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toc - Physics 461 Quantum Mechanics I P.E Parris Department...

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Physics 461 / Quantum Mechanics I P.E. Parris Department of Physics University of Missouri-Rolla Rolla, Missouri 65409 January 10, 2005
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CONTENTS 1 Introduction 7 1.1 What is Quantum Mechanics? . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1.1 What is Mechanics? . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1.2 Postulates of Classical Mechanics: . . . . . . . . . . . . . . . . . . 8 1.2 The Development of Wave Mechanics . . . . . . . . . . . . . . . . . . . . . 10 1.3 The Wave Mechanics of Schrödinger . . . . . . . . . . . . . . . . . . . . . 13 1.3.1 Postulates of Wave Mechanics for a Single Spinless Particle . . . . 13 1.3.2 Schrödinger’s Mechanics for Conservative Systems . . . . . . . . . 16 1.3.3 The Principle of Superposition and Spectral Decomposition . . . . 17 1.3.4 The Free Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.3.5 Superpositions of Plane Waves and the Fourier Transform . . . . . 23 1.4 Appendix: The Delta Function . . . . . . . . . . . . . . . . . . . . . . . . 26 2 The Formalism of Quantum Mechanics 31 2.1 Postulate I: Speci fi cation of the Dynamical State . . . . . . . . . . . . . . 31 2.1.1 Properties of Linear Vector Spaces . . . . . . . . . . . . . . . . . . 31 2.1.2 Additional De fi nitions . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.1.3 Continuous Bases and Continuous Sets . . . . . . . . . . . . . . . . 34 2.1.4 Inner Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.1.5 Expansion of a Vector on an Orthonormal Basis . . . . . . . . . . 38 2.1.6 Calculation of Inner Products Using an Orthonormal Basis . . . . 39 2.1.7 The Position Representation . . . . . . . . . . . . . . . . . . . . . 40 2.1.8 The Wavevector Representation . . . . . . . . . . . . . . . . . . . . 41 2.2 Postulate II: Observables of Quantum Mechanical Systems . . . . . . . . . 43 2.2.1 Operators and Their Properties . . . . . . . . . . . . . . . . . . . . 43 2.2.2 Multiplicative Operators . . . . . . . . . . . . . . . . . . . . . . . . 45 2.2.3 Di ff erential Operators . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.2.4 Ket-Bra Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.2.5
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