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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, September 2017 (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 . 11 1.6 Relativistic quantum mechanics . 13 1.7 Quantum field theory . 14 1.8 Beyond quantum field theory? . 16 1.9 Experiment and theory . 18 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 Uncertainty principles 71 4.1 The Heisenberg uncertainty relations . 71 4.2 The time and energy uncertainty relation . 74 4.3 The impact of quantum uncertainty on the dynamics . 77 5 Pictures of the time evolution 81 5.1 The time-evolution operator . 81 5.1.1 Calculating U^(t) .......................... 82 5.1.2 Significance of U^(t)......................... 85 5.2 The pictures of time evolution . 87 5.2.1 The Heisenberg picture . 87 5.2.2 The Schr¨odingerpicture . 87 5.2.3 The Dirac picture . 91 6 Mixed states 95 6.1 Density matrix . 95 6.2 Dynamics of a mixed state . 97 6.3 How to quantify the mixedness . 97 6.4 Shannon entropy . 98 7 Thermal states 103 7.1 Methods for determining density matrices . 103 7.2 Thermal equilibrium states . 105 7.3 Thermal states are states of maximum ignorance . 106 8 Measurements and state collapse 111 8.1 Ideal measurements . 111 8.2 State collapse . 111 CONTENTS 5 8.3 Probability for finding measurement outcomes: the Born rule . 113 8.4 Projection operators . 114 8.5 Degenerate eigenvalues . 115 8.6 States versus state vectors . 117 8.7 Measurements and mixed states . 118 9 Composite quantum systems 119 9.1 The Heisenberg cut . 119 9.2 Combining quantum systems via tensor products . 121 9.2.1 The tensor product of Hilbert spaces . 122 9.2.2 The tensor product of operators . 123 9.2.3 Bases . 125 10 Heisenberg cuts 127 10.1 Review: traces . 127 10.2 Partial traces . 128 10.3 How to calculate the state of a subsystem . 129 10.4 A simple measure of a state's purity . 130 10.5 Entangled subsystems are mixed subsystems . 131 11 Interactions and measurements 135 11.1 How interactions generate entanglement . 135 11.2 Interactions without measurements . 138 11.3 Entanglement entropy . 140 12 Composition of identical systems 145 12.1 Swapping identical subsystems . 145 12.2 Examples of composite boson and fermion systems . 147 12.3 Bose statistics versus Fermi statistics . 148 13 The position and momentum representations 151 13.1 Review: eigenbasis of the harmonic oscillator . 152 13.2 From the fjnig basis to the fjxig basis . 154 13.3 The position representation . 157 13.4 Shorthand notation for operators acting on wave functions . 159 13.5 The Schr¨odingerequation in the position representation . 160 13.6 Integration by differentiation . 161 13.7 The momentum representation . 163 13.8 Green's functions . 165 13.9 The gauge principle . 168 6 CONTENTS 14 Symmetries, Noether's theorem and spin 173 14.1 Noether's theorem . 173 14.2 Spatial translation symmetry and momentum conservation . 175 14.3 Rotation symmetry and angular momentum conservation . 176 14.4 Spin . 177 A Functional analysis of self-adjoint operators 181 A.1 Self-adjointness . 181 A.2 The spectrum of an operator . 182 A.3 Stieltjes Integration . 184 A.4 The spectral theorem for self-adjoint operators . 187 A.4.1 Case 1: The self-adjoint operator f^ possesses only a point spectrum187 A.4.2 Case 2: The spectrum of f^ is purely continuous . 189 A.4.3 Definition using the Stieltjes integral . 191 A.4.4 Case 3: The self-adjoint operator f^ has a point spectrum and a continuous spectrum . 192 A.4.5 The spectral theorem for unitary operators . 192 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. We appear to originate in quantum fluctuations. New satellite-based telescopes are being planned. The aim of this course is to explain the mathematical structure of all quantum the- ories and to apply it to nonrelativistic quantum mechanics. Nonrelativistic quantum mechanics is the quantum theory that replaces Newton's mechanics and it is the sim- plest quantum theory. 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 relativistic quantum mechanics to quantum field theory and on to the modern day quest 7 8 CONTENTS 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. I will next teach that course in the term Winter 2018. 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. In addition, heat had been understood as a form of energy. Together, these 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. And so, one of Max Planck's teachers, von Jolly, advised him against a career in physics. But a few years later, classical physics was overthrown. 1.2 Planck and the \Ultraviolet Catastrophe" The limits to the validity of classical physics first became apparent in measurements of the spectrum of heat radiation. It had been known that very hot objects, such as a smith's hot iron, are emitting light. They do because matter consists of charged particles which can act like little antennas that emit and absorb electromagnetic waves.
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