Dimensional perturbation theory for vibration-rotation spectra of linear triatomic molecules

A very efficient large-order perturbation theory is formulated for the nuclear motion of a linear triatomic molecule. To demonstrate the method, all of the experimentally observed rotational energies, with values of $J$ almost up to 100, for the ground and first excited vibrational states of CO$_2$ and for the ground vibrational states of N$_2$O and of OCS are calculated. All coupling between vibration and rotation is included. The perturbation expansions reported here are rapidly convergent. The perturbation parameter is $D^{-1/2}$, where $D$ is the dimensionality of space. Increasing $D$ is qualitatively similar to increasing the angular momentum quantum number $J$. Therefore, this approach is especially suited for states with high rotational excitation. The computational cost of the method scales only as $JN_v^{5/3}$, where $N_v$ is the size of the vibrational basis set.Comment: submitted to Journal of Chemical Physics, 23 pages, REVTeX, no figure

Existence of Bound States in Continuous 0<D<\infty Dimensions

In modern fundamental theories there is consideration of higher dimensions, often in the context of what can be written as a Schr\"odinger equation. Thus, the energetics of bound states in different dimensions is of interest. By considering the quantum square well in continuous $D$ dimensions, it is shown that there is always a bound state for $0<D \le 2$. This binding is complete for D \to 0 and exponentially small for D \to 2_-. For D>2, a finite-sized well is always needed for there to be a bound state. This size grows like D^2 as D gets large. By adding the proper angular momentum tail a volcano, zero-energy, bound state can be obtained.Comment: 15 pages, 5 figures. Section added on square-well volcanos in arbitrary dimension

Power of a determinant with two physical applications

An expression for the kth power of an n×n determinant in n2 indeterminates (zij) is given as a sum of monomials. Two applications of this expression are given: the first is the Regge generating function for the Clebsch-Gordan coefficients of the unitary group SU(2), noting also the relation to the  3 F2 hypergeometric series; the second is to the even powers of the Vandermonde determinant, or, equivalently, all powers of the discriminant. The second result leads to an interesting map between magic square arrays and partitions and has applications to the wave functions describing the quantum Hall effect. The generalization of this map to arbitrary square arrays of nonnegative integers, having given row and column sums, is also given

Combinatorial aspects of representations of U(n)

The boson operator theory of the representations of the unitary group, its Wigner-Clebsch-Gordan, and Racah coefficients is reformulated in terms of the ring of polynomials in any number of indeterminates with the goal of bringing the theory, as nearly as possible, under the purview of combinatorial oriented concepts. Four of the basic relations in unitary group theory are interpreted from this viewpoint

New relations and identities for generalized hypergeometric coefficients

AbstractGeneralized hypergeometric coefficients 〈pFq(a; b)¦λ〉 enter into the problem of constructing matrix elements of tensor operators in the unitary groups and are the expansion coefficients of a multivariable symmetric function generalization pFq(a; b; z), z = (z1, z2,…, zt), of the Gauss hypergeometric function in terms of the Schur functions eλ(z), where λ = (λ1, λ2,…, λt) is an arbitrary partition. As befits their group-theoretic origin, identities for these generalized hypergeometric coefficients characteristically involve series summed over the Littlewood-Richardson numbers g(μνλ). Identities that may be interpreted as generalizations of the Bailey, Saalschütz,… identities are developed in this paper. Of particular interest is an identity which develops in a natural way a group-theoretically defined expansion over new inhomogeneous symmetric functions. While the relations obtained here are essential for the development of the properties of tensor operators, they are also of interest from the viewpoint of special functions

Unitary symmetry, combinatorics, and special functions

From 1967 to 1994, Larry Biedenham and I collaborated on 35 papers on various aspects of the general unitary group, especially its unitary irreducible representations and Wigner-Clebsch-Gordan coefficients. In our studies to unveil comprehensible structures in this subject, we discovered several nice results in special functions and combinatorics. The more important of these will be presented and their present status reviewed

Extended MacMahon-Schwinger's Master Theorem and Conformal Wavelets in Complex Minkowski Space

We construct the Continuous Wavelet Transform (CWT) on the homogeneous space (Cartan domain) D_4=SO(4,2)/(SO(4)\times SO(2)) of the conformal group SO(4,2) (locally isomorphic to SU(2,2)) in 1+3 dimensions. The manifold D_4 can be mapped one-to-one onto the future tube domain C^4_+ of the complex Minkowski space through a Cayley transformation, where other kind of (electromagnetic) wavelets have already been proposed in the literature. We study the unitary irreducible representations of the conformal group on the Hilbert spaces L^2_h(D_4,d\nu_\lambda) and L^2_h(C^4_+,d\tilde\nu_\lambda) of square integrable holomorphic functions with scale dimension \lambda and continuous mass spectrum, prove the isomorphism (equivariance) between both Hilbert spaces, admissibility and tight-frame conditions, provide reconstruction formulas and orthonormal basis of homogeneous polynomials and discuss symmetry properties and the Euclidean limit of the proposed conformal wavelets. For that purpose, we firstly state and prove a \lambda-extension of Schwinger's Master Theorem (SMT), which turns out to be a useful mathematical tool for us, particularly as a generating function for the unitary-representation functions of the conformal group and for the derivation of the reproducing (Bergman) kernel of L^2_h(D_4,d\nu_\lambda). SMT is related to MacMahon's Master Theorem (MMT) and an extension of both in terms of Louck's SU(N) solid harmonics is also provided for completeness. Convergence conditions are also studied.Comment: LaTeX, 40 pages, three new Sections and six new references added. To appear in ACH

INFO3333 GROUP ASSIGNMENT Group 61 (Prac 13 Tue 3pm CC)

This literature review will analyse, examine, and confirm all the information included in the literature from 2013 to 2021 is recent, relevant, and correct. Our project, classroom AR integration, aims to allow remote students to join the class through VR imaging and to project the remote students into the classroom as in-campus students through AR. The main objective is to enable two modes (online and on-campus) of teaching to work generally as conventional, allowing diverse and flexible learning styles

Differences in post-traumatic growth: Individual quarantine, COVID-19 duration and gender

ObjectiveThis study focuses on positive effects of the COVID-19 pandemic and aims to identify associations between gender, individual quarantine and duration of the COVID-19 (short- medium- and long-term pandemic), and posttraumatic growth (PTG).MethodThe data was collected via an online survey in Israel, and included 1,301 participants, 543 participants experienced short-term pandemics, 428 participants experienced medium-term pandemics and 330 participants experienced long-term pandemics. Most of the participants were female (73.6%), ranging from 18 to 89 years-old. The participants answered questions about their demographic background, individual quarantine experiences and ranked their PTG level.ResultsThe results indicate a significant main effect of gender and pandemic duration (short-, medium- and long-term pandemic). Women reported higher PTG levels than men, and participants experiencing short-term pandemic reported significantly lower PTG levels than participants experiencing medium- or long-term pandemic. There was also a significant interaction between gender and pandemic duration regarding PTG and a significant interaction in PTG by gender, pandemic duration and individual quarantine.ConclusionThe discussion addresses the findings in the context of traditional gender roles and gender differences in finding meaning and worth in home confinement situations

Quantum four-body system in D dimensions

By the method of generalized spherical harmonic polynomials, the Schr\"{o}dinger equation for a four-body system in $D$-dimensional space is reduced to the generalized radial equations where only six internal variables are involved. The problem on separating the rotational degrees of freedom from the internal ones for a quantum $N$-body system in $D$ dimensions is generally discussed.Comment: 19 pages, no figure, RevTex, Submitted to J. Math. Phy