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World Scientific lecture notes in physics

A Hamiltonian for Nonrelativistic, Spinless Charged Particles in the Presence of Electromagnetic Fields

Published Jun 1, 1988• 0 citations
Publisher
DOI
10.1142/9789812799333_0006

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Gauge freedom is used to reduce the Hamiltonian of a nonrelativistic charged particle with conservative potential energy that is in a time-dependent external electromagnetic field to kinetic energy only. Nevertheless, the energy of the particle is shown to be kinetic energy plus conservative potential energy. Besides simplifying the Hamiltonian, this gauge dramatically illustrates the conceptual differences between the Hamiltonian and energy for a particle in a time-dependent external electromagnetic field.
Spinor wave functions; classical Lagrangian and Hamiltonian of a charged particle in an electromagnetic field, and its quantum Hamiltonian; gauge invariance; spin magnetic moment in a uniform magnetic field; magnetic resonance; the Stern--Gerlach experiment; neutron interferometry and rotations of spinor wave functions; treatment of a particle in a uniform magnetic field with and without the inclusion of spin degrees of freedom; Ahronov--Bohm effect for a charged spinless particle confined in a cylindrical shell.
We describe here the coherent formulation of electromagnetism in the nonrelativistic quantummechanical many-body theory. We use the mathematical frame of the field theory and its quantization in the spirit of the QED. This is necessary because of the manifold of misinterpretations emerging from the hystorical development of quantum mechanics, starting from the Schrödinger equation of a single particle in the presence of given electromagnetic fields, followed by the many-body theories of many charged identical particles having just Coulomb interactions inspired from the classical electromagnetic theory of point-like charges. However, this later is known to be inconsistent due to the self-interaction. This way could not be continued further to include properly the magnetic forces between the charged particles and lead to a lot of confusion about the interpretation of the magnetic field in the Hamiltonian, as well as about the gauge invariance. We emphasize the importance of the distinction between the applied (external fields) and the field in the matter. All these problems are length properly solved within the non-relativistic QED, nevertheless the confusion dominates in all the problems related to the magnetic properties of the solid state.