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arXiv:1305.5583

Some Solvable Quantum Mechanical Models and Their Symmetries

Bikashkali Midya

Published May 23, 2013• 0 citations
arXivHTMLPDF

Abstract

This thesis is focused on some solvable quantum mechanical models and their associated symmetries.
DOI
10.48550/arxiv.1305.5583

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The thesis is divided in two parts. In the first part of the thesis, I present an updated version of the theory of covariant quantum mechanics and of the associated classical background. This theory has originally been proposed by Jadczyk and Modugno and further developed in cooperation with other people. It provides a geometric model for quantum mechanics of a scalar and spin particle in a curved spacetime with absolute time equipped with a given classical gravitational and electromagnetic field. Starting from minimal axioms, this model yields in a covariant way the Schrödinger equation by means of a global Lagrangian formalism and the quantum operators associated with classical quantisable functions by means of classification of distinguished quantum vector fields. In the second part of the thesis, I investigate systematically the infinitesimal symmetries of the classical and quantum geometric structures. This part deals with the main original contributions of the thesis. Some results fit well-known facts of other mathematical models of quantum mechanics, but are derived by innovative procedures. Other results are completely new. The most important original contributions can be summarized by the following two statements: The quantum symmetries constitute Lie algebras, which are naturally isomorphic to subalgebras of the new classical Lie algebra of quantisable functions. Let me stress that this algebra is not a Poisson subalgebra. On the other hand, by means of the quantum Lagrangian formalism, we can associate a conserved quantum current with any element of a distinguished classical subalgebra. These results might be the prequantum source of the probabilistic interpretation of quantum mechanics.
The exactly solvable systems---the harmonic oscillator, the hydrogen atom, and the angular momentum algebra---are quantum mechanics' solved ground: the spectra computed in closed form that calibrated the theory, taught its methods, and revealed the symmetries that make solvability itself a theorem. This article presents a narrative review of the primary literature of that ground, from de Broglie's 1924 thesis---the wave hypothesis the solvable problems confirmed---through Heisenberg's 1925 umdeutung, Schrödinger's 1926 eigenvalue problem, whose oscillator and rotator opened the solvable list, Dirac's 1926 theory of quantum mechanics and Pauli's 1926 hydrogen solution by the new algebra, Darwin's 1927 vector waves, Dirac's 1928 electron---the relativistic hydrogen's fine structure---Wigner's 1931 group theory and its application to the atomic spectra, Condon and Shortley's 1935 atomic spectra canon, Fock's 1935 hydrogen symmetry---the O(4) hidden behind the spectrum---to the factorization method's systematization in Infeld and Hull's 1951 review and its supersymmetric completion in Cooper, Khare, and Sukhatme's 1995 physics reports. The synthesis is organized around three themes: the solved spectra, in which the oscillator's and hydrogen's levels fixed the new mechanics' validity; the hidden symmetries, in which Runge-Lenz and spherical symmetry explained the solvabilities; and the factorization program, in which the ladder operators' algebra became a method---the exactly solvable potentials classified. It is concluded that the solvable systems are quantum mechanics' load-bearing examples---and that their symmetries remain the theory's deepest explanation of why some problems close.
It is shown that braid matrices and link polynomials can be systematically constructed from exactly solvable models in statistical mechanics. Through symmetry breaking transformations, different braid matrices are derived from a solvable model. By associating the Markov traces with multi-variable representations, multi-variable link polynomials are obtained. Infinitesimal operators for braid matrices are constructed. Connection of our approach to the conformal field theories and the topological quantum field theory is discussed.