Despite conventional wisdom that spin-1/2 systems have no classical analog, we introduce a set of classical coupled oscillators with solutions that exactly map onto the dynamics of an unmeasured electron spin state in an arbitrary, time-varying, magnetic field. While not addressing the quantum measurement problem (discrete outcomes and their associated probabilities), this new classical analog yields a classical, physical interpretation of Zeeman splitting, geometric phase, the electron's doubled gyromagnetic ratio, and other quantum phenomena. This Lagrangian-based model can be used to clarify the division between classical and quantum systems, and might also serve as a guidepost for certain approaches to quantum foundations.
Spin has long been regarded as a fundamentally quantum phenomena that is incapable of being described classically. To bridge the gap and show that aspects of spin's quantum nature can be described classically, this work uses a classical Lagrangian based on the coupled oscillations of Foucault's pendulum as an analog for the electron spin state in an external magnetic field. With this analog it is possible to demonstrate that Foucault's pendulum not only serves as a basis for explaining geometric phase, but is also a basis for reproducing a broad range of behavior from Zeeman-like frequency splitting to precession of the spin state. By demonstrating that unmeasured electron spin states can be fully described in classical terms, this research opens the door to using the tools of classical physics to examine an inherently quantum phenomenon.
We extend a recent classical mechanical analog of Bohr's atom consisting of a scalar field coupled to a massive point-like particle [P. Jamet and A. Drezet, "A mechanical analog of Bohr's atom based on de Broglie's double-solution approach," Chaos 31, 103120 (2021)] by adding and studying the contribution of a uniform weak magnetic field on their dynamics. In doing so, we are able to recover the splitting of the energy levels of the atom called Zeeman's effect within the constraints of our model and in agreement with the semiclassical theory of Sommerfeld. This result is obtained using Larmor's theorem for both the field and the particle, associating magnetic effects with inertial Coriolis forces in a rotating frame of reference. Our work, based on the old "double solution" theory of de Broglie, shows that a dualistic model involving a particle guided by a scalar field can reproduce the normal Zeeman effect.
The quantum dynamics of arbitrary
N-level systems, including dissipative systems, are modeled exactly here by the dynamics of classical coupled oscillators. A one-to-one correspondence is established between the quantum states and the positions of the oscillators. Quantum coherence, expectation values, and measurement probabilities for system observables can therefore be realized from the corresponding classical states. Although the well-known equivalence [SU(2), SO(3) homomorphism] of two-level quantum dynamics to a rotation in real, physical space cannot be generalized to arbitrary
N-level systems, the representation of quantum dynamics by a system of coupled harmonic oscillators in one physical dimension
is general for any
N. The time evolution of an
N-level system [generated by a complex element of the SU(
N) group], is first represented as the rotation of a real state vector in (unphysical) hyperspace, as previously known for density matrix states and also extended here to include Schr\"odinger states. The resulting rotor in
n Euclidean dimensions [the rotation group SO(
n)] is then mapped directly to
n oscillators in one physical dimension, which significantly reduces the level of abstraction required to visualize quantum dynamics compared to vector models or generalized Bloch spheres in higher dimensions. The number of such oscillators needed to represent
N-level systems scales as
N2 for the density matrix formalism but increases only linearly with
N for Schr\"odinger states. Values for the classical coupling constants are readily derived from the system Hamiltonian, allowing construction of classical mechanical systems that provide insight into the dynamics of abstract quantum systems (new dynamical invariants) as well as a metric for characterizing the interface between quantum and classical mechanics. A distinctive attribute of the quantum-classical connection as presented here is the necessity for both positive and negative couplings and, in the case of dissipative systems, antisymmetric couplings.
The electron spin-1/2, a cornerstone of quantum mechanics traditionally viewed as an intrinsic property, is reinterpreted here through a revolutionary classical spiral orbit model in molecular hydrogen (H
2). By modeling the electron's motion as a three-dimensional spiral trajectory, we derive the total spin
S=0.866ℏ from x-y plane oscillations and the discrete
Sz=±0.5ℏ from a classical quantization of the z-axis projection, eliminating the need for an inherent quantum attribute. The model accurately predicts H
2's ground state (
S=0) and triplet state (
S=1), matches spectroscopic transitions (e.g., 4.18~eV, 8.69~eV), and reproduces magnetic properties (diamagnetic ground state, paramagnetic excited state) with a refined g-factor
g≈1.97, closely approaching the quantum value of 2. Validated against experimental data, this approach challenges the foundational assumption of intrinsic spin, offering a dynamic, classical explanation that bridges quantum phenomena with intuitive mechanics. This work not only redefines spin's origin but also paves the way for a new paradigm in understanding quantum-classical interfaces.
Contrary to popular belief, the Zeeman effect can be treated exactly in single-electron systems, for arbitrary magnetic field strengths, as long as the term quadratic in the magnetic field can be ignored. These formulas were actually derived already around 1927 by Darwin, using the classical picture of angular momentum, and presented in their proper quantum-mechanical form in 1933 by Bethe, although without any proof. The expressions have since been more or less lost from the literature; instead, the conventional treatment nowadays is to present only the approximations for weak and strong fields, respectively. However, in fusion research and other plasma physics applications, the magnetic fields applied to control the shape and position of the plasma span the entire region from weak to strong fields, and there is a need for a unified treatment. In this paper we present the detailed quantum-mechanical derivation of the exact eigenenergies and eigenstates of hydrogen-like atoms and ions in a static magnetic field. Notably, these formulas are not much more complicated than the better-known approximations. Moreover, the derivation allows the value of the electron spin gyromagnetic ratio g s to be different from 2. For completeness, we then review the details of dipole transitions between two hydrogenic levels, and calculate the corresponding Zeeman spectrum. The various approximations made in the derivation are also discussed in details.
The trajectory of a charged-massive point-particle in presence of an external magnetic field, at the classical level, is here obtained by using spinorial equations of motion. Within this representation the solution is given by means of two 2-component Weyl-spinors which contain global phase-factors coinciding with the quantum representation of a 1/2 spin particle in its rest-frame. Each individual spinor may be then associated to the up- and -down state respectively. The formalism then appears to reveal classical properties of the spin as well as quantum ones.