Possible interference efFect in the Stern-Gerlach phenomenon

We show that if it is possible to manufacture a beam of spin-1/2 heavy atoms corresponding to a quantum-mechanical pure state, then the dark lines of the Stern-Gerlach effect should display a fine structure. The two lips of the open-mouth pattern will each be a doublet; the lighter area between the two lines of the doublet arises from an interference between silver atoms on opposite sides of the incoming beam. We obtain a simple classical model giving exactly the same angular distribution around the open mouth. However, this classical model does not give the fine structure


I. INTRODUCTION
The Stern-Gerlach phenomenon is what occurs when a beam of spin--, ' atoms (preferably heavy ones; the original experiment was with silver) pass through a transverse in- homogeneous magnetic field.A plate, placed some dis- tance beyond the exit from the magnet, shows a distribu- tion of atoms resembling the pair of lips of a half-open mouth.
The phenomenon is important from the point of view of the quantum theory of measurement for two reasons.
The first, which we intend to discuss in a subsequent arti- cle, arises from the proposition that the two lips of the distribution contain atoms whose spins are polarized ei- ther "up" or "down" with respect to the principal trans- verse field component.According to such a proposition, it may be possible to design the beam and the magnet so that the two lips become completely separated.We ~ould then have a perfect "spin meter, " a device which would enable us to demonstrate [1],possibly in the most convincing manner so far, the property known as quantum nonlocality.The second, which we discuss in this ar- ticle, is that if the properties of the deflected beam are correctly described by the Pauli-Schrodinger wave equa- tion, then the phases of atoms in different parts of the beam should be correlated, thus giving us a particularly striking illustration of wave interference for such a large composite object as the silver atom.
Bohm [2] argues that, even with the Pauli-Schrodinger description, the interaction with the magnetic field is essentially a "measurement" which destroys the phase coherence.This is because the magnetic field is not deterministic; it is an interaction between the silver atom and the very large array of atoms constituting the mag- net, and the average field of the latter is necessarily ac- companied by uncontrollable Auctuations.To test his hy- pothesis Bohm suggested that the beam be passed through a second magnet which is a mirror image of the first.Then on the assumption of a deterministic" Pauli- Schrodinger evolution (that is one which converts an in- cident pure state into a final pure state through a unitary transformation), the initial state should be perfectly reproduced after the interaction with the two magnets.
For example, if all the incident silver atoms have their spins in the direction of the beam then so do the atoms emerging from the second magnet.Wigner [3], and En- glert, Schwinger, and Scully [4] have argued that, in prin- ciple, such a process does occur; the latter authors have called it "putting Humpty Dumpty (who is a broken egg) together again." On the Bohm hypothesis the emerging atoms would be completely unpolarized; the pure state evolves into a mixture.The latter authors also find that the degree of exactness with which the second magnet needs to duplicate the first is a technological impossibility, so it seems rather unlike- ly that anyone will attempt the miraculous reconstruction of Humpty Dumpty by this method.However, we shall show in the present article that, at least for an inhomo- geneous field with a certain geometry, an interference effect is already exhibited, according to the Paul- Schrodinger theory, by the fine structure of the distribu- tion after the first magnet.

II. THE MODEL HAMILTONIAN
We base our study on the magnetic field B(x, y, x) =( -B'x, O, Bo+B'z) (0&y & 1')   46 2265 1992 The American Physical Society where the beam of silver atoms is traveling along the y axis, and the field is zero for values of y outside the inter- val (0, Y).We remark that this is the simplest inhomo- geneous field which satisfies the (static) (except of course at the transition points y =0, Y; we post- pone discussion of these to a later article), and that, since we may move the origin to the point (0,0, Bo/-B'), there is no loss of generality in putting Bo =0.Then the Pauli-Schrodinger equation is where g(x, z) is a two-component spinor and (2.4) (2.5) so that g+ (P ) may be regarded as the component of g with spin parallel (antiparallel) to the local field, whose direction is (sing, 0, cosg).Then the solution of (3.1) is cos(8/2) p(r)8)t)=g+.(r)8)e'" . (g/2) p being the magnetic moment and M the mass of the silver atom.We have assumed that the initial state is an eigenstate of the momentum p, and so the y-dependent part of the wave function has been factored out.Now, defining b'= f g (x,z, t)P(x, z, t)(x'+z')dx dz, ( we rescale Eq. ( 2.3) by expressing x and z in units of b"t in units of (R/pB'i() ), and M in units of (R /pB'i(), ).This If the plate is placed a large distance away from the mag- net, it is legitimate to regard the beam issuing from the magnet as a point source, so that the trace on the plate is a picture of the momentum (p) distribution in the outgo- ing beams, that is i p= (Ho+ H, )g, where where g is the two-dimensional Fourier transform of g, that is (2.9) We shall confine ourselves, in the present article, to con- sidering the limiting case M~~, so that H =Ho.This approximation is equivalent to assuming that the silver atoms spend a sufficiently short time in the magnetic field for us to take account of changes in their momentum while neglecting the consequent changes in position.It may be called a nonconvective approximation and has been made many times before.For example, Bohm [2]   and, more recently, Scully, Lamb, and Barut [5] (who also considered a magnetic field similar to ours) made this approximation in a quantum context, while Singh and Sharma [6) made it in an attempted classical treatment.P(R, B, t)= f ™r dr f dgg(r, g, t)e 277 0 0 and t, which should be put equal to MY/p, is the transit time of the silver atoms through the magnetic field.Now we anticipate, and will shortly confirm, that, for large values of t, the distribution 8' will be zero except for large R. We therefore use the stationary-phase asymptotic approximation [7] for the phase factor in this latter equation, that is III.(3.9) be written h(8)= f dylan (g, 8}+y ((, 8)l' -f" dykey (g, 8)+y (g, 8)l'.
By Parseval's theorem, therefore, h(8)-f dr~/+(r, 8)+P (r, 8)= f rdr[~@ (r, 8)~+(f (r, 8+m)) ], 0 since the cross term is zero.Formally we expect h to be a function of t, but it is asymptotically independent of t since, from (3.8), it may The form of (3.8) confirms what we anticipated; given that 1Lis zero for r »1 (or r »6 in the original units) then 1{i is small for ~R t~&&1 -(or ~R pB't ~&-&Pi/b in the original units}.So, crudely speaking, the distribution on the plate is confined to a region close to the circle R = t (the half-open mouth is now fully open).This is, of course, all in the asymptotic regime t &&1.In this same regime there is a perfect correlation between spin and momentum; the spin pattern around the circle R =t is shown in Fig. 1.
The fine structure in (3.8) is exhibited by the interfer- ence between P+ and P, and we shall discuss it in Sec.
V. But first we propose to examine the coarse structure, that is the distribution around the circle R =t.To this end let us form the integral In spite of the title which Stern gave to his original pa- per [8] on the Stern-Gerlach phenomenon ("Experimen- tal demonstration of directional quantization in a mag- netic field" ) his discussion was really classical, as has been pointed out in the more recent literature [9].However, this more recent literature has established that Stern's classical model, at least for the values of 80 and 8' used in the Stern-Gerlach experiment [10] and its modern refinements [11], does not give a satisfactory explanation of the phenomenon.
We now propose a model, which is as classical in spirit as Stern's, but which reproduces the coarse-structure dis- tribution h(8) of (3.13), and also the associated spin dis- tribution of Fig. 1.We suppose that, more or less im- mediately on entering the magnetic field, a silver atom has position x, momentum p and spin parallel [antiparal- lel] to the local magnetic field with probability density W+(x, p) [W (x, p)].To establish the correspondence with the model in Sec.III we shall suppose that f ~(x,p)ld'p=lg (x)l'.This means that a kind of "collapse" occurs to the silver atoms as they enter the magnetic field, their spins aligning either parallel or antiparallel to the local field.As a consequence of this alignment, no subsequent precession of the spins occurs inside the magnetic field, and the momentum change is determined by the standard classi- cal magnetic force (It, V)B.
The magnetic force acting on a parallel (antiparallel) aligned atom is of a constant unit magnitude (in the units we are using) and in the direction x(x), that is the unit vector in the outward (inward) radial direction.Hence the momentum distribution at time t is 8 -f [8'+(rp, p -xt)+ W ( -rp, p -xt)]d x as t~~.This last step follows because, for large t, the argument pxt(p+xt ) has large modulus, and therefore negligible probability, unless x=p (orp).Hence where J is the Jacobian for the change of coordinates p'=pxt, for which we find W(R, 6,t) -(24rR) '[1+ cos@(6)] X [D "(g)]2e -"/2)w here, as before, g=Rt, and (5.4} where D, /2 is the parabolic cylinder function, (3.8) gives J=R't '~s ec(8 -6)~.
But, again for large t, the integrand is small except for R close to t and 0 close to e. Hence J-R'/R and f W"(p, t)R dR 0 f r dr f [W+(rp, p')+ W ( -rp, p')]d p' .

(4.7)
With the identification made in (4.1), this is just Eq. (3.13).Now as to the organization of the spin distribution, leading to Fig. 1, this is clearly obtained from the latter equation, because the antiparallel spin direction atrp is the same as the parallel direction at rp.The corre- sponding process in the quantum evolution is more complex; it is explained as a phase cancellation of those com- ponents of g having the "wrong" spin direction.We con- clude that there is nothing really wavelike in the coarse structure of (3.13) nor in the spin pattern of Fig. 1; any wavelike properties which silver atoms may have must be sought in the fine structure of (3.8).
f R dR f d8[W+(rp, p -xt) POSSIBLE INTERFERENCE EFF FECT IN THE STERN-. . . the circle depicted in Fig. 1, ac- del (~) of Sec.II an cording to the quantum mo e madel (cl) of Sec.IV.

10 'd 1
Stern-Gerlach expenment, ac-" hoto late" of the istri u d' ribution of deflected silve ver atoms in an i ea e It has been suppose t a d hat the incident beam a p erlach lines which arise in the m (left-hand plate) and classica rig of the two separate Stern-Ger ac ine e c h fi d cos@(B)]~D, & (g+ia cosB t t n of silver atoms at the p late for the