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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, October 2016 (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 . 12 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 19 2.1 Newton's laws for classical mechanics cannot be upgraded . 19 2.2 Levels of abstraction . 20 2.3 Classical mechanics in Hamiltonian formulation . 21 2.3.1 The energy function H contains all information . 21 2.3.2 The Poisson bracket . 23 2.3.3 The Hamilton equations . 25 2.3.4 Symmetries and Conservation laws . 27 2.3.5 A representation of the Poisson bracket . 29 2.4 Summary: The laws of classical mechanics . 30 2.5 Classical field theory . 31 3 Quantum mechanics in Hamiltonian form 33 3.1 Reconsidering the nature of observables . 34 3.2 The canonical commutation relations . 35 3.3 From the Hamiltonian to the Equations of Motion . 38 3.4 From the Hamiltonian to predictions of numbers . 41 3.4.1 A matrix representation . 42 3.4.2 Solving the matrix differential equations . 44 3.4.3 From matrix-valued functions to number predictions . 46 3.5 Initial conditions . 48 3 4 CONTENTS 3.6 Emergence of probabilities . 49 3.7 The Hilbert space and Dirac's notation . 52 3.7.1 Hilbert spaces . 53 3.7.2 Hilbert bases . 55 3.7.3 Discrete wave functions and matrix representations . 56 3.7.4 The domain of operators . 57 3.7.5 Changes of basis . 58 4 Uncertainty principles 61 4.1 The Heisenberg uncertainty relations . 61 4.2 The impact of quantum uncertainty on the dynamics . 65 4.3 The time and energy uncertainty relation . 66 5 Pictures of the time evolution 69 5.1 The time-evolution operator . 69 5.1.1 Calculating U^(t) .......................... 70 5.1.2 Significance of U^(t)......................... 73 5.2 The pictures of time evolution . 75 5.2.1 The Heisenberg picture . 75 5.2.2 The Schr¨odingerpicture . 75 5.2.3 The Dirac picture . 78 6 Measurements and state collapse 83 6.1 Ideal measurements . 83 6.2 The state collapse . 85 6.3 Simultaneous measurements . 86 6.4 States versus state vectors . 87 7 Quantum mechanical representation theory 89 7.1 Self-adjointness . 89 7.2 The spectrum of an operator . 90 7.3 Stieltjes Integration . 92 7.4 The spectral theorem . 95 7.4.1 Case 1: The operator f^ possesses only a point spectrum . 95 7.4.2 Case 2: The spectrum of f^ is purely continuous . 97 7.4.3 Case 3: The self-adjoint operator f^ has a point spectrum and a continuous spectrum . 100 7.4.4 The spectral theorem for unitary operators . 100 7.5 Example representations . 101 7.5.1 The spectrum and eigenbasis of the Hamiltonian of a harmonic oscillator . 101 7.5.2 From the fjnig basis to the fjxig basis . 103 CONTENTS 5 7.5.3 The position representation . 105 7.5.4 Shorthand notation for operators acting on wave functions . 107 7.5.5 The Schr¨odingerequation in the position representation . 108 7.5.6 The momentum representation . 109 7.6 Integration by differentiation . 112 7.7 Green's functions . 113 7.8 The gauge principle . 116 8 Mixed states 119 8.1 Density matrix . 120 8.2 How to quantify the mixedness . 121 8.3 Shannon entropy . 122 8.4 Von Neumann entropy . 124 8.5 Dynamics of a mixed state . 125 9 Thermal states 127 9.1 Thermalization . 127 9.2 The density operator of thermal states . 129 9.3 Infrared cutoff . 130 9.4 Thermal states are states of maximum ignorance . 130 10 Composite quantum systems 135 10.1 The Heisenberg cut . 135 10.2 Combining quantum systems via tensor products . 137 10.2.1 The tensor product of Hilbert spaces . 137 10.2.2 The tensor product of operators . 139 10.2.3 Bases . 140 10.2.4 Traces and partial traces . 141 11 Entanglement 143 11.1 How to reduce a Heisenberg cut . 144 11.2 A simple test of purity . 145 11.3 Purity versus entanglement . 145 11.4 Entanglement entropy . 147 11.5 Measurement interactions . 149 11.6 Interactions without measurements . 151 6 CONTENTS 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 is also a refresher course for beginning graduate students, as AMATH673 at the University of Waterloo. Graduate do the same homework and write the same midterm and final but write also an essay. If you are a grad student taking this course, talk with me about the topic. Here at Waterloo, there are a number of graduate courses that build on this course. For example, I normally teach every other year Quantum Field Theory for Cosmology (AMATH872/PHYS785). 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, Max Planck's teacher, Scholli, advised his student against a career in physics. Soon after 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. This means that also cold objects emit and absorb electromagnetic radiation. Their heat radiation is not visible because it too weak and too red for our eyes to see. Black objects are those that absorb electromagnetic radiation (of whichever frequency range under consideration) most easily and by time reversal symmetry they are therefore also the objects that emit electromagnetic radiation of that frequency range most readily. Tea in a black tea pot cools down faster than tea in a white or reflecting tea pot. Now at the time that Planck was a student, researchers were ready to apply the laws of classical physics to a precise calculation of the radiation spectrum emitted by black bodies.
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