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arXiv:2407.08025

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

Lihong V. Wang

Published Jul 10, 2024• 0 citations
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Abstract

The transition from classical physics to quantum mechanics has been mysterious. Here, we derive the space-independent von Neumann equation for electron spin mathematically from the classical Bloch or Majorana--Bloch equation, which is also derived. Subsequently, the space-independent Schrödinger--Pauli equation is derived in both the quantum mechanical and recently developed co-quantum dynamic frameworks. An additional appendix incorporates the CQD signum branching function into the nonlinear von Neumann induction equation and develops its explicit double-commutator and state-dependent effective-Hamiltonian forms.

DOI
10.48550/arxiv.2407.08025

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The transition from classical physics to quantum mechanics has been mysterious. Here, we derive the space-independent von Neumann equation for electron spin mathematically from the classical Bloch or Majorana--Bloch equation, which is also derived. Subsequently, the space-independent Schrödinger--Pauli equation is derived in both the quantum mechanical and recently developed co-quantum dynamic frameworks. An additional appendix incorporates the CQD signum branching function into the nonlinear von Neumann induction equation and develops its explicit double-commutator and state-dependent effective-Hamiltonian forms.
After publishing the derivation from the classical Bloch equation to the quantum von Neumann equation to the Schrdinger-Pauli equation for spin-12\tfrac{1}{2}21​, we proposed renaming the Bloch equation to the Majorana-Bloch equation because Majorana's work predated Bloch's in the presentation of the Bloch equation by 14 years. Here, we first generalize our previous derivation to higher spins or angular momenta in coherent pure states. Using the polynomial representation of the coherent-state projector, we derive an invertible mapping from the Majorana-Bloch equation to the von Neumann equation, establishing a one-to-one correspondence between these two formalisms. Application of the Ehrenfest theorem also shows that expectation values in these states reproduce the classical equation of motion as expected. Then, we obtain arbitrary spin-sss states by symmetrizing tensor products of spin-12\tfrac{1}{2}21​ primitives, in accordance with the Majorana construction or the Schur-Weyl duality.
The derivation of the time dependent Schrödinger equation with transversal and longitudinal relaxation, as the quantum mechanical analog of the classical Landau-Lifshitz-Bloch equation, has been described. Starting from the classical Landau-Lifshitz-Bloch equation the transition to quantum mechanics has been performed and the corresponding von-Neumann equation deduced. In a second step the time Schrödinger equation has been derived. Analytical proofs and computer simulations show the correctness and applicability of the derived Schrödinger equation.
After recognizing that Majorana first formulated what later became known as the Bloch equation fourteen years before Felix Bloch, we propose the name Majorana–Bloch equation. Building on our previous work for spin-½, we have extended the derivation of time evolution from the Majorana–Bloch equation through the von Neumann equation to systems of arbitrary angular momentum in pure coherent states. By exploiting the polynomial form of the coherent-state projector, we establish a one-to-one correspondence between classical and quantum dynamics on the coherent-state manifold. Taking expectation values in coherent states also recovers the classical equation through the Ehrenfest theorem.
In a previous article [H. Bergeron, J. Math. Phys. 42, 3983 (2001)], we presented a method to obtain a continuous transition from classical to quantum mechanics starting from the usual phase space formulation of classical mechanics. This procedure was based on a Koopman-von Neumann approach where classical equations are reformulated into a quantumlike form. In this article, we develop a different derivation of quantum equations, based on purely classical stochastic arguments, taking some elements from the Bohm-Fenyes-Nelson approach. This study starts from a remark noticed by different authors [M. Born, Physics in My Generation (Pergamon Press, London, 1956); E. Prugovecki, Stochastic Quantum Mechanics and Quantum Spacetime (Reidel, Dordrecht, 1986)], suggesting that physical continuous observables are stochastic by nature. Following this idea, we study how intrinsic stochastic properties can be introduced into the framework of classical mechanics. Then we analyze how the quantum theory can emerge from this modified classical framework. This approach allows us to show that the transition from classical to quantum formalism (for a spinless particle) does not require real postulates, but rather soft generalizations.