Among the various techniques for the solution of the Schr{\"o}dinger equation, supersymmetric methods have recently gained much attention. The concept of supersymmetry was originally introduced in high energy field theory. It soon became evident that, when coupled with the concept of shape invariance, the supersymmetry yields a large set of algebraically solvable quantum mechanical models, which are neatly classified according to the type of shape invariance. Growing out of several lectures and a well-regarded review article by the authors, this eminently readable book is a welcome
introductory treatise on supersymmetric methods in quantum mechanics. Here, readers will not find the elegance associated with mathematical sophistication, nor the broadest generality such high mathematics would bring. Nonetheless, the elementary mathematical language employed throughout the book makes it accessible to anybody who has gone through an introductory quantum mechanics course. The problems at the end of each chapter, to which the answers are given in the last appendix, enhance the pedagogical value of this book.
The contents of each chapter could be summarized as follows.
After the introduction in chapter 1, essential elements of non-relativistic one-demensional quantum mechanics are recalled in chapter 2.
In chapter 3, the authors introduce the concepts of supersymmetry and super algebra, and show how they can be applied to the one-dimensional Schr\"{o}dinger equation: When the ground state solution is known, one can obtain the factorized form of the Hamiltonian, which then leads to another factorized Hamiltonian of inverted order, the supersymmetric partner Hamiltonian, whose spectra are identical to the original problem. If the obtained partner Hamiltonian belongs to the same functional class as the original Hamiltonian with shifted ground state energy and possibly with altered parameters, the new Hamiltonian yields yet another partner Hamiltonian, leading to a whole hierarchy of Hamiltonians.
This is the concept of shape invariance, the backbone of chapter 4, that enables the algebraic construction of complete solutions of the problem. An extensive classification of potentials with ``translational shift'' type shape invariance is given, and attempts for the extension to more general shape invariance are outlined in this chapter.
In chapter 5, the Pauli equation and Dirac equation for the electrons under electric and magnetic fields are re-examined from the perspective of supersymmetry, and several solvable examples both in 1+1 and 3+1 dimensions are studied.
In chapter 6, the authors take a slight detour to cover the application of supersymmetric methods to the soliton solution of the Korteweg-de Vries equation, which is related to the solutions of n-parameter set of isospectral Hamiltonian problems. A hurried narration in this chapter makes it less than self-contained, but interested readers can consult the references quoted there. The rest of the book covers less well known applications of supersymmetric methods.
In chapter 7, with the supersymmetric methods, a set of new solvable models of one-dimensional periodic potentials are constructed out of standard Lam{\'e} potentials and their properties are analyzed.
In chapter 8, supersymmetric methods are applied to the semiclassical WKB approximation, and better convergences are obtained for low energy states.
In chapter 9, one finds other examples of supersymmetric methods that give faster perturbative convergences. The central idea is to find partner potentials with reduced coupling strengths. The anharmonic oscillator problems, double well tunneling problems, and large-N expansions are treated with this technique.
There are three appendices (apart from the last one already mentioned) each covering the supersymmetric path-integral formalism, the operator transformations and the logarithmic perturbation technique.
|
Goto:
HOME