Practical quantum mechanics : modern tools and applications / Efstratios Manousakis.

Format:

Book

Author/Creator:

Manousakis, Efstratios, author.

Series:

Oxford graduate texts

Language:

English

Subjects (All):

Quantum theory.

Physical Description:

xv, 332 pages ; 26 cm.

Edition:

First edition.

Place of Publication:

Oxford, United Kingdom : Oxford University Press, 2016.

Summary:

Quantum mechanics forms the foundation of all modem physics, including atomic, nuclear, and molecular physics, the physics of the elementary particles, condensed matter physics. Modern astrophysics also relies heavily on quantum mechanics. Quantum theory is needed to understand the basis for new materials, new devices, the nature of light coming from stars, the laws which govern the atomic nucleus, and the physics of biological systems. As a result the subject of this book is a required course for most physics graduate students. While there are many books on the subject, this book targets specifically graduate students and it is written with modern advances in various fields in mind. Many examples treated in the various chapters as well as the emphasis of the presentation in the book are designed from the perspective of such problems. For example, the book begins by putting the Schrodinger equation on a spatial discrete lattice and the continuum limit is also discussed, inspired by Hamiltonian lattice gauge theories. The latter and advances in quantum simulations motivated the inclusion of the path integral formulation. This formulation is applied to the imaginary-time evolution operator to project the exact ground state of the harmonic oscillator as is done in quantum simulations. As an example of how to take advantage of symmetry in quantum mechanics, one-dimensional periodic potentials are discussed, inspired by condensed matter physics. Atoms and molecules are discussed within mean-field like treatment (Hartree-Fock) and how to go beyond it. Motivated by the recent intense activity in condensed matter and atomic physics to study the Hubbard model, the electron correlations in the hydrogen molecule are taken into account by solving the two-site Hubbard model analytically. Using the canonical Hamiltonian quantization of quantum electrodynamics, the photons emerge as the quanta of the normal modes, in the same way as the phonons emerge in the treatment of the normal modes of the coupled array of atoms. This is used later to treat the interaction of radiation with atomic matter. Book jacket.

Contents:

1 Schrödinger equation on a lattice 1

1.1 Discretizing the spatial continuum 1

1.2 The Schrodinger equation in a matrix form 2

1.3 Problems 4

2 Dirac notation 6

2.1 The bit and the q-bit 6

2.2 Dirac notation 7

2.3 Outer product 8

2.4 Matrices and matrix elements 9

2.5 Quantum gates 9

2.6 Rotation 10

2.7 Functions of operators 11

2.8 Generator of planar rotations 12

2.9 Problems 12

3 Back to the Schrodinger equation on a lattice 15

3.1 Lattice states 15

3.2 Transformation of basis: momentum states 15

3.3 Continuum limit of space 17

3.4 Continuum limit of k-space 19

3.5 Generalization in d-dimensions 21

3.6 Problems 22

4 Operator mechanics 23

4.1 Operators and observables 23

4.2 Representation of operators 25

4.3 The Hamiltonian operator 26

4.4 Problems 27

5 Time evolution and wavepackets 28

5.1 Time-independent Hamiltonian 28

5.2 An example of time evolution 29

5.3 Wavepackets 30

5.4 Time-dependent Hamiltonian 31

5.5 Problems 34

6 Simultaneous observables 36

6.1 Uncertainty principle 36

6.2 Commuting operators 37

6.3 Symmetries of the Hamiltonian 39

6.4 Problems 42

7 Continuity equation and wavefunction properties 44

7.1 Continuity equation 44

7.2 Conditions on the wavefunction and its derivative 46

7.3 Non-negative kinetic energy expectation value 47

7.4 Variational theorem 48

7.5 Practical use of the variational theorem 49

7.6 Problems 52

8 Bound states in one dimension 54

8.1 Square-well potential 54

8.2 Delta function potential 58

8.3 Problems 60

9 Scattering in one dimension 68

9.1 Step barrier potential 68

9.2 Tunneling 70

9.3 Attractive square-well potential 72

9.4 Poles of 3(E) 74

9.5 Resonances 75

9.6 Analytic structure of S(E) 79

9.7 Meaning of the resonance 79

9.8 Problems 81

10 Periodic potentials 83

10.1 Bloch's theorem 83

10.2 The Kronig-Penney model 85

10.3 Problems 89

11 The harmonic oscillator 91

11.1 Why is it useful? 91

11.2 One-dimensional harmonic oscillator 92

11.3 Eigenatates 95

11.4 Problems 96

12 WKB approximation 99

12.1 The approximation 99

12.2 Region of validity of WKB 101

12.3 Exact solution near a turning point 102

12.4 Matching of WKB and exact solution: right turning point 104

12.5 Two turning points: bound states 105

12.6 Tunneling within the WKB approximation 106

12.7 Problems 109

13 Quantum mechanics and path integrals 110

13.1 Derivation of the path integral in ID 110

13.2 Classical mechanics as limit of quantum mechanics 114

13.3 WKB from path integrals 116

13.4 Further reading 117

14 Applications of path integrals 118

14.1 Bohm-Aharonov effect 118

14.2 The harmonic oscillator and path integrals 120

14.3 Projection of the ground state 122

14.4 Application to many-body physics 124

14.5 Quantum statistical mechanics 125

14.6 Further reading 126

15 Angular momentum 127

15.1 Angular momentum operators 127

15.2 The spectrum of angular momentum operators 129

15.3 Eigenstates of angular momentum 132

15.4 Legendre polynomials and spherical harmonics 134

15.5 Problems 135

16 Bound states in spherically symmetric potentials 139

16.1 Spherical Bessel functions 140

16.2 Relation to ordinary Bessel equation 145

16.3 Spherical Bessel functions and Legendre polynomials 146

16.4 Bound states in a spherical well 147

16.5 Expansion of a plane wave 148

16.6 Problems 149

17 The hydrogen-like atom 153

17.1 The general two-body problem 153

17.2 The relative motion 155

17.3 Wavefunctions 158

17.4 Associated Laguerre polynomials 159

17.5 Problems 160

18 Angular momentum and spherical symmetry 161

18.1 Generators of infinitesimal rotations 161

18.2 Invariant subspaces and tensor operators 162

18.3 Wigner-Eckart theorem for scalar operators 164

18.4 Wigner-Eckart theorem: general case 165

18.5 Problems 166

19 Scattering in three dimensions 167

19.1 Scattering cross section 167

19.2 Quantum mechanical scattering 168

19.3 Scattering amplitude and differential cross section 171

19.4 Born series expansion 173

19.5 Partial wave expansion 174

19.6 Examples: phase shift calculation 177

19.7 Problems 182

20 Time-independent perturbation expansion 184

20.1 Statement of the problem 184

20.2 Non-degenerate case 185

20.3 Degenerate perturbation theory 188

20.4 Quasi-degenerate perturbation theory 191

20.5 Problems 192

21 Applications of perturbation theory 195

21.1 Stark effect 195

21.2 Origin of the Van der Walls interaction 197

21.3 Electrons in a weak periodic potential 201

21.4 Problems 205

22 Time-dependent Hamiltonian 207

22.1 Time-dependent perturbation theory 207

22.2 Adiabatic processes 214

22.3 Problems 218

23 Spin angular momentum 220

23.1 Spin and orbital angular momentum 220

23.2 Spin-1/2 222

23.3 Coupling of spin to a uniform magnetic field 223

23.4 Rotations in spin space 224

23.5 Problems 225

24 Adding angular momenta 227

24.1 Coupling between angular momenta 227

24.2 Spin-orbit coupling 228

24.3 The angular momentum coupling 229

24.4 Problems 235

25 Identical particles 236

25.1 Symmetry under particle permutations 236

25.2 Second quantization 238

25.3 Hilbert space for identical particles 239

25.4 Operators 241

25.5 Creation and annihilation operators 242

25.6 Problems 245

26 Elementary atomic physics 246

26.1 Helium atom 246

26.2 Hartree and Hartree-Fock approximation 249

26.3 Hartree equations 253

26.4 Hartree-Fock and non-locality 254

26.5 Beyond mean-field theory 255

26.6 Characterization of atomic states 257

26.7 Spin-orbit interaction in multi-electron atoms 260

26.8 Problems 263

27 Molecules 265

27.1 H⁺₂ and the Born-Oppenheimer approximation 265

27.2 Hybridization 268

27.3 Tight-binding approximation 269

27.4 The hydrogen molecule 274

27.5 Problems 278

28 The elasticity field 280

28.1 Monoatomic chain 280

28.2 Diatomic chain 284

28.3 Problems 289

29 Quantization of the free electromagnetic field 292

29.1 Classical treatment 292

29.2 Quantization 295

29.3 Problems 297

30 Interaction of radiation with charged particles 299

30.1 The total Hamiltonian 299

30.2 Absorption and emission processes 301

30.3 Problems 304

31 Elementary relativistic quantum mechanics 306

31.1 Klein-Gordon equation 306

31.2 Continuity equation 308

31.3 Solutions of Klein-Gordon equation 308

31.4 First-order Klein-Gordon equation 309

31.5 The Dirac equation 310

31.6 Rotational invariance 312

31.7 Free-particle solution of the Dirac equation 312

31.8 Non-relativistic limit 314

31.9 Spin-orbit coupling 315

31.10 Covariant form 317

31.11 Coupling to external electromagnetic fields 318

31.12 Continuity equation 318

31.13 Interpretation of the Dirac equation 319

31.14 Problems 320.

Notes:

Includes index.

ISBN:

9780198749349

0198749341

OCLC:

932856241

Publisher Number:

99968757475