␶ ␴ ប,z0=B0 ␴ B1.共8兲 The adiabaticity parameter Ais the ratio of the interaction time ␶ to the natural time of expansion of the Gaussian packet. The separation parameter Sis the ratio of the momentum change induced by the magnetic field gradient divided by the momentum width of the Gaussian packet. The inhomogeneity parameter z0determines the relative change of the magnetic field in the range of the Gaussian. Note that in the position 共x=0,z=−z0兲, the magnetic field vanishes. Note that the product AS= ␮ B1 ␶ 2/M ␴ is independent of ប. This magnitude is related to the deviation of the beam in the magnet. For a given trajectory, which is determined by a fixed value of the product AS, the classical limit is reached as S→⬁and A→0. Note that this corresponds to making ប→0 in Eqs. 共8兲. Validity of the semiclassical descriptions We will now discuss the validity of the semiclassical and textbook descriptions of the Stern-Gerlach experiment. It should be noticed that, in general, a beam of particles is not given by a pure quantum mechanical state, but rather by a POTEL et al. PHYSICAL REVIEW A 71, 052106 共2005兲 052106-2 mixture of small quantum wave packets. For definiteness, we consider that initially one has a distribution of particles described as a Gaussian mixture, of range ␴ m, of small Gaussian wave packets of range ␴ . The beam profile will then be characterized by a Gaussian of range ␴ t=冑 ␴ m 2+ ␴ 2. The conditions required, in order to justify the semiclassical description done in 关7兴are the following. 共a兲The inhomogeneity of the magnetic field over the quantum size of the wave packet should be small: ␴ B1ⰆB0. This implies that z0Ⰷ1. 共b兲The momentum change should be large compared to the quantum spread of the beam momentum: ␮ B1 ␶ Ⰷប/ ␴ . This implies that SⰇ1. Note that these conditions are very well satisfied in realistic situations for Stern-Gerlach experiments. However, the validity of the textbook description requires also the far more stringent condition ␴ tB1ⰆB0, which requires a very strong field B0or, alternatively, a very thin beam. The purpose of this work is to investigate the full quantum solution of this problem for values of the parameters z0 and Swhich are not necessarily very large, so that the semiclassical and textbook descriptions become dubious. Nevertheless, in order to have a reference to compare the quantum calculation, we recall the expected results in the textbook description. The trajectory of the center of the wave packet inside the magnet is given by the expression zm共t兲= 1/2共SA兲mt2,共9兲 which depends on the spin projection m. Note that, after the interaction 共t=1兲, the positions of the center of the wave packets for each spin projection are given by zm共1兲 =SAm/2 and their velocities are z ˙m共1兲=SAm/2. If, after the interaction, the beam evolves freely during a time td, then the positions of the center of the wave packets are expected to be given by zm共td兲=共1/2 + td兲共SA兲m.共10兲 As a typical value of the drift time tdwe will consider the time necessary to reach the position zm=−z0, for the spin projection m=−1/2, td=2z0/共SA兲− 1/2. 共11兲 Thus we would expect that, after a drift time td, particles with spin projection m=1/2 should appear around z=z0,x =0, and particles with spin projection m=−1/2 should appear around z=−z0,x=0. III. NUMERICAL CALCULATIONS We consider the scattering of a spin-1/2 particle. We expand the wave function into two components, which have definite spin projections along the zaxis, 具xz;m= 1/2兩⌽共t兲;m0典= ␣ 共x,z,t兲eitSz0/2, 具xz;m= − 1/2兩⌽共t兲;m0典= ␤ 共x,z,t兲e−itSz0/2,共12兲 and the Schrödinger equation for the 共x,z兲plane can be written as 冤 A 2共px 2+pz 2兲−S 2zS 2x S 2xA 2共px 2+pz 2兲+S 2z 冥 冋 ␣ 共x,z,t兲 ␤ 共x,z,t兲 册 =id dt 冋 ␣ 共x,z,t兲 ␤ 共x,z,t兲 册 ,共13兲 where ␣ 共x,z,t兲and ␤ 共x,z,t兲are the components of the spinor in the basis of the eigenstates of Iz. The numerical solution of this equation has already been performed by Garraway and Stenholm 关9兴. However, they considered the case in which z0was large, so their numerical result corresponded to the textbook interpretation. A similar problem has been addressed by Franca et al. 关10兴, but they made use of the adiabatic approximation, neglecting the kinetic energy during the interaction time. To follow our approach we must first write both components of the spinor as linear combinations of harmonic oscillator functions, so that ␣ 共x,z,t兲=兺 nm anm共t兲 ␾ n共x兲 ␾ m共z兲, FIG. 1. Probability distribution for an unpolarized wave packet after going through an inhomogeneous magnetic field. Note the focusing effect of the lower component, which corresponds predominantly to m=−1/2. The upper figure corresponds to A=0.5, S=4. The lower figure is for A=0.1, S=20, which is closer to the classical limit. QUANTUM MECHANICAL DESCRIPTION OF STERN-…PHYSICAL REVIEW A 71, 052106 共2005兲 052106-3 ␤ 共x,z,t兲=兺 nm bnm共t兲 ␾ n共x兲 ␾ m共z兲,共14兲 where ␾ n共x兲and ␾ m共z兲are the harmonic oscillator eigenstates of order nand min the xand zdirections, respectively. To calculate the time-dependent coefficients anm共t兲and bnm共t兲of the expansion, it is natural to rewrite Eq. 共13兲in terms of the well-known creation and destruction operators ax=1 冑2共x+ipx兲,ax †=1 冑2共x−ipx兲, az=1 冑2共z+ipz兲,az †=1 冑2共z−ipz兲.共15兲 Thus, substituting the operators 共15兲into Eq. 共13兲, we obtain the desired system of ordinary coupled differential equations for the coefficients of the expansion of ␣ 共x,z,t兲and ␤ 共x,z,t兲: a ˙nm =iA 4关an+2,m冑共n+1兲共n+2兲+an−2,m冑n共n−1兲+an,m+2冑共m+1兲共m+2兲+an,m−2冑m共m−1兲−2anm共n+m+1兲兴 +iS 2冑2关an,m+1冑m+1+an,m−1冑m−共bn+1,m冑n+1+bn−1,m冑n兲e−iSz0t兴, b ˙nm =iA 4关bn+2,m冑共n+1兲共n+2兲+bn−2,m冑n共n−1兲+bn,m+2冑共m+1兲共m+2兲+bn,m−2冑m共m−1兲−2bnm共n+m+1兲兴 +iS 2冑2关−bn,m+1冑m+1−bn,m−1冑m−共an+1,m冑n+1+an−1,m冑n兲eiSz0t兴,共16兲 where the overdot stands for differentiation with respect to the dimensionless parameter t. This system is solved using a fourth-order Runge-Kutta method. The number of harmonic oscillator basis functions needed in the calculation was typically of the order of 40 in each coordinate. We have performed calculations using typical values of A=0.5, S=4, and z0=4. This corresponds to a case in which the magnetic field vanishes at a distance of 4 ␴ . The time of the interaction is such that the width of the beam would increase by a factor of 冑1+A2. The magnetic field gradient is such that each component of the magnetic field will acquire a momentum of Sប/2 ␴ , in opposite directions. As a comparison, we have also considered calculations with A=0.1, S =20, and z0=4, which produce the same deviation of the beam, but are closer to the classical limit. After the interaction, we consider a drift time td, given by Eq. 共11兲, during which the system evolves in the free Hamiltonian, so that the center of the m=±1/2wave packet would reach the point z=±z0, according to the textbook description. In Fig. 1 we represent the probability distribution of a wave packet, corresponding initially to an unpolarized beam. This is given by P0共x,z兲=1 2兺 mm0 兩具x,z;m兩⌽共t兲;m0典兩2.共17兲 The focusing effect can be clearly seen by comparing the shape of the distributions for the upper and lower components, which correspond predominantly to m=1/2 and m =−1/2, respectively. The effect of the focusing is increased as Adecreases and Sincreases. So we have confirmed that the focusing effect that was predicted in the semiclassical calculation in 关8兴is a genuine result that appears in the quantum mechanical calculation, although it is diffused if the adiabaticity parameter Ahas a sizable value. It should be noticed that this focusing effect was also found in the calculations presented in 关9兴. In contrast to the textbook description, even if the initial beam has a definite spin projection along the zaxis, after the scattering process this spin projection can change. We have evaluated the probability that the particles change their spin projection along the zaxis. It should be noticed that the probability of going from spin up to spin down is not exactly the same as that of going from spin down to spin up. For the reference case 共A=0.5, S=4, z0=4兲, we obtain that p共1/2, −1/2兲=0.0166 and p共−1/2,1/2兲=0.0198. The spin-flip phenomenon also appears in the semiclassical description, because not all the particles that compose the beam see the magnetic field along the zaxis. The semiclassical spin-flip probability is p共1/2,−1/2兲=p共−1/2,1/2兲 =0.0156, which depends only on the value of z0. This is in good qualitative agreement with the quantum calculations. In Fig. 2 we represent the spatial distribution of the spin-flip probability. Note that the spin-flip probability vanishes for particles coming out along the zaxis. The spatial distribution of the spin-flip probability is in qualitative agreement with the semiclassical calculation, which becomes more accurate as one makes the limit A→0, S→⬁, with AS constant. The results of our calculations can be summarized as follows: When a beam of particles, described by a Gaussian POTEL et al. PHYSICAL REVIEW A 71, 052106 共2005兲 052106-4 wave function and with a given spin projection along the z axis, goes through an inhomogeneous magnetic field, most of the particles scatter as expected in the textbook description. However, a sizable fraction of them, which depends on z0 共about 2% for z0=4兲, suffer a change of the spin projection 共spin flip兲. From these particles that suffer spin flip, about half scatter in the same direction as the majority of the particles and the other half scatter in the opposite direction. We can conclude that the spin-flip effect described in the semiclassical description, which was not present in the textbook description of Stern-Gerlach experiments, is supported by the full quantum mechanical calculations. Also, we confirm that the Stern-Gerlach experiment, when considered as a measurement apparatus of the spin projection, is not an ideal measurement 共because there is spin flip兲and it is not fully reliable 共because there is not an exact correlation between the initial spin projection and the final position of the particle兲. However, there are qualitative features of the full quantum mechanical result, such as the difference between updown and down-up spin-flip probabilities, that are not present in the semiclassical description and require further investigation. IV. APPROXIMATE TREATMENTS Having solved numerically the problem, we will consider several approximate treatments to improve our understanding of the phenomena under consideration. The starting point is the exact evolution operator and the free evolution operator U共t兲= exp关−i共h0+v兲t兴,U0共t兲= exp共−ih0t兲.共18兲 It should be noticed that h0and vdo not commute. Thus, a priori there is not a single basis of spin states where the evolution operator is diagonal. Nevertheless, it can be argued that the interaction vdominates over the free Hamiltonian h0. That would indicate that the eigenstates of v, which are states with definite spin projection along the magnetic field 共and hence coherent internal states兲, should play an important role in the approximate solution of this problem. In this section we derive several approximate expressions which make use of expansions of the exact evolution operator in terms of vand h0and its commutators. We can use the coordinates ␳ =冑共z+z0兲2+x2, ␤ = arctan x 共z+z0兲,共19兲 and refer the spin components to the direction of the magnetic field at each position: IB=Izcos共 ␤ 兲−Ixsin共 ␤ 兲,IT=Izsin共 ␤ 兲+Ixcos共 ␤ 兲. 共20兲 In terms of these variables, the initial state can be expressed as 具 ␳ ␤ ;m兩⌽共t=0兲;m0典 =Nexp 冉 − ␳ 2−2 ␳ z0cos ␤ +z0 2 2 冊 ␦ 共m,m0兲 共21兲 and h0and vtake the expressions h0=A 2共p ␳ 2+ ␳ −2p ␤ 2兲,v=−S ␳ IB,共22兲 where p ␳ and p ␤ are the momenta associated with ␳ and ␤ . The relevant commutators are the following: 关h0,v兴=iAS共p ␳ IB−兵p ␤ ,IT其/2 ␳ 兲,共23兲 关关h0,v兴,v兴=−AS2共IB 2+IT 2−兵p ␤ ,Iy其/2兲.共24兲 Note that 关关h0,v兴,h0兴=0 and 关关关h0,v兴,v兴,h0兴=0. For spin1/2 particles, I=1/2, IB 2=IT 2=1/4. A. Adiabatic approximation The simplest approximation for the evolution operator consists in neglecting completely the effect of h0. This leads to the adiabatic approximation, given by U共t兲⯝exp共−itv兲= exp共itS ␳ IB兲.共25兲 Note that this expression conserves the projection of the spin along the direction of the magnetic field. Thus, it is convenient to expand the initial spin state into states 兩n典which fulfill IB兩n典=n兩n典. This can be done considering the rotation of an angle ␤ around the yaxis which takes the z axis to the direction of the magnetic field. Thus, the adiabatic expression for the wave function after the interaction becomes 具 ␳ ␤ ;m兩⌽共t兲;m0典=Nexp 冉 − ␳ 2−2 ␳ z0cos ␤ +z0 2 2 冊 ⫻兺 ndnm 1/2共 ␤ 兲exp共in ␳ St兲dnm0 1/2 共 ␤ 兲. 共26兲 Note that this expression is equivalent to Eq. 共3.3兲in 关10兴, FIG. 2. Contour plot of the probability distribution of the spinflip component 共spin up to spin down兲of the wave function. The maximum is 3.3⫻10−4. QUANTUM MECHANICAL DESCRIPTION OF STERN-…PHYSICAL REVIEW A 71, 052106 共2005兲 052106-5 where they expanded the wave function in components that had definite spin projections along the local magnetic field. This expression contains the qualitative features described in the numerical calculation. There is a spin-flip probability, as m⫽m0. The focusing effect appears when this adiabatic wave function undergoes a free evolution during a time td after the interaction. However, during the interaction time, the probability distribution is frozen. B. Pseudoadiabatic approximation The next approximation consists in neglecting the commutator 关h0,v兴. This leads to the pseudoadiabatic approximation, given by U共t兲⯝exp共−itv兲exp共−ith0兲= exp共itS ␳ IB兲U0共t兲.共27兲 This expression also conserves the projection of the spin along the direction of the magnetic field, but starting from a wave function that has evolved freely during the interaction time t. The wave function has an analytic expression given by 具 ␳ ␤ ;m兩⌽共t兲;m0典=Nexp 冉 − ␳ 2−2 ␳ z0cos ␤ +z0 2 2共1+iAt兲 冊 ⫻兺 ndnm 1/2共 ␤ 兲exp共in ␳ St兲dnm0 1/2 共 ␤ 兲. 共28兲 The difference of this expression with the adiabatic one lies in the fact that the Gaussian wave packet gets wider during the interaction time, by a factor 冑1+A2, which is the widening of the free wave packet during the interaction time. C. Coherent-state approximation We consider the expansion of the evolution operator up to the third order commutator. The following relations can be derived: U共t兲⯝exp 冉 共−it兲3 6关关关h0,v兴,v兴兴 冊 exp共−itv兲 ⫻exp 冉 共−it兲2 2关h0,v兴 冊 U0共t兲.共29兲 This expression is the basis for an analytic treatment of the wave function. For that purpose, we note that the dominant terms in the evolution operator are those which conserve the spin projection along the direction of the magnetic field. The strongly oscillating factor exp共−itv兲tends to cancel the terms that do not conserve IB. We retain in the expansion only those terms which commute with IB. This leads to the expression U共t兲⯝exp共−it3AS2/12兲exp共itS ␳ IB兲exp共−it2ASp ␳ IB兲U0共t兲. 共30兲 The operator exp共−it2ASp ␳ IB兲, when acting on eigenstates of IB, generates a displacement in ␳ , which is given by ␳ f= ␳ i +t2ASIB. This leads to an analytic expression for the wave function, given by 具 ␳ ␤ ;m兩⌽共t兲,m0典 = exp共iAS2t3/12兲N兺 ndnm 1/2共 ␤ 兲exp共in ␳ St兲冑 ␳ n ␳ ⫻exp 冉 − ␳ n 2−2 ␳ nz0cos ␤ −z0 2 2共1+iAt兲 冊 dnm0 1/2 共 ␤ 兲,共31兲 where ␳ n= ␳ −nASt2/2. This wave function conserves the spin projection along the direction of the magnetic field. Thus, the states with a definite spin projection along the magnetic field in each position correspond to the coherent internal states introduced in Ref. 关7兴. So we call this approximation the coherent-state approximation. Note that in this approximation the wave function not only gets wider during the interacting region, but the components with different values of IBseparate. D. Symmetrized approximation We can approximate the evolution operator by the following expression, which is correct up to commutators of fourth order: U共t兲⯝U0共t/2兲exp兵−itv−共−it兲3关关h0,v兴,v兴/12其U0共t/2兲. 共32兲 Neglecting the terms that do not commute with IB,we have U共t兲⯝exp共it3AS2/24兲U0共t/2兲exp共itS ␳ IB兲U0共t/2兲.共33兲 The wave function can be written as 兩⌽共t兲;m0典= exp共iAS2t3/24兲U0共t/2兲兩⌽⬘共t兲;m0典,共34兲 where 具 ␳ ␤ ;n兩⌽⬘共t兲;m0典=Nexp 冉 − ␳ 2−2 ␳ z0cos ␤ +z0 2 2共1+iAt/2兲 冊 ⫻兺 ndnm 1/2共 ␤ 兲exp共−in ␳ St兲dnm0 1/2 共 ␤ 兲, 共35兲 which, although it is not completely analytic, it can be applied to evaluate the expansion of the wave function in a harmonic oscillator basis. This approximation corresponds to split the effect of U0共t兲during the interaction symmetrically, taking half of it before and half of it after the interaction. Note that here also the evolution associated with the interaction conserves the spin projection along the magnetic field. We call this the symmetrized approximation. E. Comparison with the exact calculation We have performed calculations with all the approximations. We find that the qualitative characteristics of the exact calculations discussed above, which are the focusing effect in the component which goes to negative zvalues and the POTEL et al. PHYSICAL REVIEW A 71, 052106 共2005兲 052106-6 presence of spin-flip components, appear in all the calculations. The quantitative differences between the different approaches arise in the momentum distribution of the spin flip component. This comes out symmetric in the adiabatic and pseudoadiabatic approximations 共same probability distribution for positive and negative momenta兲and not fully symmetric in the coherent-state or symmetrized approximations, in closer agreement with the exact calculations. To evaluate the quality of these approximations, we have calculated the average of the overlap between the exact and approximate calculations. This overlap is defined as O=1 2冏兺 m0 具⌽ex共t=1兲;m0兩⌽ap共t=1兲;m0典冏.共36兲 They are displayed in Fig. 3, as a function of the adiabaticity parameter A, for a fixed value of the product AS=2, which determines the deviation of the center of the wave packet in the magnetic field, as shown in Eq. 共9兲. The quantity 1−Ois about 10% for a wide range of values of A. In particular, for A=0.5 and S=4, 1−0=0.088 for the adiabatic calculation and 1−O=0.064 for the pseudoadiabatic calculation. On the contrary, the symmetrized and coherent-state approximations are much better, so that 1−Ois about 0.1%. In particular, for A=0.5 and S=4, 1−0=0.0015 for the coherent-state and 1−0=0.0006 for the symmetrized calculations. The reason for this better agreement arises from the fact that the coherent-state and symmetrized calculations allow for the distortion in the wave function produced by the magnetic field gradient, while for the adiabatic and pseudoadiabatic calculations the effect of the field contributes only to a phase. In all the calculations that we have performed, the quality of the approximated calculations improves as one goes from the adiabatic to the pseudoadiabatic to the coherent state and finally to the symmetrized approximations. Globally considered, the approximations deteriorate as the product SA gets larger, because then there is more distortion introduced in the wave function due to the combined effect of the interaction and the free Hamiltonian. A very interesting case is the limit A→0, S→⬁for fixed values of AS. Naively, one would expect that the adiabatic approximation would be adequate here, as the free Hamiltonian h0is negligible compared to v. However, this is not the case. As shown in Fig. 3, the adiabatic and pseudoadiabatic approximations are rather poor, giving values of 1−Oof about a few percent. The coherent-state and symmetrized approximations are very good for A=0.015, but then they become worse for smaller values of A. Numerical calculations are very difficult when Sis large, because a large oscillator basis is needed. An analytic solution of this limiting case would be desirable. The interest of this limit case 共A→0, AS constant兲is not only formal. In nuclear physics there are cases in which weakly bound nuclei interact strongly with targets during a very short time, so that the quantum state is significantly distorted. The validity of the adiabatic approximation in these situations is open to debate 关12兴. Note that in the definition of the overlap we allow for an overall phase difference between the exact and approximate wave functions. This overall phase difference does not affect any observable. We find that the best approximate calculations 共coherent state and symmetrized兲only reproduce accurately the phase of the exact wave function when both Aand Sare small. We think that this is related to the effect of higher-order terms in the commutator series of the evolution operator, which seem to affect only a global phase in the wave function. So we see from these approximations that a crucial feature of them is the fact that the most relevant terms in the evolution operator conserve the spin projection along the local direction of the magnetic field. This is the basis of the semiclassical calculation performed in 关7兴, in which the states with definite spin projections along the local magnetic field were taken as coherent internal states, and hence their motion could be described in terms of trajectories. Despite the fact that the approximations discussed here, especially the coherent-state and symmetrized approximations, are very accurate, they do not describe an important effect of the exact evolution operator. In all the approaches described here, the scattering amplitudes for given spin projections along the yaxis 共the beam axis兲are equal, up to a phase factor, to the amplitudes in which the spin projections are reversed. This is a result of the fact that only terms which commute with IBare allowed in the expansion of the evolution operator. V. REEXAMINING THE STERN-GERLACH EXPERIMENTS In the textbook description of the Stern-Gerlach experiment, the deflection of the beam gives information of the spin projection along the zaxis, which is the one that points along the magnetic field at the center of the beam. The deflection of the beam is not sensitive to the spin components along other directions. If, for a spin-1/2 particle, the initial FIG. 3. Overlaps of the approximate wave functions with the exact one, as a function of the adiabaticity parameter, for SA=2. The value 1−O=0 correspond to perfect agreement. The solid line is the adiabatic approximation, the dashed line is the pseudoadiabatic approximation, the dotted line is the coherent-state approximation, and the dot-dashed line is the symmetrized approximation. QUANTUM MECHANICAL DESCRIPTION OF STERN-…PHYSICAL REVIEW A 71, 052106 共2005兲 052106-7 spin points along the xaxis, mx=+1/2,thetextbook description would indicate that the pattern of scattered particles would be completely equivalent to that one produced by a mixture of 50% mz=+1/2 and 50% mz=−1/2 particles. The same would be true for mx=−1/2. So a Stern-Gerlach experiment is not expected to give any asymmetry between different spin projections perpendicular to the zaxis. To investigate this question, we define the asymmetry for a given axis as the difference in the probabilities of finding the scattered particles in a given position in the 共z,x兲plane for the two spin projections. Thus, we have Az共x,z兲=兺 mm0m0 ⬘ 具x,z;m兩⌽共t兲;m0典具x,z;m兩⌽共t兲;m0 ⬘典*具m0兩 ␴ z兩m0 ⬘典, 共37兲 Ax共x,z兲=兺 mm0m0 ⬘ 具x,z;m兩⌽共t兲;m0典 ⫻具x,z;m兩⌽共t兲;m0 ⬘典*具m0兩 ␴ x兩m0 ⬘典,共38兲 Ay共x,z兲=兺 mm0m0 ⬘ 具x,z;m兩⌽共t兲;m0典 ⫻具x,z;m兩⌽共t兲;m0 ⬘典*具m0兩 ␴ y兩m0 ⬘典.共39兲 Note that, in the standard description of the Stern-Gerlach experiment, the spin projection along the zaxis is conserved, and thus the asymmetries Axand Ayshould vanish at all points. This is not the case. As shown in Fig. 4共b兲, there is a difference in the pattern of particles scattered depending on the spin projection along the xaxis. This effect is found to depend on the inhomogeneity of the magnetic field, which is determined by z0=B0/B1 ␴ .Ifz0is large, the inhomogeneity of the magnetic field explored by the beam is small and so is Ax. This asymmetry can be calculated, with various degrees of accuracy, making use of the approximate treatments discussed here. It can also be calculated with the semiclassical treatment of 关7兴. The origin of this asymmetry can be understood by arguing that the motion in an inhomogeneous magnetic field conserves the spin projection along the local magnetic field, which has a different direction for the different parts of the wave function. This links with the concept of coherent internal states, which were introduced in Ref. 关8兴. The calculations in Fig. 4共a兲show also that there is an asymmetry Aywhich means that there is a dependence of the spin projection along the yaxis. This is a dynamical effect, which does not appear in the semiclassical description. In fact, in the analytic approximations presented here, the value of Ayvanishes after the interaction. Only after allowing for some time of free evolution do nonvanishing values of Ay develop. The origin of this asymmetry arises from the term AS2p ␤ Iywhich appears in the double commutator 关关h0,v兴,v兴. The effect of this term can be understood because p ␤ is the generator of rotations in the 共x,z兲plane, around the point x =0, z=−z0, where the field vanishes. The effect of this term in the expansion of the evolution operator would generate a rotation in the wave function around the point where the field vanishes, which will be opposite for the different spin projections along the zaxis. Indeed, this effect competes with the interaction v=S ␳ IB, which tends to preserve the spin projection along the direction of the field. The result of this competition is that the magnitude of the asymmetry depends on the ratio AS/z0. Note that the asymmetry Ayis associated to the dynamically generated term AS2p ␤ Iy. This term depends on the spin projection Iybut is independent of Yor PY. So the motion in the Ydirection is unaffected by the dynamics, and hence it is given by the free evolution of the Y component of the initial wave packet. FIG. 4. Asymmetries for particles polarized along the y共a兲,x 共b兲, and z共c兲directions. Note that the maximum asymmetry occurs for particles polarized along the zaxis, but that there are important asymmetries for particles polarized along the xand yaxes. POTEL et al. PHYSICAL REVIEW A 71, 052106 共2005兲 052106-8 The fact that all the asymmetries are nonvanishing and also that they have different behavior as a function of 共x,z兲 leads to an exciting possibility. Consider that we have a beam of particles, so that we do not know their polarization state. We can make the beam go through an inhomogeneous field, as described here, and detect the pattern of scattered particles. Let the polarization state be described initially as a density matrix ␳ =1/2共I+px ␴ x+py ␴ y+pz ␴ z兲, where p ជ is a vector which measures the degree and direction of the beam polarization. Then, the density of particles detected in the 共x,z兲plane will be proportional to P共x,z兲=P0共x,z兲+1 2关pxAx共x,z兲+pyAy共x,z兲+pzAz共x,z兲兴. 共40兲 This allows us to obtain all the components of the polarization vector from the pattern of scattered particles, when a sufficient number of particles are detected. Note that, in contrast to expression 共40兲, the textbook description of the Stern-Gerlach experiment would be consistent with a probability density given by P共x,z兲=P0共x,z兲+1 2pzAz共x,z兲, Az共x,z兲=2P0共x,z兲,z⬎0, Az共x,z兲=−2P0共x,z兲,z⬍0. 共41兲 This expression, when applicable, would allow one to obtain information only on the value of pz. VI. SUMMARY AND CONCLUSIONS We have investigated the motion of a particle with spin in an inhomogeneous magnetic field using a quantum mechanical framework. Our aim is to investigate in detail the limitations of the usual textbook approach to Stern-Gerlach experiments, which assumes that the spin projection along the direction of the magnetic field is conserved, while different spin components acquire a momentum which depends on the gradient of the field. We find that, consistently with a previous semiclassical analysis, there is a sizable probability of spin flip, which depends on the inhomogeneity of the field. Besides, there is a focusing effect in the component that deviates towards the direction in which the modulus of the field decreases. These characteristics are very robust and occur in dynamical situations which are far from the semiclassical limit. Thus, we can conclude that the Stern-Gerlach experiment is not, even in principle, an ideal experiment, which would “project” the internal state into the eigenvalues of the measurement operator. Moreover, the experiment is not fully reliable, as the positions or momenta of the particles do not give unequivocal information on the spin projection. The magnitude that determines how close a Stern-Gerlach experiment is to an ideal reliable measurement is z0= ␴ B0/B1. Only when the magnetic field B0is very large compared to its gradient or when the size of the beam ␴ is very small would the Stern-Gerlach experiment approximate to an ideal reliable measurement. We have investigated different approximate treatments of the exact quantum mechanical problem. We find that, to a good approximation, the interaction occurs as if the spin projection along the magnetic field at each position was conserved. This indicates that, for each position in the inhomogeneous field, the states with a given spin projection along the magnetic field are coherent internal states. Then, provided that the quantum size of the wave function is small compared to the inhomogeneity of the magnetic field, it is meaningful to approximate the motion of these states in terms of classical trajectories. This justifies the treatment performed in 关7兴. It is interesting to note that the adiabatic approximation is not accurate, even in the limit of small A共large mass or short interaction time兲, if, at the same time, the interaction is large so that it generates a fixed deflection angle. This observation can be relevant to cases, such as in nuclear physics 关12兴,in which, although the collision times are short to guarantee the validity of the adiabatic approximation, the forces are so strong to produce a finite deflection. Our calculations indicate that the Stern-Gerlach experiment is not an ideal measuring apparatus, in the sense of Ref. 关5兴. However, this does not mean that one cannot acquire accurate knowledge from the spin state of the projectile by observing the statistical results of the experiment. On the contrary, while an idealized Stern-Gerlach experiment will not give any information of the spin projection along the xor yaxis, the analysis of a realistic Stern-Gerlach experiment, such as modeled in our calculations, can give the value of all the components of the density matrix that describes the polarization of the beam. Our analysis supports the idea that the interpretation of realistic experiments does not require the use of the reduction principle, as discussed by several authors in 关11兴. Thus, the interaction between the spin and magnetic field, which is described in a purely quantum mechanical framework, generates a correlation between the spin polarization of the beam and the final position of the particles of the beam. A measurement of a sufficiently large number of these positions allows one to determine the components of the density matrix of the beam with sufficient statistical accuracy. The reduction principle is not required in this argument. ACKNOWLEDGMENT This work has been partially supported by the Spanish MCyT, Projects No. FPA2002-04181-C04-04 and BFM200203315. QUANTUM MECHANICAL DESCRIPTION OF STERN-…PHYSICAL REVIEW A 71, 052106 共2005兲 052106-9