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Brazilian Journal of Physics

Construction of exact solutions for the Stern-Gerlach effect

J.D. Bulnes, Ivan S. Oliveira

Published Sep 1, 2001• 8 citations
Publisher

Abstract

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.

DOI
10.1590/s0103-97332001000300023

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American Journal of Physics•Oct 13, 2003•3 citations
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.
The Stern-Gerlach experiment is the fundamental experiment in order to exhibit the quantization of spin and understand the measurement problem in quantum mechanics. However, although the Stern-Gerlach experiment plays an essential role in the teaching of quantum mechanics, no complete analysis of this experiment using Pauli spinors is presented in the pedagogical literature. This paper presents such an analysis and develops implications for the theory of quantum measurement. We first propose an analytic expression of both the wave function and the probability density in the Stern-Gerlach experiment. Our explicit solution is obtained via a complete integration of the Pauli equation over time and space. The probability density evolution describes a slipping of the wave packet into two separate packets due to the measurement device, but it cannot account for impacts. We therefore calculate the de Broglie-Bohm trajectories, which not only explain impacts naturally, but also accounts for the spin quantization following the magnetic field gradient. It is then possible to propose a clear explanation of measurement effects in the Stern-Gerlach experiment.
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.
The splitting of a beam of potassium atoms in the classical Stern-Gerlach scheme under varying inhomogeneity of the magnetic field is experimentally investigated in this work. First, the basic shape of the beam in the absence of an effective field is recorded, which makes it possible to introduce and calibrate the geometrical parameters of the channel. Then, when the current in the magnet windings increases and the field gradient grows, a systematic shift of the beam density maxima is observed, described by the model function F(u), which includes the parameter q, which characterizes the strength of interaction of atoms with the field. Theoretical calculations based on this function showed good agreement with the experimental results, including the asymmetry of the distribution due to the nonideal symmetry of the magnetic system. The obtained dependences of the position of the intensity maxima on ∇B confirmed the validity of both linear and asymptotic approximation for different modes of magnet operation. These conclusions have both fundamental importance for understanding the quantum mechanical aspects of beam splitting and applied significance in the development of methods for precise control of spin-polarized atomic beams in spectroscopy and spintronics.
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 consists of a beam composed by neutral spin 1/2 particles that enters on a non-uniform magnetic field, which splits the beam into two, one of which consists of spin +1/2 particles and the other one consists of spin -1/2 particles. We initially consider that the particles are independent and do not interact. The objective of this work is to study and compare two cases. The first one is the case in which the particle with spin and spatial degrees of freedom initially independents is subjected to two consecutive Stern-Gerlach. The second one is similar to the first, differing only by it's initial state, which is an entangled state of spin with a second spin 1/2 particle.
A Stern–Gerlach experiment constructed in the spirit of the original experiment is reported that demonstrates spatial quantization for a beam of silver atoms directly to the unaided eye. Splitting of the atomic beam is seen both parallel to the magnetic field direction, as in the “usual” experiment, and in a direction mutually perpendicular to the field and atomic trajectory, which is a new transverse Stern–Gerlach effect. Splitting of up to 4 mm can be recorded visually. Calculations based on the force components acting on the atomic beams reproduce the experimental results within the experimental accuracy of ±0.3 mm.