© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

Physical Review A

Adiabatic motion of a neutral spinning particle in an inhomogeneous magnetic field

Robert G. Littlejohn, Stefan Weigert

Published Aug 1, 1993• 61 citations

Abstract

The motion of a neutral particle with a magnetic moment in an inhomogeneous magnetic field is considered. This situation, occurring, for example, in a Stern-Gerlach experiment, is investigated from classical and semiclassical points of view. It is assumed that the magnetic field is strong or slowly varying in space, i.e., that adiabatic conditions hold. To the classical model, a systematic Lie-transform perturbation technique is applied up to second order in the adiabatic-expansion parameter. The averaged classical Hamiltonian contains not only terms representing fictitious electric and magnetic fields but also an additional velocity-dependent potential. The Hamiltonian of the quantum-mechanical system is diagonalized by means of a systematic WKB analysis for coupled wave equations up to second order in the adiabaticity parameter, which is coupled to Planck's constant. An exact term-by-term correspondence with the averaged classical Hamiltonian is established, thus confirming the relevance of the additional velocity-dependent second-order contribution.

DOI: 10.1103/physreva.48.924

Related papers

Quantum mechanical description of Stern-Gerlach experiments

G. Potel, F. Barranco, Sara Cruz-Barrios +1 more

The motion of neutral particles with magnetic moments in an inhomogeneous magnetic field is described in a quantum mechanical framework. The validity of the semiclassical approximations which are generally used to describe these phenomena is discussed. Approximate expressions for the evolution operator are derived and compared to the exact calculations. Focusing and spin-flip phenomena are predicted. The reliability of Stern-Gerlach experiments to measure spin projections is assessed in this framework.

arXiv:quant-ph/0409206•Physical Review A•May 27, 2005•28 citations

Phase-space study of the Stern-Gerlach experiment

C W Muller, F W Metz

We study the motion of neutral spinning particles in static inhomogeneous magnetic fields, where the orbital motion and spin dynamics do not factorize. Using the phase-space representation of quantum mechanics, the Moyal propagator is obtained for the case of a linearized field. As an example, the evolution of particles in coherent rotational states is calculated. In order to interpret the Wigner state function in a classical mechanical way, we calculated the Husimi functions.

Journal of Physics A Mathematical and General•May 21, 1994•5 citations

Semiclassical description of Stern-Gerlach experiments

Sara Cruz-Barrios, Joaquín Gómez-Camacho

The motion of neutral particles with magnetic moments in an inhomogeneous magnetic field is described in a semiclassical framework. The concept of coherent internal states is used in the formulation of the semiclassical approximation from the full quantum-mechanical expression. The classical trajectories are defined only for certain spin states, that satisfy the conditions for being coherent internal states. The reliability of Stern-Gerlach experiments to measure spin projections is assessed in this framework.

arXiv:quant-ph/0010079•Physical Review A•Nov 30, 2000•16 citations

1D toy model for magnetic trapping

S. Gov, S. Shtrikman, H. Thomas

We study, both classically and quantum mechanically, the problem of a neutral particle with a spin angular momentum S, mass m, and magnetic moment μ, moving in one dimension in an inhomogeneous magnetic field given by B=B0ẑ+B⊥′xŷ. This problem serves for us as a toy model to study the trapping of neutral particles. We identify K≡[S2(B⊥′)2/μmB03], which is the ratio between the precessional frequency of the particle and its vibrational frequency, as the relevant parameter of the problem. Classically, we find that when μ is antiparallel to B, the particle is trapped provided that K<0.5. We also find that viscous friction, be it translational or precessional, destabilizes the system. Quantum mechanically, we study the problem of a spin S=ℏ/2 particle in the same field. Treating K as a small parameter for the perturbation from the adiabatic Hamiltonian, we find that the lifetime Tesc of the particle in its trapped ground state is Tesc=(Tvib/2π)(1/8πK)exp(2/K), where Tvib=2πmB0/μ(B⊥′)2 is the classical period of the particle when placed in the adiabatic potential V=μ|B|.

American Journal of Physics•Apr 1, 2000•10 citations

An occurrence of an effective anharmonic velocity dependent potential

J M F Gunn

An integrable dynamical problem of a classical neutral particle interacting with an inhomogeneous magnetic field via a magnetic dipole moment is considered; this has been studied quantum mechanically (1978). The equation for the centre-of-mass velocity is of second order resembling that of an anharmonic oscillator with velocity replacing position in the equation. The velocity undergoes curious oscillations, where the spin and centre-of-mass degrees of freedom exchange energy, although there is also, in general, a uniform component to the velocity. An application to the motion of a classical 'neutron' in a magnetic domain wall is discussed, and comparison with previous quantum treatments made.

Journal of Physics A Mathematical and General•Aug 1, 1985•5 citations

Phenomenological Theory of the Stern-Gerlach Experiment

Sergey A. Rashkovskiy

Abstract We propose a phenomenological theory of spin behavior in a magnetic field, which explains from the point of view of classical physics the two-valued result of the Stern-Gerlach experiment. The behavior of the spin and intrinsic magnetic moment of an electron wave of an atom in an external magnetic field is considered. We show that in a weak magnetic field, the intrinsic magnetic moment of an electron wave is always oriented parallel to the magnetic field strength vector, while in a strong magnetic field, depending on the initial orientation of the intrinsic magnetic moment, two orientations are realized: either parallel or antiparallel to the magnetic field strength vector. Within the framework of classical electrodynamics, the calculation of the motion of an atomic beam in an inhomogeneous magnetic field is carried out, which reproduces the results of the Stern-Gerlach experiment.

•Oct 31, 2022•2 citations

The motion of a neutral particle with a magnetic moment in an inhomogeneous magnetic field is considered. This situation, occurring, for example, in a Stern-Gerlach experiment, is investigated from classical and semiclassical points of view. It is assumed that the magnetic field is strong or slowly varying in space, i.e., that adiabatic conditions hold. To the classical model, a systematic Lie-transform perturbation technique is applied up to second order in the adiabatic-expansion parameter. The averaged classical Hamiltonian contains not only terms representing fictitious electric and magnetic fields but also an additional velocity-dependent potential. The Hamiltonian of the quantum-mechanical system is diagonalized by means of a systematic WKB analysis for coupled wave equations up to second order in the adiabaticity parameter, which is coupled to Planck's constant. An exact term-by-term correspondence with the averaged classical Hamiltonian is established, thus confirming the relevance of the additional velocity-dependent second-order contribution.

The motion of neutral particles with magnetic moments in an inhomogeneous magnetic field is described in a quantum mechanical framework. The validity of the semiclassical approximations which are generally used to describe these phenomena is discussed. Approximate expressions for the evolution operator are derived and compared to the exact calculations. Focusing and spin-flip phenomena are predicted. The reliability of Stern-Gerlach experiments to measure spin projections is assessed in this framework.

The motion of neutral particles with magnetic moments in an inhomogeneous magnetic field is described in a semiclassical framework. The concept of coherent internal states is used in the formulation of the semiclassical approximation from the full quantum-mechanical expression. The classical trajectories are defined only for certain spin states, that satisfy the conditions for being coherent internal states. The reliability of Stern-Gerlach experiments to measure spin projections is assessed in this framework.

We study, both classically and quantum mechanically, the problem of a neutral particle with a spin angular momentum S, mass m, and magnetic moment μ, moving in one dimension in an inhomogeneous magnetic field given by B=B0ẑ+B⊥′xŷ. This problem serves for us as a toy model to study the trapping of neutral particles. We identify K≡[S2(B⊥′)2/μmB03], which is the ratio between the precessional frequency of the particle and its vibrational frequency, as the relevant parameter of the problem. Classically, we find that when μ is antiparallel to B, the particle is trapped provided that K<0.5. We also find that viscous friction, be it translational or precessional, destabilizes the system. Quantum mechanically, we study the problem of a spin S=ℏ/2 particle in the same field. Treating K as a small parameter for the perturbation from the adiabatic Hamiltonian, we find that the lifetime Tesc of the particle in its trapped ground state is Tesc=(Tvib/2π)(1/8πK)exp(2/K), where Tvib=2πmB0/μ(B⊥′)2 is the classical period of the particle when placed in the adiabatic potential V=μ|B|.

An integrable dynamical problem of a classical neutral particle interacting with an inhomogeneous magnetic field via a magnetic dipole moment is considered; this has been studied quantum mechanically (1978). The equation for the centre-of-mass velocity is of second order resembling that of an anharmonic oscillator with velocity replacing position in the equation. The velocity undergoes curious oscillations, where the spin and centre-of-mass degrees of freedom exchange energy, although there is also, in general, a uniform component to the velocity. An application to the motion of a classical 'neutron' in a magnetic domain wall is discussed, and comparison with previous quantum treatments made.

Abstract We propose a phenomenological theory of spin behavior in a magnetic field, which explains from the point of view of classical physics the two-valued result of the Stern-Gerlach experiment. The behavior of the spin and intrinsic magnetic moment of an electron wave of an atom in an external magnetic field is considered. We show that in a weak magnetic field, the intrinsic magnetic moment of an electron wave is always oriented parallel to the magnetic field strength vector, while in a strong magnetic field, depending on the initial orientation of the intrinsic magnetic moment, two orientations are realized: either parallel or antiparallel to the magnetic field strength vector. Within the framework of classical electrodynamics, the calculation of the motion of an atomic beam in an inhomogeneous magnetic field is carried out, which reproduces the results of the Stern-Gerlach experiment.