Application Of Operator Algebraic Approach In Quantum Mechanical Models

This dissertation studies mainly on the application of operator algebraic approachin quantum mechanical models. Operator algebraic approach is an important and usefultool in quantum mechanics. With the help of it, people can directly and exactly solve theenergy spectrum and wave functions of the physical systems, without dealing with thesecond-order differential Schro¨dinger equations. Operator algebraic approach relates tothe raising and the lowering operators, which may be divided into two kinds: the firstone is called raising and lowering operator approach; the second is called factorizationmethod, or the supersymmetry method in quantum mechanics.Exactly solvable models are very important in quantum mechanics. For instance,for the case of one dimension, the exactly solvable potentials have one-dimensionalharmonic oscillator, Po¨schl-Teller potential, Morse potential and so on; for the caseof three dimensions, the exactly solvable potentials have three-dimensional harmonicoscillator and hydrogen atom etc. The energy spectrum and wave functions of theseexactly solvable systems have become the main parts of the modern course of quan-tum mechanics. Because the energy spectrum and the wave functions of these physicalsystem are known, based on some of their important properties people may learn betterof the physical world. In general, any one-dimensional exactly solvable potential V (x)possesses a hidden algebraic structure, some literatures have pointed out that almostall one-dimensional exactly solvable potentials possess the SU(1, 1) algebraic struc-ture, except the one-dimensional Morse potential. In this dissertation, first we applythe raising and lowering operator approach to a physical system with one-dimensionalMorse potential, we derive its raising and lowering operators, and determine its energyspectrum as well as wave functions, and we point out its hidden algebraic structure isthe SU(2) algebra. Second, we apply the raising and lowering operator approach tothe physical systems in one-dimensional curved space, we find a wider class of exactlysolvable models in one-dimensional curved space, and that the usual quantum nonlin-ear harmonic oscillator studied in the literature is a special solution of the models in the curved space. Then, we study these exactly solvable Hamiltonians in one-dimensionalcurved space from the viewpoint of the factorization method, and the same results areobtained. Due to the Hellmann-Feynman Theorem we also discuss the Virial Theoremfor the physical systems. Finally, we study the energy spectrum of the modified Heisen-berg spin chain model, we also discuss the degeneracy of the ground state energy, andalso the change of the energy spectrum in the presence of the magnetic field.