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arXiv:0902.2029 · Journal of Physics A Mathematical and Theoretical

Position-dependent mass oscillators and coherent states

Sara Cruz y Cruz, Óscar Rosas-Ortiz

Published Apr 9, 2009• 88 citations
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Abstract

The solving of the Schrodinger equation for a position-dependent mass quantum system is studied in two ways. First, it is found the interaction which must be applied on a mass m(x) in order to supply it with a particular spectrum of energies. Second, given a specific potential V(x) acting on the mass m(x), the related spectrum is found. The method of solution is applied to a wide class of position-dependent mass oscillators and the corresponding coherent states are constructed. The analytical expressions of such position-dependent mass coherent states preserve the functional structure of the Glauber states.

DOI
10.1088/1751-8113/42/18/185205

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The solving of the Schrodinger equation for a position-dependent mass quantum system is studied in two ways. First, it is found the interaction which must be applied on a mass m(x) in order to supply it with a particular spectrum of energies. Second, given a specific potential V(x) acting on the mass m(x), the related spectrum is found. The method of solution is applied to a wide class of position-dependent mass oscillators and the corresponding coherent states are constructed. The analytical expressions of such position-dependent mass coherent states preserve the functional structure of the Glauber states.
Position-dependent e?ective mass systems are of great signi?cance 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 simpli?ed model. However, the quantization of position-dependent e?ective mass systems and ?nding their solutions, involve some conceptual and mathematical di?culties 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 di?erent 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. Inthisthesis, position-dependente?ectivemasssystemsarestudiedinthecontextoftheir quantization, ?nding the solutions, construction of the algebraic structure and associated coherent states. In the ?rst 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 ?nal 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 e?ective mass system, this factorization technique is used to determine the energy spectrum and the corresponding wave functions. For the sake of completeness the \nmethod 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 con?ning system. A general recipe for the construction of the ladder operators and inherent algebra for the position-dependent e?ective 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 e?ective mass is considered. Explicit expressions for the ladder operators and the associated algebra are presented. Usingtheladderoperatorsandtheunderlyingalgebra,thecoherentstatesforthepositiondependent e?ective 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
Using a recently developed technique to solve Schrödinger equation for constant mass, we studied the regime in which mass varies with position i.e position dependent mass Schrödinger equation(PDMSE). We obtained an analytical solution for the PDMSE and applied our approach to study a position dependent mass m(x)m(x)m(x) particle scattered by a potential V(x)\mathcal{V}(x)V(x). We also studied the structural analogy between PDMSE and two-level atomic system interacting with a classical field.
We study the generalized harmonic oscillator that has both the position-dependent mass and the potential depending on the form of mass function in a more general framework. The explicit expressions of the eigenvalue and eigenfunction for such a system are given, they have the same forms as those for the usual harmonic oscillator with constant mass. The coherent state and its properties for the system with PDM are also discussed. We give the corresponding effective potentials for several mass functions, the systems with such potentials are isospectral to the usual harmonic oscillator.
Using the coordinate transformation method, we solve the one-dimensional Schrödinger equation with position-dependent mass. The explicit expressions for the potentials, energy eigenvalues, and eigenfunctions of the systems are given. The eigenfunctions can be expressed in terms of the Jacobi, Hermite, and generalized Laguerre polynomials. All potentials for these solvable systems have an extra term V m , which is produced from the dependence of mass on the position, compared with those for the systems of constant mass. The properties of V m for several mass functions are discussed.