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Advanced Quantum Theory AMATH473/673, PHYS454 Advanced Quantum Theory AMATH473/673, PHYS454 Achim Kempf Department of Applied Mathematics University of Waterloo Canada c Achim Kempf, November 2020 (Please do not copy: textbook in progress) 2 Contents 1 A brief history of quantum theory 9 1.1 The classical period . 9 1.2 Planck and the \Ultraviolet Catastrophe" . 9 1.3 Discovery of h ................................ 10 1.4 Mounting evidence for the fundamental importance of h . 11 1.5 The discovery of quantum theory . 12 1.6 Relativistic quantum mechanics . 13 1.7 Quantum field theory . 15 1.8 Renormalization . 17 1.9 Beyond quantum field theory? . 17 1.10 Experiment and theory . 19 2 Classical mechanics in Hamiltonian form 21 2.1 Newton's laws for classical mechanics cannot be upgraded . 21 2.2 Levels of abstraction . 22 2.3 Classical mechanics in Hamiltonian formulation . 23 2.3.1 The energy function H contains all information . 23 2.3.2 The Poisson bracket . 25 2.3.3 The Hamilton equations . 27 2.3.4 Symmetries and Conservation laws . 29 2.3.5 A representation of the Poisson bracket . 31 2.4 Summary: The laws of classical mechanics . 32 2.5 Classical field theory . 33 3 Quantum mechanics in Hamiltonian form 35 3.1 Reconsidering the nature of observables . 36 3.2 The canonical commutation relations . 37 3.3 From the Hamiltonian to the equations of motion . 40 3.4 From the Hamiltonian to predictions of numbers . 44 3.4.1 Linear maps . 44 3.4.2 Choices of representation . 45 3.4.3 A matrix representation . 46 3 4 CONTENTS 3.4.4 Example: Solving the equations of motion for a free particle with matrix-valued functions . 49 3.4.5 Example: Solving the equations of motion for a harmonic oscil- lator with matrix-valued functions . 50 3.4.6 From matrix-valued functions to number predictions . 52 3.5 Initial conditions . 55 3.6 Emergence of probabilities . 56 3.7 The Hilbert space of quantum mechanics, and Dirac's notation . 59 3.7.1 Hilbert spaces . 60 3.7.2 Hilbert bases . 63 3.7.3 Discrete wave functions and matrix representations . 64 3.7.4 The domain of operators . 66 3.7.5 Changes of basis . 67 4 Eigenbases of Observables and the Spectral Theorem 71 4.1 Self-adjointness . 71 4.2 The spectrum of an operator . 72 4.3 The spectral theorem for self-adjoint operators . 75 4.3.1 Case 1: The self-adjoint operator f^ possesses only a point spectrum 76 4.3.2 Case 2: The spectrum of f^ is purely continuous . 77 4.3.3 Case 3: The self-adjoint operator f^ has a point spectrum and a continuous spectrum . 79 4.4 The spectral theorem for unitary operators . 81 5 The position, momentum and energy representations 83 5.1 The eigenbasis of the position operator . 83 5.2 The energy eigenbasis of a harmonic oscillator . 85 5.3 The position representation . 87 5.4 Shorthand notation for operators acting on wave functions . 89 5.5 The momentum representation . 90 5.6 Energy representations: bound states and scattering states . 92 6 Continuous versus discrete in quantum mechanics 95 6.1 It takes only a countable infinite amount of information to specify a quantum state . 95 6.2 Stieltjes integration explains the Dirac delta . 96 6.3 The Stieltjes integral explains operators such as jxihxj. 99 7 Uncertainty principles 101 7.1 The Heisenberg uncertainty relations . 101 7.2 The time and energy uncertainty relation . 104 7.3 The impact of quantum uncertainty on the dynamics . 107 CONTENTS 5 8 Pictures of the time evolution 111 8.1 The time-evolution operator . 111 8.1.1 Calculating U^(t) . 112 8.1.2 Significance of U^(t) . 115 8.2 The pictures of time evolution . 117 8.2.1 The Heisenberg picture . 117 8.2.2 The Schr¨odingerpicture . 117 8.2.3 The Dirac picture . 122 8.2.4 The Feynman picture . 126 9 Measurements and state collapse 131 9.1 Ideal measurements . 131 9.2 State collapse . 131 9.3 Probability for finding measurement outcomes: the Born rule . 133 9.4 1-bit measurements . 134 9.5 The case of degenerate eigenspaces . 135 9.6 Successive versus simultaneous measurements . 136 9.7 States versus state vectors . 139 10 Mixed states 141 10.1 Density matrix . 142 10.2 Dynamics of a mixed state . 144 10.3 How to quantify the mixedness . 144 10.4 Shannon entropy . 145 11 Thermal states 151 11.1 Methods for determining density matrices . 151 11.2 Thermal equilibrium states . 153 11.3 Thermal states are states of maximum ignorance . 154 12 Composite quantum systems 159 12.1 Beyond thermal baths . 159 12.2 Mixedness arising from interaction . 160 12.3 The collapse picture of interactions . 161 12.4 Heisenberg cuts . 162 12.5 Combining quantum systems via tensor products . 164 12.5.1 The tensor product of Hilbert spaces . 165 12.5.2 The tensor product of operators . 167 12.5.3 Bases . 168 6 CONTENTS 13 Entanglement vs. mixedness 171 13.1 Review: traces . 171 13.2 Partial traces . 172 13.3 How to calculate the state of a subsystem . 173 13.4 Entangled subsystems are mixed subsystems . 174 14 Amounts of entanglement and mixedness 179 14.1 Entanglement entropy . 179 14.2 Purity: a simpler measure of mixedness and entanglement . 183 15 Measurements are interactions 185 15.1 How interactions generate entanglement . 185 15.2 Interactions without measurements . 188 16 Composition of identical systems 191 16.1 Swapping identical subsystems . 191 16.2 Examples of composite boson and fermion systems . 193 16.3 Bose statistics versus Fermi statistics . 194 17 Feynman quantum mechanics 197 17.1 Basic rules of probabilities . 198 17.2 The rules of probability amplitudes . 199 17.3 The path integral . 200 17.4 How the path integral avoids noncommutative operators . 201 17.5 A useful tool: Trotterization . 202 17.6 The probability amplitudes for paths . 203 17.7 Relationship to classical mechanics . 205 Introduction Quantum theory, together with general relativity, represents humanity's so-far deepest understanding of the laws of nature. And quantum phenomena are not rare or difficult to observe. In fact, we experience quantum phenomena constantly! For example, the very stability of the desk at which you are sitting now has its origin in a quantum phenomenon. This is because atoms are mostly empty space and the only reason why atoms don't collapse is due to the uncertainty relations. Namely, the uncertainty relations imply that it costs plenty of momentum (and therefore energy) to compress atoms. Also, for example, the spectrum of sunlight is shaped by quantum effects - if Planck's constant were smaller, the sun would be bluer. Over the past century, the understanding of quantum phenomena has led to a number of applications which have profoundly impacted society, applications ranging from nuclear power, lasers, transistors and photovoltaic cells, to the use of MRI in medicine. Ever new sophisticated applications of quantum phenomena are being de- veloped, among them, for example, quantum computers which have the potential to revolutionize information processing. Also on the level of pure discovery, significant progress is currently being made, for example, in the field of cosmology, where both quantum effects and general relativistic effects are important: high-precision astronomical data obtained by satellite telescopes over the past 15 years show that the statistical distribution of matter in the universe agrees with great precision with the distribution which quantum theory predicts to have arisen from quantum fluctuations shortly after the big bang. All structure in the universe, and ultimately therefore also we, appear to have originated in quantum fluctuations. The aim of this course is to explain the mathematical structure underlying all quan- tum theories and to apply it to the relatively simple case of nonrelativistic quantum mechanics. Nonrelativistic quantum mechanics is the quantum theory that replaces Newton's mechanics. The more advanced quantum theory of fields, which is necessary for example to describe the ubiquitous particle creation and annihilation processes, is beyond the scope of this course, though of course I can't help but describe some of it. For example, the first chapter of these notes, up to section 1.5, describes the history of quantum theory as far as we will cover it in this course. The introduction goes on, however, with a historical overview that outlines the further developments, from 7 8 CONTENTS relativistic quantum mechanics to quantum field theory and on to the modern day quest for a theory of quantum gravity with applications in quantum cosmology. Quantum theory is still very much a work in progress and original ideas are needed as much as ever! Note: This course prepares for a number of graduate courses, for example, the grad- uate course Quantum Field Theory for Cosmology (AMATH872/PHYS785) that I normally teach every other year. Chapter 1 A brief history of quantum theory 1.1 The classical period At the end of the 19th century, it seemed that the basic laws of nature had been found. The world appeared to be a mechanical clockwork running according to Newton's laws of mechanics. Light appeared to be fully explained by the Faraday-Maxwell theory of electromagnetism which held that light was a wave phenomenon. Also, for example, heat had been understood as a form of energy. Together, the then known theories constituted \Classical Physics". Classical physics was so successful that it appeared that theoretical physics was almost complete, the only task left being to add more digits of precision. At that time, young Max Planck, for example, was an undergraduate student at the University of Munich.
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