No Access Submitted: 07 May 2013 Accepted: 22 November 2013 Published Online: 25 March 2014
American Journal of Physics 82, 337 (2014); https://doi.org/10.1119/1.4848217
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The alternative pilot-wave theory of quantum phenomena—associated especially with Louis de Broglie, David Bohm, and John Bell—reproduces the statistical predictions of ordinary quantum mechanics but without recourse to special ad hoc axioms pertaining to measurement. That (and how) it does so is relatively straightforward to understand in the case of position measurements and, more generally, measurements, whose outcome is ultimately registered by the position of a pointer. Despite a widespread belief to the contrary among physicists, the theory can also account successfully for phenomena involving spin. The main goal of this paper is to explain how the pilot-wave theory's account of spin works. Along the way, we provide illuminating comparisons between the orthodox and pilot-wave accounts of spin and address some puzzles about how the pilot-wave theory relates to the important theorems of Kochen and Specker and Bell.
Thanks to George Greenstein for organizing the wonderful series of meetings at which I had the opportunity to first try out some of this material on a live audience. Philip Pearle, John Townsend, Roderich Tumulka, and two anonymous referees provided helpful comments on an earlier draft.
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12. 12. Note that this expression for the field is simplified and idealized: A more precise expression would involve field components perpendicular to $ẑ$—which would allow the divergence of $B→$ to vanish, as it of course should—as well as a large constant field in the z-direction. As it is usually explained, this large constant field causes rapid precession about the z-axis: This makes any transverse forces (that would otherwise arise from the transverse magnetic field gradients) average to zero. It is much simpler, though, to simply look the other way in regard to the violation of Maxwell's equations, and thus avoid needing to talk about any of these subtleties, by simply pretending that Eq. (6) accurately describes the field to which the particles are subjected.
13. 13. If the reflected wave term from Eq. (14) is retained, the detailed particle trajectories are a little more complicated. During the time period when the incident and reflected packets overlap, the particle velocity in that (y < 0) overlap region is slightly oscillatory, with an average velocity—in precisely, the sense explained and used in the earlier paper—that is slightly less than $ℏk/m$ by an amount proportional to $(κz)2$. Thus, particles very near the trailing edge of the incident packet will in fact be overtaken by the trailing edge of the packet before reaching y = 0 and will hence be swept away, back out to the left, by the reflected packet. In short, particles that happen to begin in a small (parabolic) slice near the trailing edge of the incident packet will not behave as described in the main text, but will instead reflect. The behavior of the particles that do transmit through to the y > 0 region, however, is (in the applicable $κw→0$ limit) unaffected. So we will continue to suppress discussion at this level of detail in the main text.
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15. 15. Actually there is one subtle point that deserves to be addressed. In analyzing the measurement of z-spin, we have assumed that the incident wave packet has an essentially rectangular cross-sectional profile, with (as mentioned explicitly) some width w along the z-direction and (as only really assumed tacitly) some width w′ along the x-direction. This makes the problem “effectively 2D” in the sense that (for the width-w′ range of x-values where the wave function has support and where the particle might therefore be) nothing depends on x. If, however, the incident wave function is the same but the SG apparatus is rotated through some arbitrary angle (not an integer multiple of 90°), this nice symmetry is spoiled. Everything still works as claimed; it's just that one has to think a little bit (about, e.g., slicing the 3D wave packet up into a stack of planes perpendicular to $n̂$) to convince oneself of this. Alternatively, one can recognize that the statistics of the measurement outcomes will not depend on the exact cross-sectional profile of the incident packet and hence imagine that the cross-sectional profile of the incident packet is rotated in tandem with the SG device so that the analysis remains transparent. Google Scholar
16. 16. Of course, the mere fact that the two sub-beams are spatially separated is not sufficient to prove that the empty packet is forevermore dynamically irrelevant to the motion of the particle. The “empty” spin-down sub-beam could be made to become dynamically relevant, for example, by arranging (by use, say, of appropriate magnetic fields) to deflect the spin-down beam back up toward the spin-up beam, at which point interference effects—including perhaps the re-constitution of a guiding wave proportional to $(10)$—could occur. But if, as suggested in Fig. 4, the spin-down sub-beam is blocked by a brick wall, then the usual process of decoherence will render it impossible, at least for all practical purposes, to arrange the appropriate overlapping of the two components of the wave function, and treating the wave as having “effectively collapsed” is entirely warranted.
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translated by W. De Baere as “ On Wave-Particle Duality,” <http://arxiv.org/abs/physics/0504043>; and still earlier by Lorentz, who discussed in 1922 a pilot-wave theory of photons, previously suggested by Einstein; , Google Scholar
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32. 32. For example, in D. Griffiths, Introduction to Electrodynamics, 2nd ed. (Prentice-Hall, Englewood Cliffs, NJ, 1989), the author writes: “ all magnetic fields are due to electric charges in motion, and in fact, if you could examine a piece of magnetic material on an atomic scale you would find tiny currents: electrons orbiting around nuclei and electrons spinning on their axes.” , Google Scholar
And later: “ Since every electron constitutes a magnetic dipole (picture it, if you wish, as a tiny spinning sphere of charge), you might expect paramagnetism to be a universal phenomenon.” Interestingly, Griffiths is more carefully orthodox about spin in his Introduction to Quantum Mechanics, 2nd ed. (Pearson Prentice Hall, Upper Saddle River, NJ, 2004). Google Scholar
33. 33. In this respect it is somewhat telling that the worst offenses against the official orthodox view tend to occur in figures and illustrations. Almost all quantum mechanics textbooks, for example, include diagrams depicting the precession, induced by the presence of a magnetic field, of a particle's spin vector about some axis. Townsend's text (Ref. 3131. J. S. Townsend, A Modern Approach to Quantum Mechanics, 2nd ed. (University Science Books, Mill Valley, CA, 2012). All quotes are from Chaps. 3 and 4. ) avoids that particular misleading suggestion of a classical picture. But his front cover art—depicting a Stern-Gerlach spin measurement much like our Fig. 1—includes little circles with arrows (pointing, respectively, up and down) in the downstream sub-beams. The circles are even yellow, inviting us to recall Bell's warning, quoted earlier in Ref. 2626. The authors of Ref. 25 describe themselves as following Bell in concluding that, for the pilot-wave theory, “spin is not real.” Bell, for example, writes (in the first essay of Ref. 10): “We have here a picture in which although the wave has two components, the particle has only position.$…$ The particle does not ‘spin,’ although the experimental phenomena associated with spin are reproduced. Thus the picture resulting from a hidden-variable account of quantum mechanics need not very much resemble the traditional classical picture that the researcher may, secretly, have been keeping in mind. The electron need not turn out to be a small spinning yellow sphere.” Gestures in the direction of this same point were made even earlier by M. Renninger, “ Zum Wellen-Korpuskel-Dualismus,” Z. Phys. 136, 251–261 (1953), https://doi.org/10.1007/BF01325679 translated by W. De Baere as “ On Wave-Particle Duality,” <http://arxiv.org/abs/physics/0504043>; and still earlier by Lorentz, who discussed in 1922 a pilot-wave theory of photons, previously suggested by Einstein; see H. A. Lorentz, Problems of Modern Physics (Dover, New York, 1967), p. 157. note.
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