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The Pilot-Wave Perspective on Spin

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The Pilot-Wave Perspective on Spin Smith ScholarWorks Physics: Faculty Publications Physics 2014 The iP lot-Wave Perspective on Spin Travis Norsen Smith College, [email protected] Follow this and additional works at: https://scholarworks.smith.edu/phy_facpubs Part of the Quantum Physics Commons Recommended Citation Norsen, Travis, "The iP lot-Wave Perspective on Spin" (2014). Physics: Faculty Publications. 19. https://scholarworks.smith.edu/phy_facpubs/19 This Article has been accepted for inclusion in Physics: Faculty Publications by an authorized administrator of Smith ScholarWorks. For more information, please contact [email protected] The pilot-wave perspective on spin Travis Norsena) Smith College, Northampton, Massachusetts 01060 (Received 7 May 2013; accepted 22 November 2013) 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. VC 2014 American Association of Physics Teachers. [http://dx.doi.org/10.1119/1.4848217] I. INTRODUCTION alone. What the critics fail to appreciate, however, is that adding the particle positions allows something to be sub- In an earlier paper, which readers are encouraged to exam- tracted elsewhere in the system. In particular, the dynamical ine first and which I refer to hereafter as “the earlier paper,” laws sketched above—namely, Eqs. (1) and (2)—constitute I attempted to give a physicists’ introduction to the alterna- the entirety of the dynamical postulates of the theory. No tive, de Broglie-Bohm pilot-wave theory of quantum 1 additional axioms or special exceptions to the usual rules— phenomena. As a so-called “hidden variable theory,” the such as the collapse postulate of ordinary QM—need to be pilot-wave theory adds something to the state descriptions of introduced in order to understand measurement or, more gen- ordinary quantum mechanics: in addition to the usual wave erally, the emergence of the familiar everyday classical function W obeying the usual Schr€odinger equation world. In the earlier paper, this point was developed in the con- @W i"h H^W; (1) text of some simple scattering experiments involving single @t ¼ particles. In such situations, the fact that the scattered parti- cle is found, whole, at some particular place at the end of the one also has definite positions for each particle in the system. experiment is explained in the simplest imaginable way: For example, for a system of N spinless nonrelativistic par- there really is a particle following a definite trajectory and ticles, the position X~n of the nth particle will evolve accord- ing to hence possessing a perfectly definite position at all times. The detector finds it at a particular place (even when its wave function is spread out across several different places) dX~ t ~j ~x1;~x2; …;~x ; t nð Þ nð N Þ ; (2) because it is already at a particular place before interacting dt ¼ q ~x1;~x2; …;~xN; t ~ ~xk Xk t k with the detector. And equivariance, in light of the QEH, ð Þ ¼ ð Þ8 guarantees that the probability for the particle to be at a cer- where ~j "h=2mi Wà ~nW W ~nWà is (what in ordi- tain place at the end of the experiment perfectly matches n ¼ð Þð r À r Þ nary QM is termed) the probability current associated with what ordinary QM would instead describe as the probability particle n and q WÃW is (what in ordinary QM is termed) that the measurement intervention triggers a collapse that ¼ the probability density. makes the particle suddenly materialize at that place. It is As explained in the earlier paper, the pilot-wave theory thus clear, in pattern, how the pilot-wave theory reproduces involves the quantum equilibrium hypothesis (QEH), accord- the statistical predictions of ordinary QM for experiments ing to which the particle positions are assumed to be random, that end with the measurement of the position of a particle. with initial (t 0) probability distribution That the pilot-wave theory makes the same predictions as ¼ ordinary QM for other kinds of measurements as well can be 2 P X~1 ~x1; …; X~N ~xN W ~x1; …;~xN; 0 : (3) understood by including the particles constituting the meas- ½ ¼ ¼ ¼j ð Þj uring device in the system under study. Consider for example It is then a purely mathematical consequence of Eqs. (1) and a simple schematic treatment of the measurement of, say, the 2 3 (2) that the probability distribution will remain W -distributed energy of a particle with initial wave function w0 x . The for all times. The family of possible particle trajectoriesj j thus measurement apparatus is imagined to have a pointerð Þ with “flows along with” q,apropertythathasbeendubbedthe spatial coordinate y and initial wave function / y /0 y , 2 2 ð Þ¼ ð Þ “equivariance” of the W probability distribution. where /0 is a narrow packet centered at the origin, corre- Critics of the theoryj oftenj argue that the addition of these sponding to a “ready” state for the apparatus. definite particle positions is pointless (or “metaphysical”) An interaction between the particle and the pointer is because at the end of the day the theory’s empirical predic- regarded as a “measurement of the energy of the particle” if tions match those of ordinary quantum mechanics, which the Schr€odinger equation (for the particle pointer system) latter predictions are of course made using the wave function produces the following sort of time-evolutionþ in the case 337 Am. J. Phys. 82 (4), April 2014 http://aapt.org/ajp VC 2014 American Association of Physics Teachers 337 This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 131.229.64.25 On: Fri, 05 Aug 2016 14:48:53 that the particle is initially in an “energy eigenstate” w x known to be impossible to do, for all of the non-commuting 0ð Þ wi x with eigenvalue Ei: components of spin, what the pilot-wave theory, as sketched ¼ ð Þ above, does for position. wi x /0 y wi x /o y kEi : (4) And yet, in fact, it is entirely false that the pilot-wave ð Þ ð Þ! ð Þ ð À Þ theory cannot deal with spin. Indeed, the truth is that the pilot- That is, at the end of the experiment, the wave function for wave theory deals with spin in an almost shockingly natural— the pointer is now sharply peaked around some new point certainly a shockingly trivial—way. The goal of the rest of y kEi proportional to the initial energy Ei of the particle. this paper is to resolve this paradox and explain how. ¼ The pointer, in short, indicates the energy of the particle. The main ideas of the paper are not new. The pilot-wave But the linearity of the Schr€odinger equation then immedi- theory was applied to spin already in 1955 by Bohm et al.,8 ately implies that, in the general case in which the particle is and numerical simulations of the theory’s account of spin initially in an arbitrary superposition of energy eigenstates, measurements were carried out by Dewdney et al. in the the evolution goes as follows: 1980s.9 Our approach here, following Bell’s more elegant treatment in Ref. 10, instead aims at simplicity and accessi- ciwi x /0 y ciwi x /0 y kEi : (5) bility. In particular, the proposed delta-function model of a i ð Þ ð Þ! i ð Þ ð À Þ Stern-Gerlach experiment (the one genuine novelty in the pa- X X per) allows one to solve the Schr€odinger equation and deter- The resulting “Schr€odinger cat” type state is of course prob- mine the pilot-wave particle trajectories (without any lematic from the point of view of ordinary QM: instead of recourse to numerical simulations) using the “plane-wave registering some one definite outcome for the experiment, the packet” methods developed in the earlier paper. pointer itself ends up in an entangled superposition. Enter the The delta-function model is developed in the following collapse postulate to save us from the troubling implications section; the associated pilot-wave particle trajectories are of universal Schr€odinger evolution! But this is no problem at then displayed and analyzed in Sec. III. Section IV explains all in the pilot-wave theory, according to which the (directly how the pilot-wave theory deals with cases of repeated spin perceivable) final disposition of the pointer is not to be found measurements, while Secs. V and VI develop examples to in the wave function, but instead in the actual position Y of illustrate the “contextuality” and “non-locality” exhibited by the pointer particle at the end of the experiment. It follows the theory. Some concluding remarks about the relationship from the equivariance property discussed earlier that, assum- between the pilot-wave and orthodox points of view are then ing the initial particle positions are random in accordance made in Sec.
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