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Optical and Quantum Sensing and Precision Metrology II

Derivation from Majorana–Bloch equation to von Neumann equation: extension to higher angular momentum in coherent states

Lihong V. Wang

Published Mar 5, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.1117/12.3085701

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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.
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 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.
We study how can an angular momentum coherent state ∣ τ 〉 keeps its form-invariant during time evolution governed by the Hamiltonian H = f ( t ) J + + f * ( t ) J − + g ( t ) J z . We discuss this topic in the context of boson realization of ∣ τ 〉 . By employing the entangled state representation ∣ ζ 〉 and deriving a new binomial theorem involving two-subscript Hermite polynomials, we derive the wave function 〈 ζ ∣ τ 〉 , which turns out to be a single-subscript Hermite polynomial. Based on this result the maintenance of angular momentum coherent state during time evolution is examined, and the value of τ ( t ) is totally determined by the parameters involved in the Hamiltonian.
We studied the interacting system of qubit with a collection of N qubits or known as spin coherent state. From the interaction of the qubit – spin coherent state system, we can observe many interesting phenomena such as ‘collapse and revival’ in the time evolution of the qubit state, the qubit of attractor state, the qubit – spin coherent state entanglement, and the occurrence of the Schrӧdinger’s cat state. The event of ‘collapse and revival’ will be mainly focused in this study. Prior studies had shown the Bloch Sphere and Majorana Sphere representations for the interaction system between qubit and a coherent state. Therefore, in this research, we will analyze the time evolution of the qubit state in the qubit – spin coherent state interacting system for the one qubit case, and represent the dynamics of the qubit state using the Majorana Sphere representation.
Abstract By representing a quantum state and its evolution with the Majorana stars on the Bloch sphere, the Majorana representation provides us an intuitive way to study a physical system with SU(2) symmetry. In this work, based on coherent states, we propose a method to establish the generalization of Majorana representation for a general symmetry. By choosing a generalized coherent state as a reference state, we give a more general Majorana representation for both finite and infinite systems and the corresponding star equations are given. Using this method, we study the squeezed vacuum states for three different symmetries, Heisenberg–Weyl, SU(2) and SU(1,1), and express the effect of squeezing parameter on the distribution of stars. Furthermore, we also study the dynamical evolution of stars for an initial coherent state driven by a nonlinear Hamiltonian, and find that at a special time point, the stars are distributed on two orthogonal large circles.