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Some Aspects of Quantum Theory SOME ASPECTS OF QUANTUM THEORY John Scales Avery August 22, 2020 INTRODUCTION1 I hope that this book will be of interest to students and researchers in mathematics, physics and theoretical chemistry. The first few chapters can be read with ease by anyone with a knowledge of calculus and differential equations. However, some later chapters, and most of the appendices, are more demanding.. Chapter 10 deals with resonance energy transfer and especially with the relativistic treatment of this phenomenon. My fascination with this topic dates back to my Ph.D. thesis work in the early 1960’s at Imperial College, which was then a part of the University of London. I had previously been working at the laboratory of Prof. Albert Szent-Gy¨orgyiand the Marine Bi- ological Laboratory at Woods Hole, Massachusetts. The problem on which we had been working was a quantum mechanical treatment of the primary process in photosynthesis, where a photon is absorbed, and its energy stabi- lized. Resonance energy transfer plays a large role in this process2. When I started my Ph.D. work in London, I decided to see whether relativistic corrections made a difference3. My calculations4 showed that while the usual non-relativistic treatment leads to transition probabilities that fall off as 1/R6, the calculated relativistic transition probabilities had a long-range component that fell off as 1/R2. Thus,if we imagine a very large sphere around an excited atom of molecule, the probability that the excitation energy will be transferred to one or another of the acceptors is independent of the size of the sphere! Is this a process that competes with spontaneous photon emission? Or is it an alternative way of treating the joint process of emission and absorption?5 Today, sixty years later, I continue to be fascinated by this question. In Chapter 10 experiments are proposed which could demonstrate that reso- nance energy transfer over macroscopic distance is possible. I am grateful to my son, Associate Professor James Emil Avery of the Niels Bohr Institute, 1This book makes some use of my previously published book chapters. 2J.S. Avery, Z. Bay and A. Szent-Gy¨orgyi, On the Energy Transfer in Biological Sys- tems, Proc. Nat. Acad. Sci. (US), 47, 1742-1744, (1961) 3J.S. Avery, Resonance energy transfer and related phenomena, Ph.D. thesis, Imperial College of Science and Technology, University of London, (1965) 4J.S. Avery, Resonance energy transfer and spontaneous photon emission, Proc. Phys. Soc. (London) 88, Part 1, (1966) 5J.S. Avery, Use of the S-Matrix in the Relativistic Treatment of Resonance Energy Transfer. Int. J. Quantum Chemistry, XXV, 79-96, (1984) University of Copenhagen, for his help and advice. He deserves to be listed as co-author of this book. However, I I don’t want him to be blamed for the book’s shortcomings, for example, in case the discussion section of Chapter 10 is seen to be too speculative. Besides the usual topics, the book also focuses on some aspects of quan- tum theory that have been of special interest to myself and to my son, James. Among these special areas of interest is the use of 4-dimensional hyperspher- ical harmonics in reciprocal-space quantum chemistry. We share this interest with Professor Vincenzo Aquilanti and his group at the University of Perugia in Italy6. Both James and I have made numerous research visits to Perugia, where we have enjoyed both the wonderful hospitality and great mathemat- ical knowledge of Prof. Aquilanti and his co-workers. I should mention that James has a number of important papers in which he uses hyperspherical harmonics to calculate 3-center and 4-center interelectron repulsion integrals for exponential-type basis sets (ETO’s).7 James and I are also co-authors of several books on hyperspherical harmonics.8 9 10 My interest in many-dimensional spaces brought me into contact with Professor Dudley R. Herschbach of Harvard University. I have been privileged to visit his brilliant research group many times, and to work closely with Prof. Herschbach and his colleagues for many years.11 12 6Aquilanti, V. and Avery, J., Sturmian expansions for quantum mechanical many-body problems and hyperspherical harmonics, Adv. Quant. Chem., 39 72-101, (2001) 74-center STO interelectron repulsion integrals with Coulomb Sturmians Avery, James Emil & Avery, J. S., (2018), In : Advances in Quantum Chemistry. 76, p. 133-146 8Generalized Sturmians and Atomic Spectra , by J.E. Avery and J.S. Avery, World Scientific Publishing (2006) 9Symmetry-Adapted Basis Sets , by J.S. Avery, S. Rettrup and J.E. Avery, World Scientific Publishing Co, (2012) 10Hyperspherical Harmonics and their Physical Applications, by J.E. Avery and J.S. Avery, World Scientific Publishing Co. (2017) 11Avery, J. and Herschbach, D. R., Hyperspherical Sturmian basis functions, Int. J. Quantum Chem., 41 673, (1992) 12J. Avery, D-Dimensional Hydrogenlike Orbitals, in Dimensional Scaling in Chemical Physics , D.R. Herschbach, J, Avery, and O. Goscinski editors, Kluwer Academic Publish- ers, Dordricht, Netherlands, (1992),pages 139-164 Figure 1: Professor Dudley R. Herschbach accepting the American Institute of Chemistry’s gold medal in 2011. He shared the 1986 Nobel Prize in Chemistry for his pioneering contributions to our understanding of the mechanisms of chemical reactions. Figure 2: Professor Vincenzo Aquilanti (born in 1939). After work- ing at Harvard with Dudley Herschbach, he returned to Italy, where he became the head of the chemistry department at the University of Perugia. He and his group have done pioneering theoretical and experimental work on the mechanism of chemical reactions, using molecular beam techniques. Professor Aquilanti and his group have also developed the use of 4-dimensional hy- perspherical harmonics in momentum-space quantum theory, an interest which they share with my son James and myself. Figure 3: Associate Professor James Emil Avery of the Niels Bohr Institute, University of Copenhagen. He is the author of a number of important papers that uses hyperspherical harmonics to calcu- late difficult 3-center and 4-center interelectron repulsion integrals for exponential-type orbitals, and is also the co-author of several books on hyperspherical harmonics and generalized Sturmians. Contents 1 ERNEST RUTHERFORD 11 1.1 Rutherford's model of the atom . 11 1.2 The Geiger-Marsden scattering experiment . 13 1.3 Rutherford's model of the atom . 16 1.4 Informality, enthusiasm and speed . 16 1.5 Artificial transmutations of elements . 20 2 NIELS BOHR 23 2.1 Christian Bohr's household . 23 2.2 Planck, Einstein and Bohr . 26 2.3 Atomic numbers . 31 2.4 Bohr's Institute of Theoretical Physics . 32 2.5 Bohr anticipates the nuclear arms race . 38 3 SCHRODINGER'S¨ WAVE EQUATION 43 3.1 A wave equation for matter . 43 3.2 Felix Bloch's story about Schr¨odinger. 46 3.3 Dirac's relativistic wave equation . 46 3.4 Some equations . 53 4 HARMONIC POLYNOMIALS AND SPHERICAL HARMONICS 55 4.1 Spherical polar coordinates . 55 4.2 The Laplacian operator in spherical coordinates . 56 4.3 Homogeneous and harmonic polynomials . 57 4.4 Harmonic polynomials and spherical harmonics . 57 4.5 An angular integration theorem . 61 5 THE SCHRODINGER¨ EQUATION FOR HYDROGEN 63 5.1 Separation of the equation . 63 5.2 Solutions to the radial equation . 64 5.3 Fock's momentum-space treatment of hydrogen . 68 5.4 The Pauli exclusion principle and the periodic table . 72 5.5 Valence bond theory . 80 7 8 CONTENTS 5.6 Molecular orbital theory . 82 5.7 The Hartree-Fock-Roothaan equations . 95 5.8 Koopmans' theorem . 97 5.9 Electron creation and annihilation operators . 99 5.10 Quantum chemistry and the development of computers . 100 6 PERIODIC SYSTEMS 119 6.1 The discovery of X-rays . 119 6.2 Bragg father and son . 121 6.3 J.D. Bernal and Dorothy Crowfoot Hodgkin . 125 6.4 The structure of DNA: Molecular biology . 128 6.5 Direct and reciprocal lattice vectors . 133 6.6 A H¨uckel calculation for a graphite plane . 135 6.7 3-dimensional crystal lattices . 136 6.8 Quantum treatment of electrons in crystals . 137 6.9 The nearly-free electron approximation . 137 6.10 Molecular crystals . 140 6.11 Periodic boundary conditions . 141 6.12 Homogeneous boundary conditions . 142 6.13 Taylor series expansion of the inter-monomer interaction . 144 6.14 X-ray diffraction experiments . 145 7 HARMONIC OSCILLATORS 147 7.1 Normal modes . 147 7.2 Molecular vibrations and rotations . 150 7.3 Commutation relations . 151 7.4 Phonon creation and annihilation operators . 152 7.5 Collections of harmonic oscillators . 154 8 THE DIRAC EQUATION 155 8.1 Lorentz invariance and 4-vectors . 155 8.2 The Dirac equation for an electron in an external electromagnetic potential 157 8.3 Time-independent problems . 158 8.4 The Dirac equation for an electron in the field of a nucleus . 159 9 INTERACTION BETWEEN MATTER AND RADIATION 163 9.1 Lagrangian densities for fields . 163 9.2 Electromagnetic potentials . 165 9.3 Separation of the longitudinal and transverse potentials . 170 9.4 Linear polarization . 171 9.5 Spontaneous photon emission . 172 9.6 Photon absorption . 173 9.7 Problems with field theories . 175 CONTENTS 9 10 RESONANCE ENERGY TRANSFER 179 10.1 Introduction . 179 10.2 Review of the Perrin-F¨orstertheory . 180 10.3 A relativistic interaction . 181 10.4 The Green's function of the Helmholtz equation . 182 10.5 Matrix elements . 183 10.6 Transition probability at macroscopic separations . 184 10.7 Comparison with spontaneous photon emission . 185 10.8 The Perrin-F¨orsterregion . 186 10.9 A proposed experiment . 187 10.10 Discussion: Direct interparticle interaction .
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