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Quantum Mechanics Mathematical Structure and Physical ... Mechanics Mathematical Structure and Physical Structure Problems John R. Boccio Professor of Physics Swarthmore College October

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Quantum Mechanics

Mathematical Structureand

Physical Structure

Problems

John R. BoccioProfessor of PhysicsSwarthmore College

October 13, 2010
Contents

3 Formulation of Wave Mechanics - Part 2 13.11 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

3.11.1 Free Particle in One-Dimension - Wave Functions . . . . . 13.11.2 Free Particle in One-Dimension - Expectation Values . . . 13.11.3 Time Dependence . . . . . . . . . . . . . . . . . . . . . . 23.11.4 Continuous Probability . . . . . . . . . . . . . . . . . . . 23.11.5 Square Wave Packet . . . . . . . . . . . . . . . . . . . . . 23.11.6 Uncertain Dart . . . . . . . . . . . . . . . . . . . . . . . . 33.11.7 Find the Potential and the Energy . . . . . . . . . . . . . 33.11.8 Harmonic Oscillator wave Function . . . . . . . . . . . . . 33.11.9 Spreading of a Wave Packet . . . . . . . . . . . . . . . . . 43.11.10 The Uncertainty Principle says ... . . . . . . . . . . . . . . 43.11.11 Free Particle Schrodinger Equation . . . . . . . . . . . . . 53.11.12 Double Pinhole Experiment . . . . . . . . . . . . . . . . . 5

4 The Mathematics of Quantum Physics:Dirac Language 74.22 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4.22.1 Simple Basis Vectors . . . . . . . . . . . . . . . . . . . . . 74.22.2 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . 74.22.3 Orthogonal Basis Vectors . . . . . . . . . . . . . . . . . . 74.22.4 Operator Matrix Representation . . . . . . . . . . . . . . 84.22.5 Matrix Representation and Expectation Value . . . . . . 84.22.6 Projection Operator Representation . . . . . . . . . . . . 84.22.7 Operator Algebra . . . . . . . . . . . . . . . . . . . . . . . 84.22.8 Functions of Operators . . . . . . . . . . . . . . . . . . . . 94.22.9 A Symmetric Matrix . . . . . . . . . . . . . . . . . . . . . 94.22.10 Determinants and Traces . . . . . . . . . . . . . . . . . . 94.22.11 Function of a Matrix . . . . . . . . . . . . . . . . . . . . . 94.22.12 More Gram-Schmidt . . . . . . . . . . . . . . . . . . . . . 94.22.13 Infinite Dimensions . . . . . . . . . . . . . . . . . . . . . . 94.22.14 Spectral Decomposition . . . . . . . . . . . . . . . . . . . 104.22.15 Measurement Results . . . . . . . . . . . . . . . . . . . . 104.22.16 Expectation Values . . . . . . . . . . . . . . . . . . . . . . 10

i
ii CONTENTS

4.22.17 Eigenket Properties . . . . . . . . . . . . . . . . . . . . . 114.22.18 The World of Hard/Soft Particles . . . . . . . . . . . . . . 114.22.19 Things in Hilbert Space . . . . . . . . . . . . . . . . . . . 114.22.20 A 2-Dimensional Hilbert Space . . . . . . . . . . . . . . . 124.22.21 Find the Eigenvalues . . . . . . . . . . . . . . . . . . . . . 134.22.22 Operator Properties . . . . . . . . . . . . . . . . . . . . . 134.22.23 Ehrenfests Relations . . . . . . . . . . . . . . . . . . . . . 134.22.24 Solution of Coupled Linear ODEs . . . . . . . . . . . . . . 144.22.25 Spectral Decomposition Practice . . . . . . . . . . . . . . 144.22.26 More on Projection Operators . . . . . . . . . . . . . . . 14

5 Probability 175.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5.6.1 Simple Probability Concepts . . . . . . . . . . . . . . . . 175.6.2 Playing Cards . . . . . . . . . . . . . . . . . . . . . . . . . 195.6.3 Birthdays . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.6.4 Is there life? . . . . . . . . . . . . . . . . . . . . . . . . . . 195.6.5 Law of large Numbers . . . . . . . . . . . . . . . . . . . . 195.6.6 Bayes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.6.7 Psychological Tests . . . . . . . . . . . . . . . . . . . . . . 205.6.8 Bayes Rules, Gaussians and Learning . . . . . . . . . . . 205.6.9 Bergers Burgers-Maximum Entropy Ideas . . . . . . . . . 215.6.10 Extended Menu at Bergers Burgers . . . . . . . . . . . . 235.6.11 The Poisson Probability Distribution . . . . . . . . . . . . 235.6.12 Modeling Dice: Observables and Expectation Values . . . 245.6.13 Conditional Probabilities for Dice . . . . . . . . . . . . . . 255.6.14 Matrix Observables for Classical Probability . . . . . . . . 25

6 The Formulation of Quantum Mechanics 276.19 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

6.19.1 Can It Be Written? . . . . . . . . . . . . . . . . . . . . . . 276.19.2 Pure and Nonpure States . . . . . . . . . . . . . . . . . . 276.19.3 Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . 276.19.4 Acceptable Density Operators . . . . . . . . . . . . . . . . 286.19.5 Is it a Density Matrix? . . . . . . . . . . . . . . . . . . . . 286.19.6 Unitary Operators . . . . . . . . . . . . . . . . . . . . . . 286.19.7 More Density Matrices . . . . . . . . . . . . . . . . . . . . 296.19.8 Scale Transformation . . . . . . . . . . . . . . . . . . . . . 296.19.9 Operator Properties . . . . . . . . . . . . . . . . . . . . . 296.19.10 An Instantaneous Boost . . . . . . . . . . . . . . . . . . . 306.19.11 A Very Useful Identity . . . . . . . . . . . . . . . . . . . . 306.19.12 A Very Useful Identity with some help.... . . . . . . . . . 306.19.13 Another Very Useful Identity . . . . . . . . . . . . . . . . 316.19.14 Pure to Nonpure? . . . . . . . . . . . . . . . . . . . . . . 316.19.15 Schurs Lemma . . . . . . . . . . . . . . . . . . . . . . . . 31
CONTENTS iii

6.19.16 More About the Density Operator . . . . . . . . . . . . . 326.19.17 Entanglement and the Purity of a Reduced Density Op-

erator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326.19.18 The Controlled-Not Operator . . . . . . . . . . . . . . . . 336.19.19 Creating Entanglement via Unitary Evolution . . . . . . . 336.19.20 Tensor-Product Bases . . . . . . . . . . . . . . . . . . . . 336.19.21 Matrix Representations . . . . . . . . . . . . . . . . . . . 346.19.22 Practice with Dirac Language for Joint Systems . . . . . 346.19.23 More Mixed States . . . . . . . . . . . . . . . . . . . . . . 356.19.24 Complete Sets of Commuting Observables . . . . . . . . . 366.19.25 Conserved Quantum Numbers . . . . . . . . . . . . . . . 36

7 How Does It really Work:Photons, K-Mesons and Stern-Gerlach 377.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

7.5.1 Change the Basis . . . . . . . . . . . . . . . . . . . . . . . 377.5.2 Polaroids . . . . . . . . . . . . . . . . . . . . . . . . . . . 377.5.3 Calcite Crystal . . . . . . . . . . . . . . . . . . . . . . . . 387.5.4 Turpentine . . . . . . . . . . . . . . . . . . . . . . . . . . 387.5.5 What QM is all about - Two Views . . . . . . . . . . . . 387.5.6 Photons and Polarizers . . . . . . . . . . . . . . . . . . . 397.5.7 Time Evolution . . . . . . . . . . . . . . . . . . . . . . . . 397.5.8 K-Meson oscillations . . . . . . . . . . . . . . . . . . . . . 397.5.9 What comes out? . . . . . . . . . . . . . . . . . . . . . . . 407.5.10 Orientations . . . . . . . . . . . . . . . . . . . . . . . . . . 407.5.11 Find the phase angle . . . . . . . . . . . . . . . . . . . . . 417.5.12 Quarter-wave plate . . . . . . . . . . . . . . . . . . . . . . 417.5.13 What is happening? . . . . . . . . . . . . . . . . . . . . . 417.5.14 Interference . . . . . . . . . . . . . . . . . . . . . . . . . . 427.5.15 More Interference . . . . . . . . . . . . . . . . . . . . . . . 437.5.16 The Mach-Zender Interferometer and Quantum Interference 437.5.17 More Mach-Zender . . . . . . . . . . . . . . . . . . . . . . 45

8 Schrodinger Wave equation1-Dimensional Quantum Systems 478.15 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

8.15.1 Delta function in a well . . . . . . . . . . . . . . . . . . . 478.15.2 Properties of the wave function . . . . . . . . . . . . . . . 488.15.3 Repulsive Potential . . . . . . . . . . . . . . . . . . . . . . 488.15.4 Step and Delta Functions . . . . . . . . . . . . . . . . . . 488.15.5 Atomic Model . . . . . . . . . . . . . . . . . . . . . . . . 498.15.6 A confined particle . . . . . . . . . . . . . . . . . . . . . . 498.15.7 1/x potential . . . . . . . . . . . . . . . . . . . . . . . . . 508.15.8 Using the commutator . . . . . . . . . . . . . . . . . . . . 50
iv CONTENTS

8.15.9 Matrix Elements for Harmonic Oscillator . . . . . . . . . 508.15.10 A matrix element . . . . . . . . . . . . . . . . . . . . . . . 518.15.11 Correlation function . . . . . . . . . . . . . . . . . . . . . 518.15.12 Instantaneous Force . . . . . . . . . . . . . . . . . . . . . 518.15.13 Coherent States . . . . . . . . . . . . . . . . . . . . . . . . 518.15.14 Oscillator with Delta Function . . . . . . . . . . . . . . . 528.15.15 Measurement on a Particle in a Box . . . . . . . . . . . . 538.15.16 Aharonov-Bohm experiment . . . . . . . . . . . . . . . . . 538.15.17 A Josephson Junction . . . . . . . . . . . . . . . . . . . . 548.15.18 Eigenstates using Coherent States . . . . . . . . . . . . . 568.15.19 Bogliubov Transformation . . . . . . . . . . . . . . . . . . 568.15.20 Harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . 568.15.21 Another oscillator . . . . . . . . . . . . . . . . . . . . . . 568.15.22 The coherent state . . . . . . . . . . . . . . . . . . . . . . 578.15.23 Neutrino Oscillations . . . . . . . . . . . . . . . . . . . . . 588.15.24 Generating Function . . . . . . . . . . . . . . . . . . . . . 598.15.25 Given the wave function ...... . . . . . . . . . . . . . . . . 598.15.26 What is the oscillator doing? . . . . . . . . . . . . . . . . 598.15.27 Coupled oscillators . . . . . . . . . . . . . . . . . . . . . . 608.15.28 Interesting operators .... . . . . . . . . . . . . . . . . . . . 608.15.29 What is the state? . . . . . . . . . . . . . . . . . . . . . . 608.15.30 Things about particle in box . . . . . . . . . . . . . . . . 608.15.31 Handling arbitrary barriers..... . . . . . . .