Factorization Method in Quantum Mechanics

This work introduces the factorization method in quantum mechanics at an
advancedleveladdressingstudentsofphysics,mathematics,chemistryandelec
trical engineering. The aim is to put the mathematical and physical concepts
and techniques like the factorization method, Lie algebras, matrix elements and
quantumcontrolatthereader’sdisposal. Forthispurpose,weattempttoprovide
a comprehensive description of the factorization method and its wide applica
tions in quantum mechanics which complements thetraditional coverage found
in theexisting quantummechanicstextbooks. Relatedtothisclassicmethodare
the supersymmetric quantum mechanics, shape invariant potentials and group
theoretical approaches. It is no exaggeration to say that this method has become
the milestone of these approaches. In fact, the author’s driving force has been
his desire to provide a comprehensive review volume that includes some new
and significant results about the factorization method in quantum mechanics
since the literature is inundated with scattered articles in this field and to pave
the reader’s way into this territory as rapidly as possible. We have made the
effort to present the clear and understandable derivations and include the nec
essary mathematical steps so that the intelligent and diligent reader should be
able to follow the text with relative ease, in particular, when mathematically
difficult material is presented. The author also embraces enthusiastically the
potential of the LaTeX typesetting language to enrich the presentation of the
formulas as to make the logical pattern behind the mathematics more transpar
ent. Additionally, any suggestions and criticism to improve the text are most
welcome since this is the first version. It should be addressed that the main
effort to follow the text and master the material is left to the reader even though
this book makes an effort to serve the reader as much as was possible for the
author.
This book starts out in Chapter 1 with a comprehensive review for the tradi
tional factorization method and builds on this to introduce in Chapter 2 a new
approachtothismethodandtoreviewinChapter3thebasicpropertiesoftheLie
FACTORIZATION METHOD IN QUANTUMMECHANICS
algebras su(2) and su(1, 1) to be used in the successive Chapters. As important
applications in non-relativistic quantum mechanics, from Chapter 4 to Chap
ter 13, we shall apply our new approach to the factorization method to study
some important quantum systems such as the harmonic oscillator, infinitely
deep square well, Morse, P¨ oschl-Teller, pseudoharmonic oscillator, noncentral
ring-shaped potential quantum systems and others. One of the advantages of
this new approach is to easily obtain the matrix elements for some related phys
ical functions except for constructing a suitable Lie algebra from the ladder
operators. In Chapter 14 we are going to study the position-dependent mass
Schr¨ odinger equation for a singular oscillator based on the algebraic approach.
We shall carry out the applications of the factorization method in relativistic
Dirac and Klein-Gordon equations with the Coulomb and hyperbolic potentials
from Chapter 15 to Chapter 18. As an important generalized application of this
methodrelatedtothegrouptheoryincontroltheory, weshallstudythequantum
control in Chapters 19 and 20, in which we briefly introduce the development
of the quantum control and some well known theorems on control theory and
then apply the knowledge of the Lie algebra generated by the system’s quan
tum Hamiltonian to investigate the controllabilities of the quantum systems for
the Morse, P¨ oschl-Teller (PT) and PT-like potentials. Some conclusions and
outlooks are given in Chapter 21.
This book is in a stage of continuing development, various chapters, e.g.,
on the group theory, on the supersymmetric quantum mechanics, on the shape
invariance, on the higher order factorization method will be added to the extent
that the respective topics expand. At the present stage, however, the work
presented for such topics should be complete enough to serve the reader.
This book shall give the theoretical physicists and chemists a fresh outlook
and new ways of handling the important quantum systems for some potentials
of interest in all branches of physics and chemistry and of studying quantum
control. The primary audience of this book shall be the graduate students and
young researchers in physics, theoretical chemistry and electric engineering.