טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentAizenshtark Nitsan
SubjectQuasiexactly Solvable Potentials
DepartmentDepartment of Physics
Supervisors Professor Emeritus Moshe Moshe
Dr. Joshua Feinberg


Abstract

Quantum mechanical potentials can be divided according to their solvability into three categories. The best known category is exactly solvable potentials (e.g. the harmonic oscillator and the hydrogen atom), which can be solved analytically. This means that all the energy levels and all the wave functions can be calculated in a finite number of algebraic steps. The second category is non-solvable potentials (e.g. the quartic potential), for which it is only possible to approximate the solution using various analytical and numerical methods. The third and least known category is quasiexactly solvable (QES) potentials, for which only part of the energy spectrum and the corresponding wave functions can be analytically calculated.

The feature distinguishing quasiexactly solvable potentials from non solvable potentials is that their Hamiltonian may be represented as a block-diagonal matrix with at least one finite block. Such a finite block may always be diagonalized, thus the energy levels and wave functions corresponding to it can always be calculated. Block diagonal Hamiltonians can be created using bilinear combinations of Lie group generators. The exactly solvable potentials may be treated as a sub-group of the quasiexactly solvable potentials since it is always possible to diagonalize their Hamiltonian.

This work deals with the solution of one dimensional quasiexactly solvable potentials. It also includes one dimensional supersymmetric quasiexactly solvable potentials which are potentials that can be represented by a 2 by 2 matrix and their Hamiltonian operates on a wave function which is a two component spinor of one spatial variable. First, expanding the results of existing works, some properties of the exact solutions of quasiexactly solvable potentials are shown to hold for the general case. One interesting property of some quasiexactly solvable potentials (which is used in the second part of this work) is the simple connection between the wave functions and the initial value solution of the Schrödinger equation. The initial value solution is the generating function of the secular polynomials, the zeros of which are the quasiexactly solvable part of the energy spectrum for the potential. This property is shown to hold for a class of quasiexactly solvable and exactly solvable potentials and not only for specific examples.

Next, the secular polynomials and the quasiexactly solvable part of the energy spectrum are approximated using a new technique which is based on the WKB approximation. The approximation is useful because calculating the energy levels exactly becomes more and more complicated as the size of the finite block in the Hamiltonian grows.

The initial value solution of the Schrödinger equation is calculated using the WKB approximation. Then approximate secular polynomials and energy levels are derived from the WKB solutions. In order to simplify this process saddle point approximation is used to obtain the secular polynomials and the energy levels from the WKB approximation. This calculation has to be carried out separately for every potential. The process is demonstrated for the harmonic oscillator and the sextic potential and the quality of the results is discussed. The combined WKB and saddle point approximations give a simple evaluation of the energy levels of quasiexactly solvable potentials with a generating function that solves the Schrödinger equation.