Quantum Physics V1 - From Basics To Symmetries And Perturbations

This two-volume set can be naturally divided into two semester courses, and contains a full modern graduate course in quantum physics. The idea is to teach graduate students how to practically use quantum physics and theory, presenting the fundamental knowledge, and gradually moving on to applications, including atomic, nuclear and solid state physics, as well as modern subfields, such as quantum chaos and quantum entanglement. The book starts with basic quantum problems, which do not require full quantum formalism but allow the student to gain the necessary experience and elements of quantum thinking. Only then does the fundamental Schrodinger equation appear. The author has included topics that are not usually covered in standard textbooks and has written the book in such a way that every topic contains varying layers of difficulty, so that the instructor can decide where to stop. Although supplementary sources are not required, "Further reading" is given for each chapter, including references to scientific journals and publications, and a glossary is also provided.
Problems and solutions are integrated throughout the text.

Vladimir Zelevinsky is Professor at the Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory at Michigan State University, USA. He graduated from Moscow University and worked for many years at the Budker Institute of Nuclear Physics in Novosibirsk where he received his PhD. Later, Professor Zelevinsky served as Head of Theory Division at the Budker Institute and Head of Theoretical Physics at Novosibirsk University. He spent three years as a visiting professor at the Niels Bohr Institute in Copenhagen, Denmark. Professor Zelevinsky is the author of over 200 scientific publications, co-editor of the EPL journal and Associate Editor of the Nuclear Physics journal.

Preface.

1 Origin of Main Quantum Concepts.

1.1 Light: Waves or Particles?

1.2 Planck Constant, Beginning of the Quantum Era.

1.3 Photons.

1.4 Spectroscopy and Stability of Atoms.

1.5 Bohr Postulates.

1.6 Hydrogen Atom.

1.7 Correspondence Principle.

1.8 Spatial Quantization.

1.9 Spin.

1.10 De BroglieWaves.

2 Wave Function and the Simplest Problems.

2.1 Free Motion.

2.2 Probability Density and Current.

2.3 Superposition Principle and Uncertainty.

2.4 Potential Wall.

2.5 Potential Barrier.

2.6 Barrier Penetration.

2.7 Tunneling.

3 Bound States.

3.1 Potential Box.

3.2 Orthogonality and Completeness.

3.3 Delta-Function*.

3.4 Time Evolution.

3.5 ShallowWell and Quantum Halo.

3.6 Resonances.

3.7 Level Density.

3.8 Periodic Boundary Conditions.

3.9 Counting Levels in a Smooth Potential.

4 Dynamical Variables.

4.1 Momentum Representation.

4.2 Introducing Operators.

4.3 Commutators.

4.4 Eigenfunctions and Eigenvalues.

4.5 Momentum as a Translation Generator.

4.6 Introduction to Groups*.

4.7 Orbital Momentum as a Rotation Generator.

4.8 Transformation of Operators.

5 Uncertainty Relations.

5.1 Uncertainty in Wave Mechanics.

5.2 Simple Examples.

5.3 Complementarity and Probability.

5.4 Wave Packet: Propagation.

5.5 Spreading of a Wave Packet.

5.6 Estimates with Uncertainty Relations.

5.7 Classification of Molecular Excitations.

5.8 Level Width.

5.9 Line Width and Mössbauer Effect.

5.10 Virtual Processes and Relativistic Effects.

5.11 Spatial Quantization Revisited.

6 Hilbert Space and Operators.

6.1 Probability Amplitude.

6.2 Superposition and Interference.

6.3 State Vectors.

6.4 Geometry of Hilbert Space*.

6.5 Linear Operators*.

6.6 Hermitian Operators*.

6.7 Properties of Hermitian Operators*.

6.8 Diagonalization*.

6.9 Basis Transformations*.

6.10 Continuous Transformations and Generators*.

6.11 Projection Operators*.

6.12 Operators of Observables.

6.13 Simultaneous Measurability.

6.14 Quantifying Uncertainty Relations.

7 Quantum Dynamics.

7.1 Hamiltonian and Schrödinger Equation.

7.2 Single-Particle Hamiltonian.

7.3 Continuity Equation.

7.4 Wigner Distribution.

7.5 Heisenberg Picture.

7.6 Operator Dynamics.

7.7 Virial Theorem.

7.8 Survival Probability.

7.9 Sum Rules.

7.10 Conservation Laws.

7.11 Path Integral Formulation.

7.12 Relation to Classical Mechanics.

7.13 Back to the Schrödinger Picture.

8 Discrete Symmetries.

8.1 Time-Reversal Invariance.

8.2 Time-Reversal Transformation of Operators.

8.3 Inversion and Parity.

8.4 Scalars and Pseudoscalars, Vectors and Pseudovectors.

8.5 Parity Conservation.

8.6 Symmetry of a Crystal Lattice.

8.7 Quasimomentum and Bloch Functions.

8.8 Energy Bands.

8.9 Symmetry of Molecules.

8.10 More Group Theory: Conjugate Classes*.

8.11 Group Representations*.

8.12 Orthogonality and Completeness*.

8.13 Characters*.

9 One-Dimensional Motion: Continuum.

9.1 Eigenvalue Problem.

9.2 Continuous Spectrum.

9.3 Degeneracy in the Continuum.

9.4 Transfer Matrix.

9.5 Delay Time.

9.6 Uniform Field.

9.7 Airy and Bessel Functions*.

9.8 Asymptotic Behavior*.

9.9 Asymptotics of the Airy Function*.

9.10 Green Function for One-Dimensional Scattering.

9.11 Potential as Perturbation.

9.12 Quasistationary States.

10 Variational Approach and Diagonalization.

10.1 Variational Principle.

10.2 Direct Variational Method.

10.3 Diagonalization in a Truncated Basis.

10.4 Two-State System.

10.5 Level Repulsion and Avoided Crossing.

10.6 Time Evolution of a Two-State System.

10.7 Bright State and Fragmentation.

10.8 Collective States.

10.9 Lanczos Algorithm.

11 Discrete Spectrum and Harmonic Oscillator.

11.1 One-Dimensional Bound States.

11.2 Linear Harmonic Oscillator.

11.3 Hermite Polynomials*.

11.4 Harmonic Oscillator in Plane: Separation of Variables.

11.5 Isotropic Oscillator.

11.6 Solving the Problem in Polar Coordinates.

11.7 Ladder Construction.

11.8 Creation and Annihilation Operators.

11.9 Operator Solution for the Harmonic Oscillator.

12 Coherent and Squeezed States.

12.1 Introducing Coherent States.

12.2 Displacements in the Phase Plane.

12.3 Properties of Coherent States.

12.4 Coherent States of the Harmonic Oscillator.

12.5 Linear Source.

12.6 Semiclassical Limit, Number of Quanta and the Phase.

12.7 Pairwise Source.

12.8 Squeezed States.

12.9 More about Squeezed States.

13 Introducing Magnetic Field.

13.1 Magnetic Field in Classical Mechanics.

13.2 Quantum Formulation and Gauge Invariance.

13.3 Are Electromagnetic Potentials Observable?

13.4 Landau Levels: Energy Spectrum.

13.5 Landau Levels: Degeneracy and Wave Functions.

13.6 Quantum Hall Effect.

13.7 Arbitrary Dispersion Law.

13.8 Symmetric Gauge.

13.9 Coherent States in the Magnetic Field.

14 Macroscopic Quantum Coherence.

14.1 Ideas of Macroscopic Coherence.

14.2 Macroscopic Wave Function.

14.3 Hydrodynamic Description.

14.4 Dynamics of the Macroscopic Coherent State.

14.5 Josephson Effects.

14.6 Quantization of Circulation and Quantum Vortices.

14.7 Magnetic Fluxoid Quantization and London Electrodynamics.

15 Semiclassical (WKB) Approximation.

15.1 Heuristic Introduction.

15.2 Semiclassical Approximation.

15.3 Asymptotic Expansion.

15.4 Stationary Phase.

15.5 Matching Conditions.

15.6 Bohr–Sommerfeld Quantization.

15.7 Semiclassical Matrix Elements.

15.8 Solutions in the Complex Plane*.

15.9 Going Around the Complex Plane*.

15.10 Connection Formulae Revisited*.

15.11 Close Turning Points*.

15.12 Path Integral Approach.

16 Angular Momentum and Spherical Functions.

16.1 Angular Momentum as a Generator of Rotations.

16.2 Spin.

16.3 Angular Momentum Multiplets.

16.4 Matrix Elements of AngularMomentum.

16.5 Realization of the Algebra for Orbital Momentum.

16.6 Constructing a Set of Spherical Functions*.

16.7 Simplest Properties of Spherical Functions*.

16.8 Scalars and Vectors*.

16.9 Second Rank Tensors*.

16.10 Spherical Functions and Legendre Polynomials*.

16.11 Angular Momentum in an External Field.

17 Motion in a Central Field.

17.1 Reduction to the One-Body Problem.

17.2 Separation of Angular Variables.

17.3 Radial Part of the Schrödinger Equation.

17.4 Free Motion.

17.5 Plane and Spherical Waves.

17.6 Spherical Well.

17.7 Short-Range Potential.

17.8 Adding the Second Center.

17.9 Three-Dimensional Harmonic Oscillator.

18 Hydrogen Atom.

18.1 Bound States.

18.2 Ground State.

18.3 Discrete Spectrum.

18.4 Operator Solution.

18.5 On the Way to Precision Spectroscopy.

18.6 Solution in Parabolic Coordinates*.

18.7 Continuum States.

19 Stationary Perturbations.

19.1 Introduction.

19.2 Perturbation Theory With No Degeneracy.

19.3 Convergence.

19.4 Case of Close Levels.

19.5 Adiabatic Approximation.

19.6 Molecular Ion of Hydrogen.

19.7 Interactions of Atoms at Large Distances.

20 Spin 1/2.

20.1 SU(2) Group.

20.2 Spin 1/2: Algebra.

20.3 Spinors.

20.4 Magnetic Resonance.

20.5 Time-Reversal Transformation and Kramers Theorem.

20.6 Time-Conjugate States.

20.7 Spinors as Qubits.

21 Finite Rotations and Tensor Operators.

21.1 Matrices of Finite Rotations.

21.2 Spherical Functions as Matrix Elements of Finite Rotations.

21.3 Addition Theorem*.

21.4 Transformation of Operators.

21.5 Introduction to Selection Rules.

21.6 Electromagnetic Multipoles.

22 Angular Momentum Coupling.

22.1 Two Subsystems.

22.2 Decomposition of Reducible Representations.

22.3 Two Particles of Spin 1/2.

22.4 Tensor Operators and Selection Rules Revisited.

22.5 Applying to Electromagnetic Multipoles.

22.6 Vector Coupling of AngularMomenta.

22.7 Wigner–Eckart Theorem.

22.8 Vector Model.

22.9 Electric Dipole Moment and Anapole Moment.

22.10 Clebsch–Gordan Series*.

23 Fine and Hyperfine Structure.

23.1 Spin–Orbit Coupling.

23.2 Spin–Orbit Splitting. 

23.3 Hydrogen Fine Structure.

23.4 Fine Structure in Complex Atoms.

23.5 Magnetic Moment with Spin–Orbit Coupling.

23.6 Magnetic Hyperfine Structure.

23.7 Example: One Valence Electron.

23.8 Quadrupole Hyperfine Structure.

24 Atom in a Static Field.

24.1 Polarizability in a Static Electric Field.

24.2 Stark Effect.

24.3 Polarizability of the Hydrogen Atom.

24.4 Stark Effect in the Hydrogen Atom.

24.5 Non-uniform Electric Field and Additional Comments.

24.6 Classical Zeeman Effect.

24.7 A Quantum System in a Magnetic Field.

24.8 Normal Quantum Zeeman Effect.

24.9 Anomalous Quantum Zeeman Effect.

24.10 Stronger Magnetic Field.

24.11 Diamagnetism.

24.12 Towards Really Strong Magnetic Fields.

References.

Further Readings.

Index.