Ladder Operators, Inherent Algebras and Associated Coherent States for Position-dependent Mass Systems

Abstract

Position-dependent effective mass systems are of great significance due to the fact that these models have numerous applications in various areas of physics. The qualitative understanding of a complicated realistic system can be acquired by analyzing the exact solutions of a related simplified model. However, the quantization of position-dependent effective mass systems and finding their solutions, involve some conceptual and mathematical difficulties of a fundamental nature. The factorization method provides us with a powerful tool for obtaining solutions and the underlying algebraic structure of the exactly solvable systems. The underlying algebra of a system has vast applications in different areas of mathematics and physics, such as it plays an important role in the theory of coherent states. Coherent states are extremely useful in various areas such as quantum mechanics, quantum optics, quantum information and group theory. In this thesis, position-dependent effective mass systems are studied in the context of their quantization, finding the solutions, construction of the algebraic structure and associated coherent states. In the first part of the thesis we mainly focus on quantizing and obtaining the exact solutions of the systems with spatially varying mass. The next part deals with the construction of the ladder operators and the inherent algebra of the pertaining systems. The associated coherent states and their properties are presented in the final part of the thesis. Beside the traditional way of obtaining exact solutions by solving the Schrödinger equation there exists another elegant method to solve the systems algebraically by factorizing the corresponding Hamiltonian. This method is based on supersymmetric quantum mechanics and the integrability condition, commonly known as shape invariance. After quantizing the position-dependent effective mass system, this factorization technique is used to determine the energy spectrum and the corresponding wave functions. For the sake of completeness the iv Abstract v method of solving a time-independent Schrödinger equation with spatially varying mass is also discussed. Considering a non-linear harmonic oscillator as an illustrative example, it is shown that both the above procedures produce the same results. The property of shape invariance enables us to obtain the ladder operators of the confining system. A general recipe for the construction of the ladder operators and inherent algebra for the position-dependent effective mass systems is presented. In order to exemplify the general formalism, a non-linear harmonic oscillator together with several other examples of the shape invariant systems with position-dependent effective mass is considered. Explicit expressions for the ladder operators and the associated algebra are presented. Using the ladder operators and the underlying algebra, the coherent states for the positiondependent effective mass systems are constructed and their properties are analyzed. In particular, we emphasize on various kinds of coherent states for a non-linear harmonic oscillator with spatially varying mass. By realizing SU(1; 1) as the dynamic group of the system, the construction of Barut-Girardello coherent states is presented. In addition, an algebraic independent kind of coherent states, namely Gazeau-Klauder coherent states, are also constructed. The statistical properties of Barut-Girardello and Gazeau-Klauder coherent states are investigated by means of the Mandel parameter and the second order correlation function. Moreover, the temporal evolution of the Gazeau-Klauder coherent states is analysed by means of autocorrelation function. It is shown that these states mimic the phenomena of quantum revivals and fractional revivals during their time evolution.