Solvable Systems And Related Problems In Quantum Mechanics

In this thesis, we shall pay attention to the solvable systems and related problems. The solvable systems are extremely important in classical and quan-tum mechanics, and play an irreplaceable role in theoretical and experimental physics. Besides, the study of solvable systems promotes the study of other theo-ries, such as the Virial Theorem and Hypervirial Theorem, the physics symmetry, perturbation theory, etc.The study of solvable systems includes the investigations of both exactly solv-able systems and quasi-exactly solvable systems. Usually, one can study solvable systems in two directions. The first is to find the exact or quasi-exact solutions of a known model in physics. The second is to search solvable models and establish connections with the actually existed systems in physics.The main results in this thesis include the following two parts. On one hand, we study the properties of some solvable systems. To be more specific, we obtain the Virial Theorems of quantum nonlinear harmonic oscillators and Higgs spherical systems, and formulate a perturbation theorem without wave functions which is analogous to the Hypervirial-Hellmann-Feynman Theorem. We also discuss the Higgs algebraic symmetry of screened systems in a spherical geometry, and give the equations of the classical orbits. This part constitutes Chapters2,3and4of this thesis. On the other hand, we construct exactly solvable systems that satisfy specific symmetry properties. More precisely, we construct a Coulomb-like system, an oscillator-like system, and anon-central potential system, whose energyspectra can be obtained by simple calculations. This part constitutes Chapter5of this thesis.We next proceed to give concrete descriptions of the five chapters of this thesis. In Chapter1, we introduce the background and some basic concepts. We review some solvable systems, including the hydrogen atom system, the harmonic oscillator system, the quantum nonlinear harmonic oscillators, and the Higgs system in spherical geometry.Hydrogen atom and harmonic oscillator are two simple and real models of solve systems. There is a common feature of them that, their orbits of motion are closed in classical mechanics. This indicates that there are more constants of motion in these systems than the orbit angular momentum. These constants have been shown as the Runge-Lenz vector in the hydrogen atom and the quadruple tensors in the harmonic oscillator, which, as well as angular momentum, become the generators of the SO(N+1) and SU(N) Lie groups respectively. Therefore, the dynamical symmetry of the N-dimensional hydrogen atom is described by the SO(N+1) Lie group, and the N-dimensional harmonic oscillator is shown to have the SU(N) dynamical symmetry.Quantum nonlinear harmonic oscillator (QNHO) is the generalization of lin-ear harmonic oscillators to nonlinear harmonic oscillators. Because of its exact solvability, the QNHO has attracted much attention. Many familiar exactly solv-able systems are contained in this class. We shall restrict our attention to a kind of one-dimensional model for the QNHO, which is the Carinena’s QNHO, whose Hamiltonian is λ-dependent. This system can be considered as a system in a spherical geometry as well as a system with a position-dependent effective mass. When λ=0, the quantum nonlinear harmonic oscillator reduces to the usual harmonic oscillator.The Higgs system is another generalization of hydrogen atom and isotrop-ic harmonic oscillators to a spherical geometry. Higgs system was introduced by P.W. Higgs in1979, which has become a well-recognized model in spherical geometry. In spherical geometry, Higgs established a gnomonic projective coor-dinate system in which the projected orbits of the motion on a sphere are the same, for a given V(r), as in Euclidean geometry. Therefore, the hydrogen atom and harmonic oscillator in the spherical space preserve the dynamical symmetry. The conserved quantities of the hydrogen atom and the harmonics oscillator in a N-dimensional sphere satisfy the Higgs algebra relations, which are polynomial generalizations of the SO(N+1) and SU(N) Lie algebra.We also review the Virial Theorem, the Hypervirial Theorem, and the Hellmann- Feynman Theorem. As a well-known theorem in physics, the Virial Theorem provides a general equation that relates the average over time of the total kinetic energy, bound by potential forces, to that of the total potential energy. The sig-nificance of the Virial Theorem is that it allows the average total kinetic energy to be calculated even for very complicated systems that defy an exact solution. Therefore, the Virial Theorem has been applied to many areas such as astro-physics, statistical physics, and physical chemistry. The Hypervirial Theorem is a generalization of the ordinary Virial Theorem, which can give the recurrence relation of the expectation values (xk). The combination of the Virial Theo-rem, the Hypervirial Theorem and Hellmann-Feynman Theorem often simplifies the computation of many complicated systems, especially in the case when the eigenvalues cannot be calculated.In Chapter2, we present the Virial Theorem for a class of quantum nonlin-ear harmonic oscillators, whose Hamiltonian is λ-dependent. The λ-dependent QNHO that we are concerned with is called the Carinena’s QNHO. We obtain the Virial Theorem for the Carinena’s QNHO. Moreover, we obtain the Virial Theorem for a more general class of exactly solvable QNHO introduced by X.H. Wang. We find that the Virial Theorem for the general class of exactly solvable QNHO has the same form as the Virial Theorem for the Carinena’s QNHO. It has to do with parameter λ and (?). When λ=0, the nonlinear harmonic oscillator naturally reduces to the usual quantum linear harmonic oscillator, and the Virial Theorem also reduces to the usual Virial Theorem.In Chapter3, the Virial Theorem in the one-and two-dimensional Hig-gs spherical geometry are presented, in both classical and quantum mechanic-s. Choosing a special class of Hypervirial operators, the quantum Hyperviri-al relations in the Higgs spherical spaces are obtained. With the aid of the Hellmann-Feynman Theorem, these relations can be used to formulate a perturba-tion theorem without wave functions, corresponding to the Hypervirial-Hellmann-Feynman Theorem of Euclidean geometry. The one-dimensional harmonic oscil-lator and two-dimensional Coulomb system in the spherical spaces are given as two sample examples to illustrate the perturbation method.In Chapter4, we discuss the dynamical symmetries for the screened Coulomb potentials and screened isotropic harmonic oscillators in the spherical geometry. Besides, we get the eigenenergy, eigenfunctions, and the orbit equations. The orbits and the dynamical symmetries for the screened Coulomb potentials and isotropic harmonic oscillators have been studied by Z.B. Wu and J.Y. Zeng. We find that similar properties to those of Zeng are obtained in the corresponding systems in a spherical space, whose dynamical symmetries are described by Hig-gs algebra. There exist the extended Runge-Lenz vector for screened Coulomb potentials and extended quadruple tensor for screened harmonic oscillators. The extended Runge-Lenz vector and extended quadruple tensor, together with angu-lar momentum, constitute a polynomial Higgs algebra in Hilbert space spanned by degenerate states with a given energy eigenvalue. Moreover, there exist an infinite number of closed orbits for suitable angular momentum values, and the equations of these closed orbits are given. The eigenenergy spectra and corre-sponding eigenstates in these systems are also derived.In Chapter5, an approach to constructing the quantum systems with dynam-ical symmetry is proposed. By analyzing the symmetries of the hydrogen atom and the harmonic oscillator, we propose some new quantum systems with dynam-ical symmetries. As a result, we derive the Coulomb-like system and Oscillator-like system as the generalizations of the hydrogen atom system and harmonic oscillator system, which can be regarded as the systems with position-dependent mass. They have the symmetries which are similar to the corresponding ones, and can be solved by using the algebraic method. We also exhibit an example of the method applied to the non-central field. As the angular momentum is no longer a special conservation, we should construct all of the three generators. We construct the conserved quantities as the polynomial of the Boson operators, and get the non-central potential system. The approach in this chapter seems to be available to give rise to new systems that cannot be dealt with by the known methods.