[Paper Review] A classical analog for the electron spin state

K. B. Wharton, R. A. Linck+1 more|arXiv (Cornell University)|Nov 14, 2011

Quantum Mechanics and Applications16 references3 citations

TL;DR

This paper proposes a classical Lagrangian model of four coupled oscillators that exactly reproduces the dynamics of an unmeasured electron spin-1/2 state in an arbitrary time-varying magnetic field. By introducing a many-to-one mapping from classical variables to quantum states, the model classically explains Zeeman splitting, geometric phase, and the doubled gyromagnetic ratio—demonstrating that the dynamics of electron spin are fully classically describable, with non-classical features confined to measurement outcomes.

ABSTRACT

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.

Motivation & Objective

• To demonstrate that the dynamics of an unmeasured electron spin-1/2 state can be exactly reproduced by a classical system.
• To clarify the boundary between classical and quantum behavior by isolating the role of measurement in non-classicality.
• To provide a Lagrangian-based classical model that naturally incorporates hidden variables corresponding to complex oscillator amplitudes.
• To extend the classical polarization analog to three-dimensional magnetic fields using a four-oscillator system.
• To offer a testable classical framework for foundational studies in quantum mechanics, particularly regarding hidden variables and action principles.

Proposed method

• Construct a Lagrangian for four coupled harmonic oscillators with time-dependent coupling coefficients matching the components of a magnetic field.
• Derive equations of motion from the Lagrangian, showing they map exactly onto the Schrödinger-Pauli equation for a spin-1/2 particle.
• Impose a global constraint (L₂ = 0) that reduces the solution space to match quantum state evolution, enforcing a many-to-one mapping.
• Use complex amplitudes A and B (or unit quaternions) as hidden variables that do not affect the quantum state but influence the classical dynamics.
• Demonstrate that the expectation value of spin angular momentum corresponds to a specific combination of oscillator positions and momenta, preserving geometric phase behavior.
• Show that a 2π rotation of the spin expectation value induces a π phase shift in the classical oscillators, mirroring the quantum geometric phase.

Experimental results

Research questions

• RQ1 Can the dynamics of an unmeasured electron spin-1/2 state be exactly reproduced by a classical system?
• RQ2 What classical structure underlies the geometric phase and Zeeman splitting in electron spin?
• RQ3 Are there classical hidden variables that correspond to the quantum state’s phase and amplitude without altering the observable evolution?
• RQ4 Does the doubled gyromagnetic ratio of the electron have a natural classical explanation in this model?
• RQ5 Can a classical Lagrangian framework reproduce the full dynamics of a two-level quantum system in arbitrary time-varying fields?

Key findings

• The four-oscillator classical model exactly reproduces the time evolution of an unmeasured electron spin-1/2 state under any time-varying magnetic field.
• The model exhibits a many-to-one mapping from classical solutions to quantum states, with the complex amplitudes A and B serving as hidden variables.
• The geometric phase shift of π after a 2π rotation of the spin expectation value is naturally reproduced via the oscillator dynamics.
• Zeeman energy splitting and the doubled gyromagnetic ratio emerge directly from the classical Lagrangian and coupling structure.
• The classical system preserves the Bloch sphere dynamics and phase evolution, including global U(1) symmetry, but cannot account for discrete measurement outcomes.
• The model suggests that non-classicality in electron spin is confined to the measurement process, not the dynamics, supporting foundational models with hidden variables.