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Derivations of Bloch (Majorana--Bloch) equation, von Neumann equation, and Schrödinger--Pauli equation

This paper mathematically derives the space-independent von Neumann and Schrödinger–Pauli equations for electron spin from the classical Bloch (Majorana–Bloch) equation, while also presenting their formulations within a co-quantum dynamic framework and extending the analysis to include nonlinear induction equations with signum branching functions.

Original authors: Lihong V. Wang

Published 2026-09-16

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CC BY 4.0

Original authors: Lihong V. Wang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The world of the very small operates under rules that seem to defy our everyday experience. In the realm of atoms and electrons, particles do not simply sit still or move in straight lines; they possess an intrinsic property called spin, which behaves like a tiny, internal compass needle. For decades, physicists have relied on two different sets of rules to describe how these microscopic needles behave. One set, rooted in classical physics, treats the spin as a physical object that can be visualized and tracked, much like a spinning top wobbling in a magnetic field. The other set, known as quantum mechanics, describes the same spin using abstract mathematical objects that represent probabilities and states of being, rather than physical shapes. For a long time, scientists have been able to translate from the quantum language to the classical one, showing how the complex quantum rules simplify into the familiar classical behavior. However, the reverse journey has remained a mystery: no one had successfully shown how to start with the classical, physical description and mathematically derive the fundamental quantum equations that govern the particle's existence. This gap left a lingering question about the true nature of the transition between the world we can see and the hidden world of the atom.

A researcher at the California Institute of Technology has now bridged this gap by demonstrating that the classical description of an electron's spin can be mathematically transformed into the core equations of quantum mechanics. The work begins with the classical equation that describes how a magnetic needle rotates when placed in a magnetic field. By treating this rotating needle not just as a physical object but as a mathematical structure built from specific complex numbers, the author shows that the classical motion naturally evolves into the quantum equation that describes how a system changes over time. This derivation is significant because it proves that the two descriptions are not just similar, but are mathematically equivalent for a single, pure state of an electron. The study establishes a two-way street between the classical and quantum views, showing that one can start with the simple, physical picture of a spinning magnet and arrive at the sophisticated quantum laws that govern it.

The paper goes further by exploring a newer framework called co-quantum dynamics, which attempts to explain how a quantum system makes a choice between different possible states, a process often called the collapse of the wave function. In standard quantum theory, this collapse is treated as a sudden, unexplained event that happens when a measurement is made. The new research incorporates a specific mechanism into the equations that describes how the spin gradually settles into one direction or another, depending on its relationship with a surrounding nuclear environment. By adding this mechanism to the classical equations, the author derives a new, more complex version of the quantum equation. This new equation includes terms that depend on the current state of the system itself, making the rules of motion change as the system evolves. This suggests that the transition from a fuzzy quantum state to a definite classical outcome is not a random jump, but a continuous process driven by physical induction, similar to how a spinning top eventually slows down and falls over due to friction and gravity.

The findings offer a fresh perspective on one of the most enduring puzzles in physics: how the definite reality we experience emerges from the probabilistic nature of the quantum world. By showing that the classical equations can be converted into the quantum ones, and by extending this to include a physical mechanism for state collapse, the work provides a mathematical pathway that connects the two worlds without relying on unexplained postulates. The research does not claim to have solved every mystery of quantum mechanics, but it successfully demonstrates that the fundamental equations of the quantum world can be built directly from the classical laws of motion. This achievement suggests that the strange behavior of electrons might be more grounded in physical reality than previously thought, offering a clearer view of how the microscopic rules shape the macroscopic world we inhabit.

Technical Summary: Derivations of Bloch (Majorana–Bloch), von Neumann, and Schrödinger–Pauli Equations

Problem Statement
The transition from classical physics to quantum mechanics remains a foundational mystery, particularly regarding the derivation of the Schrödinger equation from classical principles. While it is well-established that the space-independent Schrödinger–Pauli equation (or the von Neumann equation) for a unitary two-level system can be converted into the classical Bloch equation, the inverse conversion—deriving the quantum mechanical equations from the classical Bloch equation—has not been explicitly demonstrated in the literature. Furthermore, the mechanism for wavefunction collapse, often treated as a separate postulate in the Copenhagen interpretation, lacks a unified derivation from classical electrodynamics within standard frameworks.

Methodology
The paper employs a mathematical derivation strategy to establish a two-way transition between classical and quantum descriptions of electron spin. The methodology proceeds in three primary stages:

Derivation of the Majorana–Bloch Equation: The author first provides a derivation of the classical Bloch equation (referred to as the Majorana–Bloch equation, acknowledging Ettore Majorana's prior formulation) using two classical models: a current-loop model and a point-particle model. These models utilize Newton's second law for rotational systems and the Lorentz force to derive the equation of motion for the magnetic dipole moment $\vec{\mu}$ in a magnetic field $\vec{B}$ .
Classical-to-Quantum Conversion (Bloch to von Neumann): The classical Bloch equation is mathematically converted into the space-independent von Neumann equation. This is achieved by:
- Expressing the magnetic moment unit vector $\hat{\mu}$ in terms of spherical coordinates.
- Utilizing the Pauli vector $\vec{\sigma}$ as a mathematical tool to map real-space vectors to complex matrix operations.
- Applying the Pauli vector identity $(\vec{a} \cdot \vec{\sigma})(\vec{b} \cdot \vec{\sigma}) = (\vec{a} \cdot \vec{b})\sigma_0 + i(\vec{a} \times \vec{b}) \cdot \vec{\sigma}$ to transform the cross-product dynamics of the Bloch equation into a commutator form.
- Defining a density matrix $\rho$ and a Hamiltonian $H$ to recover the standard von Neumann equation $i\hbar \frac{d\rho}{dt} = [H, \rho]$ .
Derivation of the Schrödinger–Pauli Equation: The paper attempts to derive the space-independent Schrödinger–Pauli equation from the von Neumann equation under two distinct frameworks:
- Standard Quantum Mechanics: By assuming the density matrix represents a pure state ( $\rho = |\hat{\mu}\rangle\langle\hat{\mu}|$ ) and analyzing the evolution of the bra and ket vectors. The derivation shows that if the bra and ket vectors originate from independent realizations, the Schrödinger–Pauli equation is a necessary and sufficient condition. However, if they originate from identical realizations, the resulting matrix $(1-\rho)$ is singular, meaning the Schrödinger–Pauli equation can be reached only as a sufficient but not necessary condition.
- Co-Quantum Dynamics (CQD): Using the CQD framework, which explicitly models independent realizations of the electron and a "co-quantum" (nucleus), the derivation is repeated. The paper clarifies that to reach the Schrödinger–Pauli equation as a sufficient and necessary condition, one must assume that the bra and ket vectors originate from independent realizations, a condition that is implicit in standard quantum mechanics but made explicit in CQD.

Key Contributions and Results

Inverse Conversion Established: The paper successfully demonstrates the inverse conversion from the classical Bloch equation to the space-independent von Neumann equation for a pure state of electron spin. This completes the two-way transition between the classical and quantum descriptions for this specific system.
Derivation of Schrödinger–Pauli Equation: The space-independent Schrödinger–Pauli equation is derived from the von Neumann equation. The author clarifies that this derivation holds as a necessary and sufficient condition only under the specific assumption that the bra and ket vectors originate from independent realizations. Without this assumption (i.e., if the vectors originate from identical realizations), the equation serves as only a sufficient but not necessary condition.
Nonlinear Extension for Wavefunction Collapse: The paper extends the derivation to include the Landau–Lifshitz–Gilbert equation, which incorporates an induction term. This leads to a nonlinear variant of the von Neumann equation. By integrating the CQD signum branching function (which dictates the direction of collapse based on the relative angles of the electron and nucleus), the author derives:
- An explicit double-commutator form of the nonlinear induction equation.
- A state-dependent effective Hamiltonian that is Hermitian but nonlinear in the density operator.
- A formulation where the branching sign is a functional of the density operator itself, allowing the collapse dynamics to be closed within the Hilbert space representation without a single state-independent linear propagator.

Significance and Claims
The paper claims to resolve a gap in the literature by showing the inverse transition from classical physics (Bloch equation) to quantum mechanics (von Neumann and Schrödinger–Pauli equations) for electron spin. It posits that the transition is mathematically rigorous for pure states, with the caveat that the Schrödinger–Pauli equation is a necessary and sufficient condition only when independent realizations are assumed.

Regarding the wavefunction collapse, the paper does not claim to solve the measurement problem entirely but suggests that the CQD framework, which incorporates the induction term and branching conditions, provides a physical mechanism compatible with the Landau–Lifshitz–Gilbert equation. The author notes that while the standard Schrödinger equation cannot model collapse, the derived nonlinear variant in the CQD framework captures the induction dynamics. The paper concludes that these transitions offer potential new insights into open questions in quantum foundations, specifically regarding the relationship between classical and quantum dynamics and the nature of wavefunction collapse, without asserting that the standard Schrödinger equation can be modified to model collapse in the traditional linear sense.

Technical Summary: Derivations of Bloch (Majorana–Bloch), von Neumann, and Schrödinger–Pauli Equations

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