Heisenberg's Quantum Mechanics

678 Pages·2010·4.04 MB·English

by  Mohsen Razavy

HEISENBERG’S
QUANTUM
MECHANICS
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HEISENBERG’S
QUANTUM
MECHANICS
Mohsen Razavy
University of Alberta, Canada
World Scientific
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HEISENBERG’S QUANTUM MECHANICS
Copyright © 2011 by World Scientific Publishing Co. Pte. Ltd.
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ISBN-13 978-981-4304-10-8
ISBN-10 981-4304-10-7
ISBN-13 978-981-4304-11-5 (pbk)
ISBN-10 981-4304-11-5 (pbk)
Printed in Singapore.
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Dedicated to my great teachers
A.H. Zarrinkoob, M. Bazargan and J.S. Levinger
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Preface
There is an abundance of excellent texts and lecture notes on quantum theory
and applied quantum mechanics available to the students and researchers. The
motivation for writing this book is to present matrix mechanics as it was first
discovered by Heisenberg, Born and Jordan, and by Pauli and bring it up to
datebyaddingthecontributionsbyanumberofprominentphysicistsinthein-
terveningyears. Theideaofwritingabookonmatrixmechanicsisnotnew. In
1965 H.S. Green wrote a monograph with the title “Matrix Mechanics” (Nord-
hoff, Netherlands)wherefromtheworksofthepioneersinthefieldhecollected
and presented a self-contained theory with applications to simple systems.
In most text books on quantum theory, a chapter or two are devoted to
the Heisenberg’s matrix approach, but due to the simplicity of the Schro¨dinger
wave mechanics or the elegance of the Feynman path integral technique, these
twomethodshaveoftenbeenusedtostudyquantummechanicsofsystemswith
finite degrees of freedom.
The present book surveys matrix and operator formulations of quantum
mechanics and attempts to answer the following basic questions: (a) — why
andwheretheHeisenbergformofquantummechanicsismoreusefulthanother
formulationsand(b)—howtheformalismcanbeappliedtospecificproblems?
To seek answer to these questions I studied what I could find in the original
literature and collected those that I thought are novel and interesting. My first
inclination was to expand on Green’s book and write only about the matrix
mechanics. Butthisplanwouldhaveseverelylimitedthescopeandcoverageof
the book. Therefore I decided to include and use the wave equation approach
where it was deemed necessary. Even in these cases I tried to choose the ap-
proach which, in my judgement, seemed to be closer to the concepts of matrix
mechanics. For instance in discussing quantum scattering theory I followed the
determinantal approach and the LSZ reduction formalism.
In Chapter 1 a brief survey of analytical dynamics of point particles is
presented which is essential for the formulation of quantum mechanics, and an
understandingoftheclassical-quantummechanicalcorrespondence. Inthispart
of the book particular attention is given to the question of symmetry and con-
servation laws.
vii
viii Heisenberg’s Quantum Mechanics
InChapter2ashorthistoricalreviewofthediscoveryofmatrixmechanics
is given and the original Heisenberg’s and Born’s ideas leading to the formu-
lation of quantum theory and the discovery of the fundamental commutation
relations are discussed.
Chapter 3 is concerned with the mathematics of quantum mechanics,
namely linear vector spaces, operators, eigenvalues and eigenfunctions. Here
an entire section is devoted to the ways of constructing Hermitian operators,
together with a discussion of the inconsistencies of various rules of association
of classical functions and quantal operators.
In Chapter 4 the postulates of quantum mechanics and their implica-
tions are studied. A detailed review of the uncertainty principle for position-
momentum, time-energy and angular momentum-angle and some applications
of this principle is given. This is followed by an outline of the correspondence
principle. The question of determination of the state of the system from the
measurement of probabilities in coordinate and momentum space is also in-
cluded in this chapter.
InChapter5connectionsbetweentheequationofmotion,theHamiltonian
operator and the commutation relations are examined, and Wigner’s argument
about the nonuniqueness of the canonical commutation relations is discussed.
In this chapter quantum first integrals of motion are derived and it is shown
that unlike their classical counterparts, these, with the exception of the energy
operator, are not useful for the quantal description of the motion.
InChapter6 thesymmetries andconservationlaws forquantummechan-
icalsystemsareconsidered. AlsotopicsrelatedtotheGalileaninvariance,mass
superselection rule and the time invariance are studied. In addition a brief dis-
cussion of classical and quantum integrability and degeneracy is presented.
Chapter 7 deals with the application of Heisenberg’s equations of motion
in determining bound state energies of one-dimensional systems. Here Klein’s
method and its generalization are considered. In addition the motion of a par-
ticle between fixed walls is studied in detail.
Chapter 8 is concerned with the factorization method for exactly solvable
potentials and this is followed by a brief discussion of the supersymmetry and
of shape invariance.
Thetwo-bodyproblemisthesubjectofdiscussioninChapter9,wherethe
properties ofthe orbitalandspinangularmomentum operators anddetermina-
tion of their eigenfunctions are presented. Then the solution to the problem of
hydrogenatomisfoundfollowingtheoriginalformulationofPauliusingRunge–
Lenz vector.
In Chapter 10 methods of integrating Heisenberg’s equations of motion
are presented. Among them the discrete-time formulation pioneered by Bender
and collaborators, the iterative solution for polynomial potentials advanced by
Znojil and also the direct numerical method of integration of the equations of
motion are mentioned.
TheperturbationtheoryisstudiedinChapter11andinChapter12other
methods of approximation, mostly derived from Heisenberg’s equations of mo-
Preface ix
tion are considered. These include the semi-classical approximation and varia-
tional method.
Chapter 13 is concerned with the problem of quantization of equations of
motion with higher derivatives, this part follows closely the work of Pais and
Uhlenbeck.
Potential scattering is the next topic which is considered in Chapter 14.
Here the Schro¨dinger equation is used to define concepts such as cross section
and the scattering amplitude, but then the deteminantal method of Schwinger
is followed to develop the connection between the potential and the scattering
amplitude. After this, the time-dependent scattering theory, thescattering ma-
trix and the Lippmann–Schwinger equation are studied. Other topics reviewed
in this chapter are the impact parameter representation of the scattering am-
plitude, the Born approximation and transition probabilities.
InChapter15anotherfeatureofthewavenatureofmatterwhichisquan-
tum diffraction is considered.
The motion of a charged particle in electromagnetic field is taken up in
Chapter16withadiscussionoftheAharonov–BohmeffectandtheBerryphase.
Quantum many-body problem is reviewed in Chapter 17. Here systems
withmany-fermionandwithmany-bosonarereviewedandabriefreviewofthe
theory of superfluidity is given.
Chapter18isaboutthequantumtheoryoffreeelectromagneticfieldwith
a discussion of coherent state of radiation and of Casimir force.
Chapter 19, contains the theory of interaction of radiation with matter.
Finallyinthelastchapter,Chapter20,abriefdiscussionofBell’sinequal-
ities and its relation to the conceptual foundation of quantum theory is given.
In preparing this book, no serious attempt has been made to cite all of
the important original sources and various attempts in the formulation and ap-
plications of the Heisenberg quantum mechanics.
I am grateful to my wife for her patience and understanding while I was
writing this book, and to my daughter, Maryam, for her help in preparing the
manuscript.
Edmonton, Canada, 2010