Exact treatment of finite-dimensional and infinite-dimensional quantum systems

EXACT TREATMENT OF FINITE-DIMENSIONAL AND INFINITE-DIMENSIONAL QUANTUM SYSTEMS......Page 2
MATHEMATICS RESEARCH DEVELOPMENTS......Page 3
CONTENTS......Page 10
PREFACE......Page 12
ACKNOWLEDGMENTS......Page 14
INTRODUCTION......Page 16
1.1. GENERAL PLAN OF THE PRESENTATION......Page 18
1.2. SOME MISSING POINTS......Page 22
REFERENCES......Page 23
FINITE-DIMENSIONAL SYSTEMS......Page 26
2.1. INTERNAL AND EXTERNAL MODIFICATIONSOF FINITE-DIMENSIONAL SYSTEMS......Page 28
2.2. LRM TREATMENT OF EXTERNAL MODIFICATIONS......Page 36
2.2.1. Interlacing of Modified Eigenvalueswith Parent Eigenvalues......Page 39
2.2.2. Cardinal Solutions of the Combined System......Page 40
2.2.3. Singular Solutions of the Combined System......Page 43
2.2.4. How Efficient Is a Finite-Dimensional LRM Approach?......Page 48
2.2.5. Normalization of Modified Eigenstates......Page 50
2.2.6. An Example of the LRM Solutionof an Externally Modified System......Page 52
2.3. LRM TREATMENT OFINTERNAL MODIFICATIONS......Page 57
2.3.2. Cardinal Solutions of the Modified System......Page 59
2.3.3. Singular Solutions of the Modified System......Page 62
2.3.4. Case = 0 a P......Page 66
2.3.5. Case = 0 a V......Page 70
2.4. LRM TREATMENT OF THE VIBRATIONAL ISOTOPEEFFECT IN THE HARMONIC APPROXIMATION......Page 74
2.4.1. Out-of-Plane Vibrations ofPlanar Molecules......Page 77
2.4.2. Interlacing Rule for out-of-PlaneVibrations of Planar Molecules......Page 81
2.4.3. Single Isotopic Substitutions......Page 83
2.4.4. Inversion Relations......Page 85
2.4.5. Interlacing of out-of-Plane Vibrational Frequencies of Thiophene (H,D)-Isotopomers......Page 87
2.4.6. Out-of-Plane Vibrations of Ethylene (H,D)-Isotopomers......Page 92
2.4.7. Out-of-Plane Vibrations of Benzene (H,D)-Isotopomers......Page 99
2.4.8. Equality of LRM and DFT out-of-Plane Frequencies of Benzene (H,D)-Isotopomers......Page 107
2.4.9 Comparison of LRM, Frequency Scaled DFTand Experimental out-of-Plane Frequencies ofBenzene (H,D)-Isotopomers......Page 110
2.4.10. Symmetry Increase and Emergence ofWeakly Singular Solutions......Page 113
2.5. EXTERNAL MODIFICATION AS A SPECIALCASE OF INTERNAL MODIFICATION......Page 114
REFERENCES......Page 116
INFINITE DIMENSIONAL SYSTEMS......Page 118
3.1. INFINITE-DIMENSIONAL PARENT SYSTEM......Page 122
3.1.1. Characteristic and DerivedOperators of the Parent System......Page 126
3.2. EXTERNAL MODIFICATIONS OF INFINITEDIMENSIONALPARENT SYSTEMS......Page 129
3.3. GENERAL PROPERTIES OF LRM SOLUTIONSTO THE INFINITE-DIMENSIONAL COMBINED SYSTEMS......Page 131
3.3.1. Fractional Shift......Page 134
3.3.2. Description of the Open Systemthat Interacts with an Infinite System......Page 137
3.4. CHARACTERISTIC AND DERIVED OPERATORS OF THE BASE SYSTEM......Page 138
3.4.1. Calculation of the Derived Operators......Page 144
3.5. ISOLATED SOLUTIONS OF THE COMBINED SYSTEM......Page 147
3.5.1. Isolated Cardinal Eigenvalues and Resonant Points......Page 150
3.5.2. Probabilities Associated with IsolatedCardinal Eigenstates......Page 151
3.6. EMBEDDED SOLUTIONS OF THE COMBINED SYSTEM......Page 153
3.6.1. Embedded Cardinal Solutionsand Fractional Shift Eigenvalue Equation......Page 154
3.6.2. Basic Properties of the Solutions to theFractional Shift Eigenvalue Equation......Page 157
3.6.3. Probabilities Associated withEmbedded Cardinal Eigenstates......Page 158
3.6.4. Emergence of Resonance in the Case of the Weak Coupling......Page 161
3.6.5. Anomal Points and Emergence of Isolated Cardinal Solutions in the Continuum......Page 165
3.6.6. Embedded Singular Solutions......Page 166
3.7. COMPLETENESS RELATIONS......Page 168
3.8. ONE-DIMENSIONAL OPEN SYSTEMS......Page 170
3.9. EXAMPLES......Page 174
3.9.1. Example 3.1: Interaction of a Base System with anInfinite System that Contains a Single Eigenvalue Band......Page 175
3.9.2. Example 3.2: Interaction of a Base System with an Infinite System that ContainsTwo Eigenvalue Bandsand One Isolated Eigenstate......Page 188
3.9.3. Example 2.3: Interaction of a Single State with One-Dimensional Solid in the Nearest-NeighborTight-Binding Approximation......Page 198
3.9.4. The Case of the Weak Coupling......Page 209
3.10. INTERNAL MODIFICATIONSOF INFINITE-DIMENSIONAL SYSTEMS......Page 210
3.11. LRM TREATMENT OF INTERNALLY MODIFIED SYSTEMS......Page 212
3.11.1. Isolated Solutions of Modified Systems......Page 213
3.11.2. Embedded Solutions of the Modified System......Page 215
REFERENCES......Page 216
TIME-DEPENDENT SYSTEMS......Page 218
4.1. ONE DIMENSIONAL TIME-DEPENDENTOPEN SYSTEMS......Page 225
4.1.1. Decay of a Local State in the Weak Coupling Limit......Page 227
4.1.2. Transition of the Local State to theEigenstates of the Parent System......Page 228
4.1.3. Transition of the Local Stateto the Eigenstates of the Parent Systemin the Weak Coupling Limit......Page 231
4.1.4. Example 4.1......Page 233
4.1.5. Example 4.2......Page 236
CONCLUSION......Page 240
REFERENCES......Page 242
INDEX......Page 244