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Canadian Journal of Physics

Reduction of the atomic wavefunction in the Stern–Gerlach magnetic field

Michael Devereux

Published Jun 1, 2015• 18 citations
Publisher

Abstract

Observation of two separated beam spots at a detection screen downstream of a Stern–Gerlach magnet does not, in fact, demonstrate that the wavefunction of a neutral spin one-half particle has remained in a spin superposition while traveling through that magnetic field. The wavefunction may have been reduced to just one spin-direction eigenfunction, as D. Bohm suggested, by immediate momentum and energy transfer with the magnet, rather than by subsequent, which-way determination at the screen. The same two beam spots at the detector screen will result. Einsteinian relativity, and the understanding of Schrödinger evolution applicability through a static potential, forbid continuation of a spin superposition through the Stern–Gerlach field. A calculation for single wavepacket development there conforms to observations from the Stern–Gerlach experiment. And several experiments corroborate immediate reduction to a single spin eigenfunction in the magnetic field. Additionally, Ramsey’s separated, oscillating fields observations, and related experiments, do not rebut this understanding.

DOI
10.1139/cjp-2015-0031

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Observation of two separated beam spots at a detection screen downstream of a Stern–Gerlach magnet does not, in fact, demonstrate that the wavefunction of a neutral spin one-half particle has remained in a spin superposition while traveling through that magnetic field. The wavefunction may have been reduced to just one spin-direction eigenfunction, as D. Bohm suggested, by immediate momentum and energy transfer with the magnet, rather than by subsequent, which-way determination at the screen. The same two beam spots at the detector screen will result. Einsteinian relativity, and the understanding of Schrödinger evolution applicability through a static potential, forbid continuation of a spin superposition through the Stern–Gerlach field. A calculation for single wavepacket development there conforms to observations from the Stern–Gerlach experiment. And several experiments corroborate immediate reduction to a single spin eigenfunction in the magnetic field. Additionally, Ramsey’s separated, oscillating fields observations, and related experiments, do not rebut this understanding.
Abstract For nearly one hundred years most quantum-mechanics texts have depicted a continuous, spin-direction superposition of the wavefunction of the silver atom traversing a Stern–Gerlach (S–G) magnet. But there are sound scientific arguments which deny that understanding. Schrodinger’s equation for that continuous wavefunction development cannot describe the transfer of energy from the magnetic field to that atom. Modern micro-fabrication techniques now make it possible to implement a simple, accessible experiment testing whether a spin superposition does continue in the magnet. Because the S–G experiment is the prototype for quantum measurement, that observation is crucial to implementing a realistic quantum theory of measurement.
A detailed analysis of the classic Stern–Gerlach experiment is presented. An analytical simple solution is presented for the quantum description of the translational and spin dynamics of a silver atom in a magnetic field with a gradient along a single z-direction. This description is then used to obtain an approximate quantum description of the more realistic case with a magnetic field gradient also in a second y-direction. An explicit relation is derived for how an initial off center deviation in the y-direction affects the final result observed at the detector. This shows that the “mouth shape” pattern at the detector observed in the original Stern–Gerlach experiment is a generic consequence of the gradient in the y-direction. This is followed by a discussion of the spin dynamics during the entry of the silver atom into the magnet. An analytical relation is derived for a simplified case of a field only along the z-direction. A central question for the conceptual understanding of the Stern–Gerlach experiment has been how an initially unpolarized spin ends up in a polarized state at the detector. It is argued that this can be understood with the use of the adiabatic approximation. When the atoms first experience the magnetic field outside the magnet, there is in general a change in the spin state, which transforms from a degenerate eigenstate in the absence of a field into one of two possible non-degenerate states in the field. If the direction of the field changes during the passage through the device, there is a corresponding adiabatic change of the spin state. It is shown that an application of the adiabatic approximation in this way is consistent with the previously derived exact relations.
The Stern-Gerlach experiment has played an important role in our understanding of quantum behavior. We propose and analyze a modified version of this experiment where the magnetic field of the detector is in a quantum superposition, which may be experimentally realized using a superconducting flux qubit. We show that if incident spin-1/2 particles couple with the two-state magnetic field, a discrete target distribution results that resembles the distribution in the classical Stern-Gerlach experiment. As an application of the general result, we compute the distribution for a Gaussian waveform of the incident fermion. This analysis allows us to demonstrate theoretically: (1) the quantization of the intrinsic angular momentum of a spin-1/2 particle, and (2) a correlation between EPR pairs leading to nonlocality, without necessarily collapsing the particle’s spin wavefunction.
A nondestructive type of Stern-Gerlach ef- fect for an individual electron is described that has been thoroughly demonstrated in experiments at the University of Washington. This Stern-Gerlach makes use of an inhomogeneous magnetic field provided by a weak auxiliary magnetic bottle and is nondestructive in contrast to all previous versions of the effect. As in the classic Stern-Gerlach effect, changes in the spin state are detected via changes in classical particle trajectories; however, what is observed now is not a deflection of the orbit but rather a change of its frequency in the storage well. A simplified model of the continuous Stern-Gerlach effect at zero temperature is introduced to develop the relation between minimum measurement time required for determination of spin state, driven oscillation amplitude, and zero-point fluctuations in the storage well. The problem of the reduction of the wavefunction by the interaction of the electron with the apparatus is addressed following W. Pauli.
We obtain exact solutions for the Schrödinger-Pauli matrix equation for a neutral particle of spin 1/2 in a magnetic eld with a field gradient. The analytical wavefunctions are written on the symmetry plane Y = 0, which contains the incident and splitted beams, in terms of the Airy functions. The time-evolution of the probability densities, | Y+| ² and | Y-|² , and the eigenenergies are calculated. These include a small contribution from the field gradient, alpha, proportional to (alpha )2/3, which amounts to equal energy displacements on both magnetic levels. The results are generalized for spin S = 3/2, and in this case we found that the m = ±1/2 and m = ±3/2 magnetic sublevels are unequaly splitted by the field gradient, being the difference in energy of the order 0.4 MHz. Replacing real experimental parameters we obtained a spatial splitting of the spin up and spin down states of the order deltaz ~ 4 mm, in accordance to a real Stern-Gerlach experiment.
Most textbook authors assume that the force on a magnetic moment sent into a Stern–Gerlach apparatus, is exclusively in the direction of the magnetic field. Hence they conclude that classically one should expect a continuous distribution of particles along a line in the field direction, on a screen placed behind the apparatus. However, it is easily shown that, because the divergence of the field is zero, there is in fact an equally large force transverse to the field direction. This would lead to an isotropic distribution of particles on the screen, in contradiction to the above conclusion. The paradox is resolved by taking into account the torque on the magnetic moment. This produces a rapid precession about the field direction, causing the transverse force to average out and thus giving the usual textbook result.