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arXiv:quant-ph/0306007

A Stern-Gerlach Experiment in Time

Ashmead, John

Published Jun 1, 2003• 0 citations
arXivHTMLPDF

Abstract

In non-relativistic quantum mechanics, path integrals are normally derived from the Schroedinger equation. This assumes the two formalisms are equivalent. Since time plays a very different role in the Schroedinger equation and in path integrals, this may not be the case. We here derive path integrals directly by imposing two requirements: correct behavior in the classical limit and the most complete practicable symmetry between time and space. With these requirements, the path integral formalism predicts quantum fluctuations over the time dimension analogous to the quantum fluctuations seen over the three space dimensions. For constant potentials there is no effect. But the coupling between rapidly varying electromagnetic fields and the quantum fluctuations in time should be detectable. We consider a variation on the Stern-Gerlach experiment in which a particle with a non-zero electric dipole moment is sent through a rapidly varying electric field, oriented parallel to the particle's trajectory. The Schroedinger equation predicts changes to the precession frequency of the wave function about the trajectory but no physical splitting of the beam. With the approach here, path integrals predict the changes to the precession frequency and in addition that the beam will be split in velocity and time.

DOI
10.48550/arxiv.quant-ph/0306007

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In non-relativistic quantum mechanics, path integrals are normally derived from the Schroedinger equation. This assumes the two formalisms are equivalent. Since time plays a very different role in the Schroedinger equation and in path integrals, this may not be the case. We here derive path integrals directly by imposing two requirements: correct behavior in the classical limit and the most complete practicable symmetry between time and space. With these requirements, the path integral formalism predicts quantum fluctuations over the time dimension analogous to the quantum fluctuations seen over the three space dimensions. For constant potentials there is no effect. But the coupling between rapidly varying electromagnetic fields and the quantum fluctuations in time should be detectable. We consider a variation on the Stern-Gerlach experiment in which a particle with a non-zero electric dipole moment is sent through a rapidly varying electric field, oriented parallel to the particle's trajectory. The Schroedinger equation predicts changes to the precession frequency of the wave function about the trajectory but no physical splitting of the beam. With the approach here, path integrals predict the changes to the precession frequency and in addition that the beam will be split in velocity and time.
Normally we quantize along the space dimensions but treat time classically. But from relativity we expect a high level of symmetry between time and space. What happens if we quantize time using the same rules we use to quantize space? To do this, we generalize the paths in the Feynman path integral to include paths that vary in time as well as in space. We use Morlet wavelet decomposition to ensure convergence and normalization of the path integrals. We derive the Schrödinger equation in four dimensions from the short time limit of the path integral expression. We verify that we recover standard quantum theory in the non-relativistic, semi-classical, and long time limits. Quantum time is an experiment factory: most foundational experiments in quantum mechanics can be modified in a way that makes them tests of quantum time. We look at single and double slits in time, scattering by time-varying electric and magnetic fields, and the Aharonov-Bohm effect in time.
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.
The century-old Stern–Gerlach setup is paradigmatic for a quantum measurement. We visualize the electron trajectories following the Bohmian zig-zag dynamics. This dynamics was developed in order to deal with the fundamentally massless nature of particles (with mass emerging from the Brout–Englert–Higgs mechanism). The corresponding trajectories exhibit a stochastic zig-zagging, as the result of the coupling between left- and right-handed chiral Weyl states. This zig-zagging persists in the non-relativistic limit, which will be considered here, and which is described by the Pauli equation for a non-uniform external magnetic field. Our results clarify the different meanings of ‘spin’ as a property of the wave function and as a random variable in the Stern–Gerlach setup, and they illustrate the notion of effective collapse. We also examine the case of an EPR-pair. By letting one of the entangled particles pass through a Stern–Gerlach device, the non-local influence (action-at-a-distance) on the other particle is manifest in its trajectory, e.g. by initiating its zig-zagging.
In this thesis we are investigating a different formalism of non-relativistic quantum mechanics called the path integral formalism.It is a generalization of the classical least action principle.The introduction to this subject begins with the construction of the path integral in terms of the idea of probability amplitudes whose absolute square gives the probability of finding a system in a particular state.Then we show that if the Lagrangian is a quadratic form one needs only to calculate the classical action besides from a time-dependent normalization constant to find the explicit expression of the path integral.We look in to the subject of two kinds of slit-experiments: The square slit, the single-and the double-Gaussian slit.Also, the propagator for constrained paths is calculated and applied to the Aharonov-Bohm effect, which shows that the vector potential defined in classical electrodynamics have a physical meaning in quantum mechanics.It is also shown that the path integral formulation is equivalent to the Schrödinger description of quantum mechanics, by deriving the Schrödinger equation from the path integral.Further applications of the path integral are discussed.
In this thesis we are investigating a different formalism of non-relativistic quantum mechanics called the path integral formalism.It is a generalization of the classical least action principle.The introduction to this subject begins with the construction of the path integral in terms of the idea of probability amplitudes whose absolute square gives the probability of finding a system in a particular state.Then we show that if the Lagrangian is a quadratic form one needs only to calculate the classical action besides from a time-dependent normalization constant to find the explicit expression of the path integral.We look in to the subject of two kinds of slit-experiments: The square slit, the single-and the double-Gaussian slit.Also, the propagator for constrained paths is calculated and applied to the Aharonov-Bohm effect, which shows that the vector potential defined in classical electrodynamics have a physical meaning in quantum mechanics.It is also shown that the path integral formulation is equivalent to the Schrödinger description of quantum mechanics, by deriving the Schrödinger equation from the path integral.Further applications of the path integral are discussed.
The Stern-Gerlach experiment (SGE) has been known for decades, but a complete theory that explains all facets of it remains enigmatic. In its classic realization, each spin state of an atom follows a unique path or trajectory in an SGE. The enigmatic part is the final output of SGE. Feynman envisaged the outcome to be an entangled state, a superposition of $|\text{spin}\phantom{\rule{4.pt}{0ex}}\text{state}\ensuremath{\rangle}\ensuremath{\bigotimes}|\text{corresponding}\phantom{\rule{4.pt}{0ex}}\text{path}\ensuremath{\rangle}$. Later works described the final state of SGE as a mixed state, a weighted average of states belonging to separate beams or trajectories. Notably, Schwinger, Englert, and Scully compared the difficulty of obtaining a coherent output (a pure state) to ``putting together Humpty Dumpty'' again. Yet recent experiments using Bose-Einstein condensates indicate the emerging beams are partly coherent, favoring Feynman's description. In this work, we revisit SGE using a recent version of the quantum master equation. We show that both outcomes are probable depending on the time of flight and the timescale of the environmental fluctuations. We predict the output is initially coherent and grows gradually into a mixed state. Moreover, we also estimate how the magnetic-field gradient helps broaden the beam waist. Moreover, our treatment implies that the wave function of an atom passing through a Stern-Gerlach device does not undergo an instantaneous collapse; rather, a gradual decay of coherences results in a mixed state, resulting in an irreversible process.