Quantum Mechanics in Rigged Hilbert Space Language

✍ Scribed by Rafael de la Madrid Modino

• Year: 2001
• Tongue: English
• Leaves: 268
• Edition: version 28 Mar 2002
• Category: Library

✦ Table of Contents

1 Introduction 1
1.1 A Brief History of the Rigged Hilbert Space . . . . . . . . . . . . . . . . . 3
1.2 Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 A Rigged Hilbert Space of the Square Barrier Potential . . . . . . . . . . . .9
1.4 Scattering off the Square Barrier Potential . . . . . . . . . . . . . . . . . 12
1.5 The Gamow Vectors of the Square Barrier Potential Resonances . . . . . . .15
1.6 Time Reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.7 Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20
2 Mathematical Framework of Quantum Mechanics 23
2.1 Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .25
2.1.2 Linear Spaces and Scalar Product . . . . . . . . . . . . . . . . .25
2.1.3 Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . .28
2.1.4 Antilinear Functionals . . . . . . . . . . . . . . . . . . . . . .32
2.2 Topological Spaces . . . . . . . . . . . . . . . . . . . . . . . . .34
2.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .34
2.2.2 Open Sets and Neighborhoods . . . . . . . . . . . . . . . . . . . 34
2.2.3 Separation Axioms . . . . . . . . . . . . . . . . . . . . . . . . 37
2.2.4 Continuity and Homeomorphic Spaces . . . . . . . . . . . . . . . .38
2.3 Linear Topological Spaces . . . . . . . . . . . . . . . . . . . . . 39
2.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .39
2.3.2 Cauchy Sequences . . . . . . . . . . . . . . . . . . . . . . . . .41
2.3.3 Normed, Scalar Product and Metric Spaces . . . . . . . . . . . . .43
2.3.4 Continuous Linear Operators and Continuous Antilinear Functionals 45
2.4 Countably Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . .49
2.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .49
2.4.2 Dual Space of a Countably Hilbert Space . . . . . . . . . . . . . . 53
2.4.3 Countably Hilbert Spaces in Quantum Mechanics . . . . . . . . . . . 54
2.5 Linear Operators on Hilbert Spaces . . . . . . . . . . . . . . . . . . 56
2.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.5.2 Bounded Operators on a Hilbert Space . . . . . . . . . . . . . . . . 57
2.5.3 Unbounded Operators on a Hilbert Space . . . . . . . . . . . . . . . 62
2.6 Nuclear Rigged Hilbert Spaces . . . . . . . . . . . . . . . . . . . . .65
2.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.6.2 Nuclear Rigged Hilbert Spaces . . . . . . . . . . . . . . . . . . . .66
3 The Rigged Hilbert Space of the Harmonic Oscillator 69
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .71
3.2 Algebraic Operations . . . . . . . . . . . . . . . . . . . . . . . . .72
3.3 Construction of the Topologies . . . . . . . . . . . . . . . . . . . .77
3.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .77
3.3.2 Hilbert Space Topology . . . . . . . . . . . . . . . . . . . . . . .78
3.3.3 Nuclear Topology . . . . . . . . . . . . . . . . . . . . . . . . . .83
3.3.4 Physical Interpretation of Ψ, Φ and H . . . . . . . . . . . . . . . 85
3.3.5 Extension of the Algebra of Operators . . . . . . . . . . . . . . . 85
3.4 The RHS of the Harmonic Oscillator . . . . . . . . . . . . . . . . . .89
3.4.1 The Conjugate Space . . . . . . . . . . . . . . . . . . . . . . . . 89
3.4.2 Construction of the Rigged Hilbert Space . . . . . . . . . . . . . .91
3.4.3 Continuous Linear Operators on the Rigged Hilbert Space . . . . . . 93
3.5 Basis Systems, Eigenvector Decomposition and the Gelfand-Maurin Theorem 94
3.5.1 Basis Systems and Eigenvector Decomposition—a Heuristic Introduction .94
3.5.2 Gelfand-Maurin Theorem . . . . . . . . . . . . . . . . . . . . . . . .104
3.6 Gelfand-Maurin Theorem Applied to the Harmonic Oscillator . . . . . . . 109
3.6.1 Spectral Theorem Applied to the Energy Operator . . . . . . . . 109
3.6.2 Spectral Theorem Applied to the Position and Momentum Operators 110
3.6.3 Realizations of the RHS of the Harmonic Oscillator by Spaces of Functions 117
3.6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
3.7 A Remark Concerning Generalizations . . . . . . . . . . . . . . . . . . . . 128
3.7.1 Realization of the Abstract RHS by Spaces of Functions . . . . . . .128
3.7.2 General Statement of the Gelfand-Maurin Theorem . . . . . . . . . . 133
3.7.3 Generalization of the Algebra of Operators . . . . . . . . . . . . .134
3.7.4 Appendix: Continuity of the Algebra of the Harmonic Oscillator . . .135
4 A Rigged Hilbert Space of the Square Barrier Potential 137
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
4.2 Sturm-Liouville Theory Applied to the Square Barrier Potential . . . . 141
4.2.1 Schrödinger Equation in the Position Representation . . . . . . . . .141
4.2.2 Self-Adjoint Extension . . . . . . . . . . . . . . . . . . . . . . . 142
4.2.3 Resolvent and Green Functions . . . . . . . . . . . . . . . . . . . .143
4.2.4 Diagonalization of H and Eigenfunction Expansion . . . . . . . . . . 146
4.2.5 The Need of the RHS . . . . . . . . . . . . . . . . . . . . . . . . . . .152
4.2.6 Construction of the Rigged Hilbert Space . . . . . . . . . . . . . . . . 154
4.2.7 Dirac Basis Vector Expansion . . . . . . . . . . . . . . . . . . . . . . 155
4.2.8 Energy Representation of the RHS . . . . . . . . . . . . . . . . . .156
4.2.9 Meaning of the δ-normalization of the Eigenfunctions . . . . . . . .157
4.3 Conclusion to Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . 159
4.4 Appendices to Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . 160
4.4.1 Appendix 1: Self-Adjoint Extension . . . . . . . . . . . . . . . . .160
4.4.2 Appendix 2: Resolvent and Green Function . . . . . . . . . . . . . .161
4.4.3 Appendix 3: Diagonalization and Eigenfunction Expansion . . . . . . 163
4.4.4 Appendix 4: Construction of the RHS . . . . . . . . . . . . . . . . .165
4.4.5 Appendix 5: Dirac Basis Vector Expansion . . . . . . . . . . . . . . 167
4.4.6 Appendix 6: Energy Representation of the RHS . . . . . . . . . . . . 168
5 Scattering off the Square Barrier Potential 171
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
5.2 Lippmann-Schwinger Equation . . . . . . . . . . . . . . . . . . . . .174
5.2.1 Lippmann-Schwinger Kets . . . . . . . . . . . . . . . . . . . . . .174
5.2.2 Radial Representation of the Lippmann-Schwinger Equation . . . . . 175
5.2.3 Solution of the Radial Lippmann-Schwinger Equation . . . . . . . . 177
5.2.4 Direct Integral Decomposition Associated to the In-States . . . . .178
5.2.5 Direct Integral Decomposition Associated to the Observables . . . .183
5.3 Construction of the Lippmann-Schwinger Kets and Dirac Basis Vector Expansion . .185
5.4 S-matrix and Møller Operators . . . . . . . . . . . . . . . . . . . . . . . . . 188
5.5 Appendices to Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
5.5.1 Appendix 7: Free Hamiltonian . . . . . . . . . . . . . . . . . . . 190
5.5.2 Appendix 8: Spaces of Hardy Functions . . . . . . . . . . . . . . .198
6 The Gamow Vectors of the Square Barrier Potential Resonances 203
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
6.2 S-matrix Resonances . . . . . . . . . . . . . . . . . . . . . . . . .206
6.3 The Gamow Vectors . . . . . . . . . . . . . . . . . . . . . . . . . .208
6.3.1 Lippmann-Schwinger Equation of the Gamow Vectors . . . . . . . . . 208
6.3.2 The Gamow Vectors in Position Representation . . . . . . . . . . . . 209
6.3.3 The Gamow Vectors in Energy Representation . . . . . . . . . . . . . 212
6.4 Complex Basis Vector Expansion . . . . . . . . . . . . . . . . . . . . 216
6.5 Semigroup Time Evolution of the Gamow Vectors . . . . . . . . . . . . .217
6.6 Time Asymmetry of the Purely Outgoing Boundary Condition . . . . . . . 219
6.6.1 Outgoing Boundary Condition in Phase . . . . . . . . . . . . . . . . 219
6.6.2 Outgoing Boundary Condition in Probability Density . . . . . . . . . 220
6.7 Exponential Decay Law of the Gamow Vectors . . . . . . . . . . . . . . 221
6.8 Conclusion to Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . .223
6.9 Appendix 9: Figures . . . . . . . . . . . . . . . . . . . . . . . . . .224
7 The Time Reversal Operator in the Rigged Hilbert Space 227
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
7.2 The Standard Time Reversal Operator (ǫT = ǫI = 1) . . . . . . . . . . .230
7.3 The Time Reversal Doubling (ǫT = ǫI = −1) . . . . . . . . . . . . . . .234
7.4 Appendix 10: Time Reversal . . . . . . . . . . . . . . . . . . . . . . 238
8 Conclusions 245