A short note about directional quantization of oscillatory magnetic dipole moments associated with moments of inertia

This note points out the possibility of fast oscillations within magnetic moments of quantum particles in the light of a recent finding (1).

Kapitza pendulum (inverted pendulum) is a pendulum where its pivot is forced to vibrate up and down and this allows the pendulum to stay upright against gravity which is not possible with a regular pendulum. It was discovered in 1908 but only explained in 1951 (2) and later by Landau (3) through the effective potential model. It is also known that Kapitza pendulum can be realized magnetically by the work of Lisovskii et al., 2007 (4) by subjecting a compass needle to the field of a Helmholtz coil driven by an AC signal.

The same behaviour can be obtained with a magnetic body having an oscillatory magnetic moment with an asymmetry, for example an electromagnet driven by AC signal and also having a small DC bias (1). Exposing such a body to a static magnetic field causes to vibrate it and as a result, this body can find stable equilibrium against the magnetic field (anti-parallel) with respect to it's time-averaged magnetic moment and also can be aligned with it in parallel as an ordinary magnet can do. This means an oscillatory magnetic dipole moment associated with a moment of inertia can find two stable orientations with respect to the static magnetic field it is subjected to.

This is similar to the result of the Stern-Gerlach experiment made in 1922 (5) which confirmed the hypothesis of directional quantization (6) (Richtungsquantelung) and showed silver atoms sent with random orientation exit the apparatus with polarizations either in parallel or antiparallel orientations with respect to a magnetic field it traverses. This result caused serious difficulties within classical physics and shaped the emerging quantum mechanics. Now, a hundred years later, the Kapitza pendulum gives the opportunity to reinterpret the Stern-Gerlach experiment with the idea of inbuilt fast oscillations of magnetic moments.

[1] Ucar, H. Directional quantization of an oscillatory magnetic dipole moment associated with a moment of inertia, 2023. https://doi.org/10.31219/osf.io/pkusx

[2] P. L. Kapitza. Pendulum with a vibrating suspension. Usp. Fiz. Nauk, 44:7–15, 1951.

[3] L.D. Landau and E.M. Lifshitz, Mechanics (Pergamon, Oxford, 1960), pp. 93-95. https://doi.org/10.1002/zamm.19610410910

[4] Lisovskii, F.V., Mansvetova, E.G. Analogue of the Kapitza pendulum based on a compass arrow in an oscillating magnetic field. Bull. Russ. Acad. Sci. Phys. 71, 1500–1502 (2007). https://doi.org/10.3103/S1062873807110044

[5] Gerlach, W., Stern, O. Das magnetische Moment des Silberatoms. Z. Physik 9, 353–355 (1922). https://doi.org/10.1007/BF01326984

[6] Schmidt-Böcking, H., Schmidt, L., Lüdde, H.J. et al. The Stern-Gerlach experiment revisited. EPJ H 41,327–364 (2016). https://doi.org/10.1140/epjh/e2016-70053-2

 I wrote a note about stability of magnetic bound state which evaluate the stability partially based on the zenithal angle of the free body. 

Preprint:  https://doi.org/10.31219/osf.io/uymc2

A note on the stability of the angular motion of a free body with a magnetic moment exposed to a rotating magnetic field

Hamdi Ucar

jxucar@gmail.com

June 29, 2023

A basic magnetic bound state solution in classical physics requires the free body to perform a distinct cyclic angular motion as a result of its interaction with a rotating magnetic field. This motion resembles that of the arm of a spherical pendulum, tracing a conical pattern. The orientation of the pendulum arm corresponds to the orientation of the free body which is identified by its magnetic moment vector. According to the model in a basic configuration, this vector rotates synchronously with the rotating field on the same axis and maintains a constant zenithal angle, referred to as φ. While this angle satisfies the equilibrium between the magnetic torque exerted on the body and its inertial response, the stability analysis of this equilibrium with respect to the angle φ has not been fully explored mathematically. The complete analytical evaluation of the motion's stability can be challenging due to the non-linear nature of the equations of motion. However, we can examine a fundamental stability requirement to determine if the equilibrium might be stable. If this requirement is not met, the equilibrium is inherently unstable; otherwise, it may exhibit stability depending on the specific conditions provided. Through the evaluation of magnetic torque and angular displacement vectors it is found the dynamics satisfy a basic stability requirement.