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QM-notes-2011 - School Of Mathematics Statistics and...

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School Of Mathematics, Statistics, and Operations Research Te Kura M¯ atai Tatauranga, Rangahau P¯unaha MATH 321/322/323 Applied Math (Quantum Physics) T1 and T2 2011 Math 322: Applied Mathematics Notes — Quantum Physics module Matt Visser School of Mathematics, Statistics, and Operations Research Victoria University of Wellington, New Zealand. E-mail: [email protected] URL: http://www.mcs.vuw.ac.nz/˜visser Version of 23 February 2011; L A T E X-ed February 24, 2011 Warning: These notes will basically be the textbook for this module. This is a reading course, so these notes, and various web resources, should be your primary source of information. I may also assign some specific supplementary reading. There are still a few rough edges in these notes. If you find errors, typos, and/or obscurities, please let me know.
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Contents 1 Overview 4 2 Introduction to quantum physics 6 2.1 Quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Quantum technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 Quantum field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4 Classifying the theories of physics: Superb , Useful , and Tentative . . . 8 2.5 Superb means “Never to be Discarded” . . . . . . . . . . . . . . . . . . . 10 2.6 Textual analysis: A warning . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.7 Filtering out the nonsense . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.7.1 The two faces of physical theory . . . . . . . . . . . . . . . . . . . . 12 2.7.2 Rules based on mathematical consistency . . . . . . . . . . . . . . . 13 2.7.3 Parameterized post classical formalism? . . . . . . . . . . . . . . . . 15 2.7.4 High weirdness in quantum field theory . . . . . . . . . . . . . . . . 16 2.7.5 The rough guide to crackpot filtering . . . . . . . . . . . . . . . . . 17 2.8 Last words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3 Heisenberg uncertainty principle 20 3.1 Fourier transforms and signal theory . . . . . . . . . . . . . . . . . . . . . 20 3.2 The de Broglie and Einstein relations . . . . . . . . . . . . . . . . . . . . . 26 3.3 The Heisenberg uncertainty principle . . . . . . . . . . . . . . . . . . . . . 26 3.4 Classical operators and commutators . . . . . . . . . . . . . . . . . . . . . 28 3.5 Proof of the classical uncertainty relation . . . . . . . . . . . . . . . . . . . 32 3.6 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4 Tunnelling 35 4.1 Sound penetrating a barrier . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.2 Radio penetrating a barrier . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.3 Frustrated total internal reflection . . . . . . . . . . . . . . . . . . . . . . . 36 4.3.1 Key idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.3.2 Reminder: Snell’s law . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.3.3 Some technicalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.3.4 Barrier penetration . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2
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Math 322: Quantum Physics 3 4.4 FTIR in acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.4.1 TIR in fluid-fluid acoustics . . . . . . . . . . . . . . . . . . . . . . . 42 4.4.2 FTIR in fluid-fluid acoustics . . . . . . . . . . . . . . . . . . . . . . 47 4.5 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5 One-dimensional scattering 50 5.1 Physical interpretation of the transfer matrix M . . . . . . . . . . . . . . . 51 5.1.1 Special case: Definite parity . . . . . . . . . . . . . . . . . . . . . . 54 5.2 Simple examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 5.2.1 Delta-function potential . . . . . . . . . . . . . . . . . . . . . . . . 55 5.2.2 Two delta-function potentials . . . . . . . . . . . . . . . . . . . . . 56 5.2.3 Two-step potential . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 5.3 Some general theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 5.3.1 Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 5.3.2 Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 5.3.3 Transmission resonances . . . . . . . . . . . . . . . . . . . . . . . . 61 5.4 Lessons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 6 The scattering matrix in one-dimension 64 6.1 Physical interpretation of the S -matrix . . . . . . . . . . . . . . . . . . . . 64 6.2 Lessons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 7 Coda 68
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Chapter 1 Overview This module investigates the mathematical structure of quantum physics. So we will be very much emphasizing mathematical features of the theory, and will be particularly interested in seeing how much can be deduced purely by mathematical reasoning without any (or with very little) physics input. To set the stage, remember the two faces of any physical theory: Physical theories have a mathematical structure that can be investigated purely by logic. This mathematical structure exists independently of whether or not the
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