Designed for a two-semester advanced undergraduate or graduate level course, this distinctive and modern textbook provides students with the physical intuition and mathematical skills to tackle even complex problems in quantum mechanics with ease and fluency. Beginning with a detailed introduction to quantum states and Dirac notation, the book then develops the overarching theoretical framework of quantum mechanics, before explaining physical quantum mechanical properties such as angular momentum and spin. Symmetries and groups in quantum mechanics, important components of current research, are covered at length. The second part of the text focuses on applications, and includes a detailed chapter on quantum entanglement, one of the most exciting modern applications of quantum mechanics, and of key importance in quantum information and computation. Numerous exercises are interspersed throughout the text, expanding upon key concepts and further developing students' understanding. A fully worked solutions manual and lecture slides are available for instructors.
Author(s): Arjun Berera, Luigi Del Debbio
Publisher: Cambridge University Press
Year: 2022
Language: English
Pages: 441
City: Cambridge
Cover
Half-title
Endorsement
Title page
Copyright information
Contents
Preface
Book Organisation
Acknowledgements
Part I Basics
1 Stories and Thoughts about Quantum Mechanics
Further Reading
2 Quantum States
2.1 States of a Quantum System
2.1.1 Kets
2.1.2 Scalar Product
2.1.3 Bras
2.1.4 Norm
2.1.5 Orthogonality
2.1.6 Operators
2.1.7 Bases
2.1.8 Tensor Product
2.2 Two-State Systems
2.3 The Wave Function of a One-Dimensional System
2.3.1 Discretised System
2.3.2 Continuum System
Summary
Problems
3 Observables
3.1 Introducing Observables
3.1.1 A First Example: Position Operator
3.1.2 Generic Observables
3.2 Observing Observables
3.3 Hermitian Operators
3.3.1 Hermitian Conjugate
3.3.2 Hermitian Operators
3.4 Properties of Hermitian Operators
3.5 Spectral Decomposition
3.6 Collapse of the State Vector
3.7 Compatible Observables
3.8 Complete Sets of Commuting Observables
3.9 Continuous Spectrum
3.9.1 Eigenvalue Equation
3.9.2 Orthonormality
3.9.3 Spectral Decomposition
3.10 Momentum Operator
3.10.1 Momentum as Generator of Translations
3.10.2 Position and Momentum Representations
3.10.3 Wave Packets
3.11 The Uncertainty Principle
Summary
Problems
4 Dynamics
4.1 Schrödinger Equation
4.1.1 Equation of Motion
4.2 Eigenstates of the Hamiltonian
4.3 Evolution of a Generic State
4.4 One-Dimensional System
4.5 Some Properties of One-Dimensional Potentials
4.5.1 General Restrictions on Ψ(x,t)
4.5.2 Energy Expectation Value
4.5.3 Ehrenfest’s Theorem
4.5.4 Degeneracy
4.5.5 Nodes
4.6 Probability Current
Summary
Problems
5 Potentials
5.1 Potential Step
5.2 Tunnelling
5.3 Infinite Potential Well
5.4 Symmetry under Parity
5.5 Finite Potential Well
Summary
Problems
6 Harmonic Oscillator
6.1 The Harmonic Oscillator in Classical Mechanics
6.2 The Quantum Harmonic Oscillator
6.3 Factorising the Hamiltonian
6.4 Creation and Annihilation
6.5 Eigensystem
6.5.1 Eigenvalues
6.5.2 Normalisation of Eigenstates
6.5.3 Wave Functions
6.6 Brute Force Solution
Summary
Problems
7 Systems in Three Spatial Dimensions
7.1 Quantum States
7.2 Observables
7.2.1 General Considerations
7.2.2 Position
7.2.3 Momentum
7.3 Dynamics
7.4 Separation of Variables: Three-Dimensional Harmonic Oscillator
7.5 Degeneracy
Summary
Problems
8 Angular Momentum
8.1 Angular Momentum Operator
8.2 Squared Norm of the Angular Momentum
8.3 Eigenfunctions
8.4 Physical Interpretation
8.5 Algebraic Solution of the Eigenvalue Equations
8.6 Nomenclature
8.7 Normalisation
8.8 Matrix Representations
8.9 Wave Functions
Summary
Problems
9 Spin
9.1 The Stern–Gerlach Experiment
9.2 Physical States
9.3 Matrix Representation
9.4 Eigenvectors
9.5 Scalar Products
9.6 Eigenvectors of [sub(x)]
9.7 The Stern–Gerlach Experiment Reloaded
Summary
Problems
10 Addition of Angular Momenta
10.1 Total Angular Momentum Operator
10.2 Addition Theorem
10.3 Example: Two Spin-1/2 Particles
Summary
Problems
11 Central Potentials
11.1 Stationary States
11.2 Physical Interpretation
11.3 Quantum Rotator
11.4 Central Square Well
Summary
Problems
12 Hydrogen Atom
12.1 Stationary States
12.2 Solution of the Radial Equation
12.3 Physical Interpretation
Summary
Problems
13 Identical Particles
13.1 Permutation Symmetry
13.2 A First Look at Helium
13.3 Two-Electron Wave Function
13.4 More on the Helium Atom
13.5 Pauli Exclusion Principle
Summary
Problems
14 Symmetries in Quantum Mechanics
14.1 Classical Symmetry
14.2 Quantum Symmetry
14.3 Symmetry Groups in Physics
14.4 Group of Translations
14.5 Group of Rotations
14.5.1 Rotations in Two Dimensions
14.5.2 Rotations in Three Dimensions
14.6 Rotations in the Space of Quantum States
14.7 Rotations Acting on Operators
14.8 Commutation Relations for a Generic Non-Abelian Group
Summary
Problems
Part II Applications
15 Quantum Entanglement
15.1 Hidden Variables and the Einstein–Podolsky–Rosen Paradox
15.1.1 The Einstein–Podolsky–Rosen Paradox
15.2 Bell’s Inequality
15.2.1 Spin Correlations
15.2.2 Experimental Tests of Bell’s Inequality
15.3 Characterising Entanglement
15.3.1 Bit and Qubit
15.3.2 Information Theory Essentials – Shannon Entropy
15.3.3 Von Neumann Entropy
15.3.4 Entanglement Entropy
15.3.5 Other Measures of Information
15.4 Quantum Communication
15.4.1 No-Cloning Theorem
15.4.2 Quantum Teleportation of a Qubit
15.4.3 Superdense Coding
15.5 Quantum Computing
15.5.1 Quantum Register and Logic Gates
15.5.2 Deutsch’s Algorithm
15.5.3 Grover’s Algorithm
15.A Appendix: Klein’s Inequality
Summary
Reference
Further Reading
Problems
16 Time-Independent Perturbation Theory
16.1 Nondegenerate Time-Independent Perturbation Theory
16.1.1 First-Order Correction
16.1.2 Second-Order Energy Correction
16.1.3 Normalisation and Orthogonality of the Eigenstates to First Order
16.1.4 Higher-Order Corrections
16.1.5 Properties of the Perturbation Expansion
16.2 Degenerate Perturbation Theory
16.2.1 First-Order Degenerate Perturbation Correction
16.2.2 Second-Order Degenerate Perturbation Energy Correction
16.3 Applications
16.3.1 Hydrogen Fine Structure
16.3.2 Helium Atom Perturbation Treatment of Electron Repulsion Term
16.A Appendix: Derivation of the Fine-Structure Terms from Relativistic Theory
Summary
References
Further Reading
Problems
17 Calculation Methods Beyond Perturbation Theory
17.1 Rayleigh–Ritz Variational Method
17.1.1 Ground State of Hydrogen
17.1.2 Ground State of Helium
17.1.3 Excited States
17.2 Atomic Systems
17.2.1 First Excited States of Helium
17.2.2 Multi-electron Atoms
17.3 Born–Oppenheimer Approximation
17.3.1 The H[sub(2)sup(+)] Ion and Bonding
17.4 Hellmann–Feynman Method
17.5 Wenzel–Kramers–Brillouin–Jeffreys Approximation
17.5.1 Potential Barrier
17.5.2 Potential Well
17.5.3 Symmetric Double Well
Summary
Further Reading
Problems
18 Time-Dependent Perturbation Theory
18.1 Time-Dependent Hamiltonians
18.1.1 Time-Dependent Perturbation Theory
18.2 Time-Dependent Perturbations
18.2.1 Time-Independent Perturbation Switched on at a Given Time
18.2.2 Fermi’s Golden Rule
18.2.3 Harmonic Perturbations
18.3 Interaction of Radiation with Quantum Systems
18.3.1 Semiclassical Treatment of Electromagnetic Radiation
18.3.2 Interaction with a One-Electron Atom
18.3.3 Selection Rules
18.4 Time-Dependent Perturbation Theory at Higher Order
18.4.1 Time Dependence in Quantum Mechanics
18.4.2 Time Evolution Operator in the Interaction Picture
Summary
Problems
19 Quantum Scattering Theory
19.1 Scattering Kinematics
19.1.1 The Differential Cross-Section
19.2 Evolution of a Wave Packet
19.3 Time-Dependent Approach to Scattering
19.3.1 The Born Approximation
19.3.2 Two-Body Scattering
19.4 Time-Independent Approach to Scattering
19.4.1 Relation of Wave Packet to Time-Independent Approach
19.4.2 The Scattering Amplitude
19.4.3 Green Function Method for Calculating f(θ, [phi])
19.4.4 Partial Wave Analysis
19.A Appendix: Solutions of the Radial Equation for a Free Particle
Summary
Further Reading
Problems
Appendix: Gaussian/SI Electromagnetic Units
Index
