An outline is given of current advances on some basic ingredients of applied quantum mechanics, that were previously developed along different lines and are now being compacted within a unifying framework. Specifically, (i) complete orthogonal expansion basis sets for the atomic and molecular orbitals of quantum chemistry are classified within angular momentum theory, presently incorporated in and generalized as spin network theory; (ii) spin-networks and the underlying theory of hypergeometrical polynomials are presented within a graphical approach; (iii) the combinatorial significance of the graphical approach is given a projective geometry foundation; (iv) emergence and role of hidden (Regge’s) symmetries are revealed and discussed in a variety of contexts.
An outline is given of current advances on some basic ingredients of applied quantum mechanics, that were previously developed along different lines and are now being compacted within a unifying framework. Specifically, (i) complete orthogonal expansion basis sets for the atomic and molecular orbitals of quantum chemistry are classified within angular momentum theory, presently incorporated in and generalized as spin network theory; (ii) spin-networks and the underlying theory of hypergeometrical polynomials are presented within a graphical approach; (iii) the combinatorial significance of the graphical approach is given a projective geometry foundation; (iv) emergence and role of hidden (Regge’s) symmetries are revealed and discussed in a variety of contexts.
The mathematical apparatus of quantum-mechanical angular momentum (re)coupling, developed originally to describe spectroscopic phenomena in atomic, molecular, optical and nuclear physics, is embedded in modern algebraic settings which emphasize the underlying combinatorial aspects. SU (2) recoupling theory, involving Wigner's 3 nj symbols, as well as the related problems of their calculations, general properties, asymptotic limits for large entries, nowadays plays a prominent role also in quantum gravity and quantum computing applications. We refer to the ingredients of this theory—and of its extension to other Lie and quantum groups—by using the collective term of 'spin networks'. Recent progress is recorded about the already established connections with the mathematical theory of discrete orthogonal polynomials (the so-called Askey scheme), providing powerful tools based on asymptotic expansions, which correspond on the physical side to various levels of semi-classical limits. These results are useful not only in theoretical molecular physics but also in motivating algorithms for the computationally demanding problems of molecular dynamics and chemical reaction theory, where large angular momenta are typically involved. As for quantum chemistry, applications of these techniques include selection and classification of complete orthogonal basis sets in atomic and molecular problems, either in configuration space (Sturmian orbitals) or in momentum space. In this paper, we list and discuss some aspects of these developments—such as for instance the hyperquantization algorithm—as well as a few applications to quantum gravity and topology, thus providing evidence of a unifying background structure.
Abstract We present a formulation of quantum mechanics based on the theory of orthogonal polynomials. The wavefunction is expanded over a complete set of square integrable basis where the expansion coefficients are orthogonal polynomials in the energy and physical parameters. Information about the corresponding physical systems (both structural and dynamical) are derived from the properties of these polynomials. We demonstrate that an advantage of this formulation is that the class of exactly solvable quantum mechanical problems becomes larger than in the conventional formulation (see, for example, table 3 in the text). We limit our investigation in this work to the Askey classification scheme of hypergeometric orthogonal polynomials and focus on the Wilson polynomial and two of its limiting cases (the Meixner–Pollaczek and continuous dual Hahn polynomials). Nonetheless, the formulation is amenable to other classes of orthogonal polynomials.
The concept of atomic and molecular orbitals has been a fundamental pillar in modern chemistry, shedding light on the structures and reactivity of chemical compounds. This article examines the evolution and significance of orbital theory, its applications in chemistry, and the ongoing debate about the existence of orbitals from both physics and chemistry perspectives. Philosophical aspects related to the ontology of orbitals are explored, emphasizing the complex interplay between mathematical abstractions and tangible reality. The multifaceted nature of orbitals, their role in quantum mechanics, and their implications for understanding the quantum realm are discussed. While the debate surrounding the ontological status of orbitals remains ongoing, it highlights the profound nature of inquiries into the fundamental essence of reality. This exploration underscores the significance of continuous research and discourse in advancing our understanding of these fundamental constituents of the quantum world.
This book offers a concise introduction to the angular momentum, one of the most fundamental quantities in all of quantum mechanics. Beginning with the quantization of angular momentum, spin angular momentum, and the orbital angular momentum, the author goes on to discuss the Clebsch-Gordan coefficients for a two-component system. After developing the necessary mathematics, specifically spherical tensors and tensor operators, the author then investigates the 3- j , 6- j , and 9- j symbols. Throughout, the author provides practical applications to atomic, molecular, and nuclear physics. These include partial-wave expansions, the emission and absorption of particles, the proton and electron quadrupole moment, matrix element calculation in practice, and the properties of the symmetrical top molecule.
We formulate a quantum phase space for rotational and nuclear-spin states of rigid molecules. For each nuclear spin isomer, we re-derive the isomer's admissible angular momentum states from molecular geometry and nuclear-spin data, introduce its angular position states using quantization theory, and develop a generalized Fourier transform converting between the two. We classify molecules into three types -- asymmetric, rotationally symmetric, and perrotationally symmetric -- with the last type having no macroscopic analogue due to nuclear-spin statistics constraints. We discuss two general features in perrotationally symmetric state spaces that are Hamiltonian-independent and induced solely by symmetry and spin statistics. First, we quantify when and how an isomer's state space is completely rotation-spin entangled, meaning that it does not admit any separable states. Second, we identify isomers whose position states house an internal pseudo-spin or "fiber" degree of freedom, and the fiber's Berry phase or matrix after adiabatic changes in position yields naturally robust operations, akin to braiding anyonic quasiparticles or realizing fault-tolerant quantum gates. We outline how the fiber can be used as a quantum error-correcting code and discuss scenarios where these features can be experimentally probed
We use the mathematical structure of group algebras and H+-algebras for describing certain problems concerning the quantum dynamics of systems of angular momenta, including also the spin systems. The underlying groups are SU(2) and its quotient SO(3,R). The scheme developed is applied in two different contexts. Firstly, the purely group-algebraic framework is applied to the system of angular momenta of arbitrary origin, e.g., orbital angular momenta of electrons and nucleons, systems of quantized angular momenta of rotating extended objects like molecules. The other promising area of applications is Schrödinger quantum mechanics of rigid body with its often rather unexpected and very interesting features. Even within this Schrödinger framework the algebras of operators related to group algebras are a very useful tool. We investigate some problems of composed systems and the quasiclassical limit obtained as the asymptotics of "large" quantum numbers, i.e., "quickly oscillating" functions on groups. They are related in an interesting way to geometry of the coadjoint orbits of SU(2).