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Author	Shankar, Ramamurti
Title	Principles of quantum mechanics / R. Shankar
Publish Info	New York : Plenum Press, ©1994
Edition	2nd ed

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Contents

1 Mathematical Introduction 1

1.1 Linear Vector Spaces: Basics 1

1.2 Inner Product Spaces 7

1.3 Dual Spaces and the Dirac Notation 11

1.4 Subspaces 17

1.5 Linear Operators 18

1.6 Matrix Elements of Linear Operators 20

1.7 Active and Passive Transformations 29

1.8 The Eigenvalue Problem 30

1.9 Functions of Operators and Related Concepts 54

1.10 Generalization to Infinite Dimensions 57

2 Review of Classical Mechanics 75

2.1 The Principle of Least Action and Lagrangian Mechanics 78

2.2 The Electromagnetic Lagrangian 83

2.3 The Two-Body Problem 85

2.4 How Smart Is a Particle? 86

2.5 The Hamiltonian Formalism 86

2.6 The Electromagnetic Force in the Hamiltonian Scheme 90

2.7 Cyclic Coordinates, Poisson Brackets, and Canonical Transformations 91

2.8 Symmetries and Their Consequences 98

3 All Is Not Well with Classical Mechanics 107

3.1 Particles and Waves in Classical Physics 107

3.2 An Experiment with Waves and Particles (Classical) 108

3.3 The Double-Slit Experiment with Light 110

3.4 Matter Waves (de Broglie Waves) 112

4 The Postulates - a General Discussion 115

4.1 The Postulates 115

4.2 Discussion of Postulates I-III 116

4.3 The Schrodinger Equation (Dotting Your i's and Crossing your h's) 143

5 Simple Problems in One Dimension 151

5.1 The Free Particle 151

5.2 The Particle in a Box 157

5.3 The Continuity Equation for Probability 164

5.4 The Single-Step Potential: a Problem in Scattering 167

5.5 The Double-Slit Experiment 175

5.6 Some Theorems 176

6 The Classical Limit 179

7 The Harmonic Oscillator 185

7.1 Why Study the Harmonic Oscillator? 185

7.2 Review of the Classical Oscillator 188

7.3 Quantization of the Oscillator (Coordinate Basis) 189

7.4 The Oscillator in the Energy Basis 202

7.5 Passage from the Energy Basis to the X Basis 216

8 The Path Integral Formulation of Quantum Theory 223

8.1 The Path Integral Recipe 223

8.2 Analysis of the Recipe 224

8.3 An Approximation to U(t) for the Free Particle 225

8.4 Path Integral Evaluation of the Free-Particle Propagator 226

8.5 Equivalence to the Schrodinger Equation 229

8.6 Potentials of the Form V = a + bx + cx[superscript 2] + dx + exx 231

9 The Heisenberg Uncertainty Relations 237

9.2 Derivation of the Uncertainty Relations 237

9.3 The Minimum Uncertainty Packet 239

9.4 Applications of the Uncertainty Principle 241

9.5 The Energy-Time Uncertainty Relation 245

10 Systems with N Degrees of Freedom 247

10.1 N Particles in One Dimension 247

10.2 More Particles in More Dimensions 259

10.3 Identical Particles 260

11 Symmetries and Their Consequences 279

11.1 Overview 279

11.2 Translational Invariance in Quantum Theory 279

11.3 Time Translational Invariance 294

11.4 Parity Invariance 297

11.5 Time-Reversal Symmetry 301

12 Rotational Invariance and Angular Momentum 305

12.1 Translations in Two Dimensions 305

12.2 Rotations in Two Dimensions 306

12.3 The Eigenvalue Problem of L[subscript z] 313

12.4 Angular Momentum in Three Dimensions 318

12.5 The Eigenvalue Problem of L[superscript 2] and L[subscript z] 321

12.6 Solution of Rotationally Invariant Problems 339

13 The Hydrogen Atom 353

13.1 The Eigenvalue Problem 353

13.2 The Degeneracy of the Hydrogen Spectrum 359

13.3 Numerical Estimates and Comparison with Experiment 361

13.4 Multielectron Atoms and the Periodic Table 369

14 Spin 373

14.2 What is the Nature of Spin? 373

14.3 Kinematics of Spin 374

14.4 Spin Dynamics 385

14.5 Return of Orbital Degrees of Freedom 397

15 Addition of Angular Momenta 403

15.1 A Simple Example 403

15.2 The General Problem 408

15.3 Irreducible Tensor Operators 416

15.4 Explanation of Some "Accidental" Degeneracies 421

16 Variational and WKB Methods 429

16.1 The Variational Method 429

16.2 The Wentzel-Kramers-Brillouin Method 435

17 Time-Independent Perturbation Theory 451

17.1 The Formalism 451

17.2 Some Examples 454

17.3 Degenerate Perturbation Theory 464

18 Time-Dependent Perturbation Theory 473

18.1 The Problem 473

18.2 First-Order Perturbation Theory 474

18.3 Higher Orders in Perturbation Theory 484

18.4 A General Discussion of Electromagnetic Interactions 492

18.5 Interaction of Atoms with Electromagnetic Radiation 499

19 Scattering Theory 523

19.2 Recapitulation of One-Dimensional Scattering and Overview 524

19.3 The Born Approximation (Time-Dependent Description) 529

19.4 Born Again (The Time-Independent Approximation) 534

19.5 The Partial Wave Expansion 545

19.6 Two-Particle Scattering 555

20 The Dirac Equation 563

20.1 The Free-Particle Dirac Equation 563

20.2 Electromagnetic Interaction of the Dirac Particle 566

20.3 More on Relativistic Quantum Mechanics 574

21 Path Integrals - II 581

21.1 Derivation of the Path Integral 582

21.2 Imaginary Time Formalism 613

21.3 Spin and Fermion Path Integrals 636

21.4 Summary 652

App. A.1. Matrix Inversion 655

App. A.2. Gaussian Integrals 659

App. A.3. Complex Numbers 660

App. A.4. The i[epsilon] Prescription 661

Answers to Selected Exercises 665

Table of Constants 669

Index 671

Description xviii, 676 pages : illustrations ; 27 cm

Note Includes bibliographical references and index

Contents 1. Mathematical Introduction. 1.1. Linear Vector Spaces: Basics. 1.2. Inner Product Spaces. 1.3. Dual Spaces and the Dirac Notation. 1.4. Subspaces. 1.5. Linear Operators. 1.6. Matrix Elements of Linear Operators. 1.7. Active and Passive Transformations. 1.8. The Eigenvalue Problem. 1.9. Functions of Operators and Related Concepts. 1.10. Generalization to Infinite Dimensions -- 2. Review of Classical Mechanics

2.1. The Principle of Least Action and Lagrangian Mechanics. 2.2. The Electromagnetic Lagrangian. 2.3. The Two-Body Problem. 2.4. How Smart Is a Particle? 2.5. The Hamiltonian Formalism. 2.6. The Electromagnetic Force in the Hamiltonian Scheme. 2.7. Cyclic Coordinates, Poisson Brackets, and Canonical Transformations. 2.8. Symmetries and Their Consequences -- 3. All Is Not Well with Classical Mechanics. 3.1. Particles and Waves in Classical Physics

3.2. An Experiment with Waves and Particles (Classical). 3.3. The Double-Slit Experiment with Light. 3.4. Matter Waves (de Broglie Waves) -- 4. The Postulates -- a General Discussion. 4.1. The Postulates. 4.2. Discussion of Postulates I-III. 4.3. The Schrodinger Equation (Dotting Your i's and Crossing your h's) -- 5. Simple Problems in One Dimension. 5.1. The Free Particle. 5.2. The Particle in a Box. 5.3. The Continuity Equation for Probability

5.4. The Single-Step Potential: a Problem in Scattering. 5.5. The Double-Slit Experiment. 5.6. Some Theorems -- 6. The Classical Limit -- 7. The Harmonic Oscillator. 7.1. Why Study the Harmonic Oscillator? 7.2. Review of the Classical Oscillator. 7.3. Quantization of the Oscillator (Coordinate Basis). 7.4. The Oscillator in the Energy Basis. 7.5. Passage from the Energy Basis to the X Basis -- 8. The Path Integral Formulation of Quantum Theory

8.1. The Path Integral Recipe. 8.2. Analysis of the Recipe. 8.3. An Approximation to U(t) for the Free Particle. 8.4. Path Integral Evaluation of the Free-Particle Propagator. 8.5. Equivalence to the Schrodinger Equation. 8.6. Potentials of the Form V = a + bx + cx[superscript 2] + dx + exx -- 9. The Heisenberg Uncertainty Relations. 9.2. Derivation of the Uncertainty Relations. 9.3. The Minimum Uncertainty Packet. 9.4. Applications of the Uncertainty Principle

9.5. The Energy-Time Uncertainty Relation -- 10. Systems with N Degrees of Freedom. 10.1. N Particles in One Dimension. 10.2. More Particles in More Dimensions. 10.3. Identical Particles -- 11. Symmetries and Their Consequences -- 11.1. Overview. 11.2. Translational Invariance in Quantum Theory. 11.3. Time Translational Invariance. 11.4. Parity Invariance. 11.5. Time-Reversal Symmetry -- 12. Rotational Invariance and Angular Momentum

12.1. Translations in Two Dimensions. 12.2. Rotations in Two Dimensions. 12.3. The Eigenvalue Problem of L[subscript z]. 12.4. Angular Momentum in Three Dimensions. 12.5. The Eigenvalue Problem of L[superscript 2] and L[subscript z]. 12.6. Solution of Rotationally Invariant Problems -- 13. The Hydrogen Atom. 13.1. The Eigenvalue Problem. 13.2. The Degeneracy of the Hydrogen Spectrum. 13.3. Numerical Estimates and Comparison with Experiment

13.4. Multielectron Atoms and the Periodic Table -- 14. Spin. 14.2. What is the Nature of Spin? 14.3. Kinematics of Spin. 14.4. Spin Dynamics. 14.5. Return of Orbital Degrees of Freedom -- 15. Addition of Angular Momenta. 15.1. A Simple Example. 15.2. The General Problem. 15.3. Irreducible Tensor Operators. 15.4. Explanation of Some "Accidental" Degeneracies -- 16. Variational and WKB Methods. 16.1. The Variational Method

16.2. The Wentzel-Kramers-Brillouin Method -- 17. Time-Independent Perturbation Theory. 17.1. The Formalism. 17.2. Some Examples. 17.3. Degenerate Perturbation Theory -- 18. Time-Dependent Perturbation Theory. 18.1. The Problem. 18.2. First-Order Perturbation Theory. 18.3. Higher Orders in Perturbation Theory. 18.4. A General Discussion of Electromagnetic Interactions. 18.5. Interaction of Atoms with Electromagnetic Radiation -- 19. Scattering Theory

19.2. Recapitulation of One-Dimensional Scattering and Overview. 19.3. The Born Approximation (Time-Dependent Description). 19.4. Born Again (The Time-Independent Approximation). 19.5. The Partial Wave Expansion. 19.6. Two-Particle Scattering -- 20. The Dirac Equation. 20.1. The Free-Particle Dirac Equation. 20.2. Electromagnetic Interaction of the Dirac Particle. 20.3. More on Relativistic Quantum Mechanics -- 21. Path Integrals -- II

21.1. Derivation of the Path Integral. 21.2. Imaginary Time Formalism. 21.3. Spin and Fermion Path Integrals. 21.4. Summary -- App. A.1. Matrix Inversion -- App. A.2. Gaussian Integrals -- App. A.3. Complex Numbers -- App. A.4. The i[epsilon] Prescription

Summary A textbook on quantum mechanics that develops the subject from its postulates, beginning with a rather lengthy chapter in which the relevant mathematics of vector spaces is developed from simple ideas on vectors and matrices students are assumed to know. This revised edition (1st ed., 1980) adds a discussion of time-reversal invariance and a new chapter, Path Integrals: Part II, which discusses many kinds of path integrals and their uses. Annotation copyright by Book News, Inc., Portland, OR

Note British Library not licensed to copy 0. Uk

Subjects Quantum theory

Quantum Theory

Link Online version: Shankar, Ramamurti. Principles of quantum mechanics. 2nd ed. New York : Plenum Press, ©1994 (OCoLC)1311565703

LC NO QC174.12 .S52 1994

Call # 530.145 S524

Dewey No 530.1/2 20

OCLC # 30811075

ISBN 0306447908

9780306447907

Isn/Std # (OCoLC)30811075 (OCoLC)971358235 (OCoLC)1172637577 (OCoLC)1191273563

LCCN 94026837

1	Mathematical Introduction		1
	1.1	Linear Vector Spaces: Basics	1
	1.2	Inner Product Spaces	7
	1.3	Dual Spaces and the Dirac Notation	11
	1.4	Subspaces	17
	1.5	Linear Operators	18
	1.6	Matrix Elements of Linear Operators	20
	1.7	Active and Passive Transformations	29
	1.8	The Eigenvalue Problem	30
	1.9	Functions of Operators and Related Concepts	54
	1.10	Generalization to Infinite Dimensions	57
2	Review of Classical Mechanics		75
	2.1	The Principle of Least Action and Lagrangian Mechanics	78
	2.2	The Electromagnetic Lagrangian	83
	2.3	The Two-Body Problem	85
	2.4	How Smart Is a Particle?	86
	2.5	The Hamiltonian Formalism	86
	2.6	The Electromagnetic Force in the Hamiltonian Scheme	90
	2.7	Cyclic Coordinates, Poisson Brackets, and Canonical Transformations	91
	2.8	Symmetries and Their Consequences	98
3	All Is Not Well with Classical Mechanics		107
	3.1	Particles and Waves in Classical Physics	107
	3.2	An Experiment with Waves and Particles (Classical)	108
	3.3	The Double-Slit Experiment with Light	110
	3.4	Matter Waves (de Broglie Waves)	112
4	The Postulates - a General Discussion		115
	4.1	The Postulates	115
	4.2	Discussion of Postulates I-III	116
	4.3	The Schrodinger Equation (Dotting Your i's and Crossing your h's)	143
5	Simple Problems in One Dimension		151
	5.1	The Free Particle	151
	5.2	The Particle in a Box	157
	5.3	The Continuity Equation for Probability	164
	5.4	The Single-Step Potential: a Problem in Scattering	167
	5.5	The Double-Slit Experiment	175
	5.6	Some Theorems	176
6	The Classical Limit		179
7	The Harmonic Oscillator		185
	7.1	Why Study the Harmonic Oscillator?	185
	7.2	Review of the Classical Oscillator	188
	7.3	Quantization of the Oscillator (Coordinate Basis)	189
	7.4	The Oscillator in the Energy Basis	202
	7.5	Passage from the Energy Basis to the X Basis	216
8	The Path Integral Formulation of Quantum Theory		223
	8.1	The Path Integral Recipe	223
	8.2	Analysis of the Recipe	224
	8.3	An Approximation to U(t) for the Free Particle	225
	8.4	Path Integral Evaluation of the Free-Particle Propagator	226
	8.5	Equivalence to the Schrodinger Equation	229
	8.6	Potentials of the Form V = a + bx + cx[superscript 2] + dx + exx	231
9	The Heisenberg Uncertainty Relations		237
	9.2	Derivation of the Uncertainty Relations	237
	9.3	The Minimum Uncertainty Packet	239
	9.4	Applications of the Uncertainty Principle	241
	9.5	The Energy-Time Uncertainty Relation	245
10	Systems with N Degrees of Freedom		247
	10.1	N Particles in One Dimension	247
	10.2	More Particles in More Dimensions	259
	10.3	Identical Particles	260
11	Symmetries and Their Consequences		279
11.1	Overview		279
	11.2	Translational Invariance in Quantum Theory	279
	11.3	Time Translational Invariance	294
	11.4	Parity Invariance	297
	11.5	Time-Reversal Symmetry	301
12	Rotational Invariance and Angular Momentum		305
	12.1	Translations in Two Dimensions	305
	12.2	Rotations in Two Dimensions	306
	12.3	The Eigenvalue Problem of L[subscript z]	313
	12.4	Angular Momentum in Three Dimensions	318
	12.5	The Eigenvalue Problem of L[superscript 2] and L[subscript z]	321
	12.6	Solution of Rotationally Invariant Problems	339
13	The Hydrogen Atom		353
	13.1	The Eigenvalue Problem	353
	13.2	The Degeneracy of the Hydrogen Spectrum	359
	13.3	Numerical Estimates and Comparison with Experiment	361
	13.4	Multielectron Atoms and the Periodic Table	369
14	Spin		373
	14.2	What is the Nature of Spin?	373
	14.3	Kinematics of Spin	374
	14.4	Spin Dynamics	385
	14.5	Return of Orbital Degrees of Freedom	397
15	Addition of Angular Momenta		403
	15.1	A Simple Example	403
	15.2	The General Problem	408
	15.3	Irreducible Tensor Operators	416
	15.4	Explanation of Some "Accidental" Degeneracies	421
16	Variational and WKB Methods		429
	16.1	The Variational Method	429
	16.2	The Wentzel-Kramers-Brillouin Method	435
17	Time-Independent Perturbation Theory		451
	17.1	The Formalism	451
	17.2	Some Examples	454
	17.3	Degenerate Perturbation Theory	464
18	Time-Dependent Perturbation Theory		473
	18.1	The Problem	473
	18.2	First-Order Perturbation Theory	474
	18.3	Higher Orders in Perturbation Theory	484
	18.4	A General Discussion of Electromagnetic Interactions	492
	18.5	Interaction of Atoms with Electromagnetic Radiation	499
19	Scattering Theory		523
	19.2	Recapitulation of One-Dimensional Scattering and Overview	524
	19.3	The Born Approximation (Time-Dependent Description)	529
	19.4	Born Again (The Time-Independent Approximation)	534
	19.5	The Partial Wave Expansion	545
	19.6	Two-Particle Scattering	555
20	The Dirac Equation		563
	20.1	The Free-Particle Dirac Equation	563
	20.2	Electromagnetic Interaction of the Dirac Particle	566
	20.3	More on Relativistic Quantum Mechanics	574
21	Path Integrals - II		581
	21.1	Derivation of the Path Integral	582
	21.2	Imaginary Time Formalism	613
	21.3	Spin and Fermion Path Integrals	636
21.4	Summary		652
	App. A.1. Matrix Inversion		655
	App. A.2. Gaussian Integrals		659
	App. A.3. Complex Numbers		660
	App. A.4. The i[epsilon] Prescription		661
	Answers to Selected Exercises		665
	Table of Constants		669
	Index		671