On the Non-relativistic Dynamics of Spin-1/2 Particles

Basak,, Ankan (2025) On the Non-relativistic Dynamics of Spin-1/2 Particles. Masters thesis, Indian Institute of Science Education and Research Kolkata.

Text (MS Dissertation of Ankan Basak (20MS060)) 20MS060_Thesis_file.pdf - Submitted Version Restricted to Repository staff only Download (939kB)

Abstract

In this document, we have followed a step-by-step approach in uncovering the dynamical equations of motion of suitable variables for systems of spin-1/2 particles under various constraints, in the non-relativistic regime. We began with a system of two such particles, identical to each other in terms of mass and gyromagnetic ratios, lacking spatial degrees of freedom, and each subjected to a stochastic magnetic field which had its orthogonal components (with respect to a chosen orientation of the Cartesian coordinate system) as zero mean Gaussian white noises (in appropriate units). The results obtained from this treatment were that if an arbitrary maximally entangled state was begun with before the magnetic fields were switched on, then the final state reached by the system always signified a partial or complete loss of entanglement, and the exact description of the asymptotic state reached, as well as the time taken by the initial state to evolve into this final state, could be controlled completely by the experimenter, by setting chosen values of some parameters defining the magnetic fields. Next, we added spatial degrees of freedom and set the stochastic Hamiltonian to zero, meaning that the two particles could now interact with each other through the magnetic dipole-dipole interaction Hamiltonian, and move in 3-dimensional free space. The dynamical equations of motion for the expectation of the separation between the particles, for that of the relative momentum, and for those of the individual spin angular momenta, were obtained purely using the equation of motion for the expectation of a quantum mechanical observable. For some specific situations, we have been able to retrieve heuristically expected classical behavior in terms of spatial motion of the particles (behaving as point dipoles). The case where the joint Hamiltonian is formed by combining the Hamiltonians of the two above mentioned scenarios has been briefly discussed at the end, setting the ground for future analytical and numerical observations.

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