notes of quantum mechanics

notes of quantum mechanics, Mechanika kwantowa

[ Pobierz całość w formacie PDF ]
Notes on Quantum Mechanics
K. Schulten
Department of Physics and Beckman Institute
University of Illinois at Urbana–Champaign
405 N. Mathews Street, Urbana, IL 61801 USA
(April 18, 2000)
Preface
i
Preface
The following notes introduce Quantum Mechanics at an advanced level addressing students of Physics,
Mathematics, Chemistry and Electrical Engineering. The aim is to put mathematical concepts and tech-
niques like the path integral, algebraic techniques, Lie algebras and representation theory at the readers
disposal. For this purpose we attempt to motivate the various physical and mathematical concepts as well
as provide detailed derivations and complete sample calculations. We have made every eort to include in
the derivations all assumptions and all mathematical steps implied, avoiding omission of supposedly `trivial'
information. Much of the author's writing eort went into a web of cross references accompanying the mathe-
matical derivations such that the intelligent and diligent reader should be able to follow the text with relative
ease, in particular, also when mathematically dicult material is presented. In fact, the author's driving
force has been his desire to pave the reader's way into territories unchartered previously in most introduc-
tory textbooks, since few practitioners feel obliged to ease access to their eld. Also the author embraced
enthusiastically the potential of the T
E
X typesetting language to enhance the presentation of equations as
to make the logical pattern behind the mathematics as transparent as possible. Any suggestion to improve
the text in the respects mentioned are most welcome. It is obvious, that even though these notes attempt
to serve the reader as much as was possible for the author, the main eort to follow the text and to master
the material is left to the reader.
The notes start out in Section 1 with a brief review of Classical Mechanics in the Lagrange formulation and
build on this to introduce in Section 2 Quantum Mechanics in the closely related path integral formulation. In
Section 3 the Schrodinger equation is derived and used as an alternative description of continuous quantum
systems. Section 4 is devoted to a detailed presentation of the harmonic oscillator, introducing algebraic
techniques and comparing their use with more conventional mathematical procedures. In Section 5 we
introduce the presentation theory of the 3-dimensional rotation group and the group SU(2) presenting Lie
algebra and Lie group techniques and applying the methods to the theory of angular momentum, of the spin
of single particles and of angular momenta and spins of composite systems. In Section 6 we present the theory
of many{boson and many{fermion systems in a formulation exploiting the algebra of the associated creation
and annihilation operators. Section 7 provides an introduction to Relativistic Quantum Mechanics which
builds on the representation theory of the Lorentz group and its complex relative Sl(2;C). This section makes
a strong eort to introduce Lorentz{invariant eld equations systematically, rather than relying mainly on
a heuristic amalgam of Classical Special Relativity and Quantum Mechanics.
The notes are in a stage of continuing development, various sections, e.g., on the semiclassical approximation,
on the Hilbert space structure of Quantum Mechanics, on scattering theory, on perturbation theory, on
Stochastic Quantum Mechanics, and on the group theory of elementary particles will be added as well as
the existing sections expanded. However, at the present stage the notes, for the topics covered, should be
complete enough to serve the reader.
The author would like to thank Markus van Almsick and Heichi Chan for help with these notes. The
author is also indebted to his department and to his University; their motivated students and their inspiring
atmosphere made teaching a worthwhile eort and a great pleasure.
These notes were produced entirely on a Macintosh II computer using the T
E
X typesetting system, Textures,
Mathematica and Adobe Illustrator.
Klaus Schulten
University of Illinois at Urbana{Champaign
August 1991
ii
Preface
Contents
1 Lagrangian Mechanics 1
1.1 Basics of Variational Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Lagrangian Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Symmetry Properties in Lagrangian Mechanics . . . . . . . . . . . . . . . . . . . . . 7
2 Quantum Mechanical Path Integral 11
2.1 The Double Slit Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Axioms for Quantum Mechanical Description of Single Particle . . . . . . . . . . . . 11
2.3 How to Evaluate the Path Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4 Propagator for a Free Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.5 Propagator for a Quadratic Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.6 Wave Packet Moving in Homogeneous Force Field . . . . . . . . . . . . . . . . . . . 25
2.7 Stationary States of the Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . 34
3 The Schrodinger Equation 51
3.1 Derivation of the Schrodinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.3 Particle Flux and Schrodinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.4 Solution of the Free Particle Schrodinger Equation . . . . . . . . . . . . . . . . . . . 57
3.5 Particle in One-Dimensional Box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.6 Particle in Three-Dimensional Box . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4 Linear Harmonic Oscillator 73
4.1 Creation and Annihilation Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.2 Ground State of the Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.3 Excited States of the Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . 78
4.4 Propagator for the Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.5 Working with Ladder Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.6 Momentum Representation for the Harmonic Oscillator . . . . . . . . . . . . . . . . 88
4.7 Quasi-Classical States of the Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . 90
5 Theory of Angular Momentum and Spin 97
5.1 Matrix Representation of the group SO(3) . . . . . . . . . . . . . . . . . . . . . . . . 97
5.2 Function space representation of the group SO(3) . . . . . . . . . . . . . . . . . . . . 104
5.3 Angular Momentum Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
iii
iv
Contents
2
and the group SU(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.8 Generators and Rotation Matrices of SU(2) . . . . . . . . . . . . . . . . . . . . . . . 128
5.9 Constructing Spin States with Larger Quantum Numbers Through Spinor Operators 129
5.10 Algebraic Properties of Spinor Operators . . . . . . . . . . . . . . . . . . . . . . . . 131
5.11 Evaluation of the Elements d
j
mm
0
() of the Wigner Rotation Matrix . . . . . . . . . 138
5.12 Mapping of SU(2) onto SO(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
6 Quantum Mechanical Addition of Angular Momenta and Spin 141
6.1 Clebsch-Gordan Coecients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
6.2 Construction of Clebsch-Gordan Coecients . . . . . . . . . . . . . . . . . . . . . . . 147
6.3 Explicit Expression for the Clebsch{Gordan Coecients . . . . . . . . . . . . . . . . 151
6.4 Symmetries of the Clebsch-Gordan Coecients . . . . . . . . . . . . . . . . . . . . . 160
6.5 Example: Spin{Orbital Angular Momentum States . . . . . . . . . . . . . . . . . . 163
6.6 The 3j{Coecients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
6.7 Tensor Operators and Wigner-Eckart Theorem . . . . . . . . . . . . . . . . . . . . . 176
6.8 Wigner-Eckart Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
7 Motion in Spherically Symmetric Potentials 183
7.1 Radial Schrodinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
7.2 Free Particle Described in Spherical Coordinates . . . . . . . . . . . . . . . . . . . . 188
8 Interaction of Charged Particles with Electromagnetic Radiation
203
8.1 Description of the Classical Electromagnetic Field / Separation of Longitudinal and
Transverse Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
8.2 Planar Electromagnetic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
8.3 Hamilton Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
8.4 Electron in a Stationary Homogeneous Magnetic Field . . . . . . . . . . . . . . . . . 210
8.5 Time-Dependent Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 215
8.6 Perturbations due to Electromagnetic Radiation . . . . . . . . . . . . . . . . . . . . 220
8.7 One-Photon Absorption and Emission in Atoms . . . . . . . . . . . . . . . . . . . . . 225
8.8 Two-Photon Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
9 Many{Particle Systems 239
9.1 Permutation Symmetry of Bosons and Fermions . . . . . . . . . . . . . . . . . . . . . 239
9.2 Operators of 2nd Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
9.3 One{ and Two{Particle Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
9.4 Independent-Particle Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
9.5 Self-Consistent Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
9.6 Self-Consistent Field Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
9.7 Properties of the SCF Ground State . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
9.8 Mean Field Theory for Macroscopic Systems . . . . . . . . . . . . . . . . . . . . . . 272
1
5.4 Angular Momentum Eigenstates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.5 Irreducible Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.6 Wigner Rotation Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.7 Spin
[ Pobierz całość w formacie PDF ]

zanotowane.pl

doc.pisz.pl

pdf.pisz.pl

anio102.xlx.pl