After publishing the derivation from the classical Bloch equation to the quantum von Neumann equation to the Schrdinger-Pauli equation for spin- $\tfrac{1}{2}$ , 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- $s$ states by symmetrizing tensor products of spin- $\tfrac{1}{2}$ primitives, in accordance with the Majorana construction or the Schur-Weyl duality.After publishing the derivation from the classical Bloch equation to the quantum von Neumann equation to the Schrdinger-Pauli equation for spin-$\tfrac{1}{2}$, 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-$s$ states by symmetrizing tensor products of spin-$\tfrac{1}{2}$ primitives, in accordance with the Majorana construction or the Schur-Weyl duality.

arXiv:2508.08414Aug 11, 20250 citations

Analytical solutions and expressions of the propagator for Bloch equations

Heung‐Ryoul Noh

In this paper, we present analytical solutions to the Bloch equations. After solving the secular equation for the eigenvalues, derived from the Bloch equations, analytical solutions for the temporal evolution of the magnetization vector are obtained at arbitrary initial conditions. Subsequently, explicit analytical expressions of the propagator for the Bloch equations and optical Bloch equations are obtained. Compared to the results of existing analytical studies, the present results are more succinct and rigorous, and they can predict the behavior of the propagator in different regions of parameter spaces. The analytical solutions to the propagator can be directly used in composite laser-pulse spectroscopy.In this paper, we present analytical solutions to the Bloch equations. After solving the secular equation for the eigenvalues, derived from the Bloch equations, analytical solutions for the temporal evolution of the magnetization vector are obtained at arbitrary initial conditions. Subsequently, explicit analytical expressions of the propagator for the Bloch equations and optical Bloch equations are obtained. Compared to the results of existing analytical studies, the present results are more succinct and rigorous, and they can predict the behavior of the propagator in different regions of parameter spaces. The analytical solutions to the propagator can be directly used in composite laser-pulse spectroscopy.

International Journal of Modern Physics BJul 1, 20201 citation

The Bloch equation

Pieters, Laura ME

Jan 1, 20170 citations

A Simple Derivation of the Bloch Equation

D. ter Haar

Elsevier eBooksJan 1, 19770 citations

Analytic solutions of Bloch and Maxwell-Bloch equations in the case of arbitrary field amplitude and phase modulation

А. В. Алексеев, N. V. Sushilov

The matrix exponent method which we applied recently for the solution of the Bloch equations [Sov. Phys. JETP 62, 1125 (1985)] allows us to obtain in the present paper the Bloch-equation solutions when the amplitude and the phase of the exciting field are arbitrary functions of time. The solution's validity conditions are derived. It is shown that one can also apply the matrix exponent method to the solution of the Maxwell-Bloch equations. The equation describing the evolution of the area of a pulse with modulated envelope and phase is obtained. This equation has an analytic solution which is a generalization of the well-known McCall-Hahn solution.The matrix exponent method which we applied recently for the solution of the Bloch equations [Sov. Phys. JETP 62, 1125 (1985)] allows us to obtain in the present paper the Bloch-equation solutions when the amplitude and the phase of the exciting field are arbitrary functions of time. The solution's validity conditions are derived. It is shown that one can also apply the matrix exponent method to the solution of the Maxwell-Bloch equations. The equation describing the evolution of the area of a pulse with modulated envelope and phase is obtained. This equation has an analytic solution which is a generalization of the well-known McCall-Hahn solution.

Physical Review AJul 1, 199217 citations

After publishing the derivation from the classical Bloch equation to the quantum von Neumann equation to the Schrdinger-Pauli equation for spin-

\tfrac{1}{2}

, 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-

s

states by symmetrizing tensor products of spin-

\tfrac{1}{2}

primitives, in accordance with the Majorana construction or the Schur-Weyl duality.