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Based on the solution of the direct vibrational-rotational problem with the Watson Hamiltonian using Van Vleck perturbation theory, employing the quantum-chemical (CCSD(T)/aug-cc-pVQZ) molecular geometry of molecule , sextic force field, and cubic dipole moment surface, absorption spectra and spectroscopic constants of effective rotational Hamiltonians (A-reduction) for the ground state and ν2 band are predicted. Theoretical sixth-order vibrational (fourth for dipole operators) and third-order rotational perturbative methods are based on a systematic procedure for calculating rotational commutators using the normal ordering of cylindrical angular momentum operators. The obtained reduced spectroscopic constants are refined and augmented by being fitted into sets of experimental energy levels. The new effective Hamiltonians significantly improve the reproduction of the experimental data for J ≤ 30, and the computed intensities accurately reproduce the observed values. [ABSTRACT FROM AUTHOR]

For two operators X and Y , which obey [ X , Y ] = τ Y + μ we shall prove e X e Y = exp X + τ Y 1 − e − τ + μ 1 − e − τ − μ τ , which means we want to merge two exponential operators' product into one exponential operator. This kind of operator identity is useful for calculating the quantum entropy S since S = − κ Tr { ρ ln ρ } , when ρ ≡ ρ 1 ρ 2 , but ln (ρ 1 ρ 2) ≠ ln ρ 1 + ln ρ 2 , thus the merging operator formula is demanded. In this way, several exponential operators' product can also be merged. We shall employ the method of integration within normally ordered product (IWOP) to derive the merging operator identity. [ABSTRACT FROM AUTHOR]

It is known that the Stirling numbers of the second kind are related to normal ordering in the Weyl algebra, while the unsigned Stirling numbers of the first kind are related to normal ordering in the shift algebra. Recently, Kim-Kim introduced a λ\lambda -analogue of the unsigned Stirling numbers of the first kind and that of the rr-Stirling numbers of the first kind. In this article, we introduce a λ\lambda -analogue of the shift algebra (called λ\lambda -shift algebra) and investigate normal ordering in the λ\lambda -shift algebra. From the normal ordering in the λ\lambda -shift algebra, we derive some identities about the λ\lambda -analogue of the unsigned Stirling numbers of the first kind.

The normal ordering of an integral power of the number operator in terms of boson operators is expressed with the help of the Stirling numbers of the second kind. As a ‘degenerate version’ of this, we consider the normal ordering of a degenerate integral power of the number operator in terms of boson operators, which is represented by means of the degenerate Stirling numbers of the second kind. As an application of this normal ordering, we derive two equations defining the degenerate Stirling numbers of the second kind and a Dobinski-like formula for the degenerate Bell polynomials.

Spivey's Bell number identity involving Stirling numbers of the second kind was derived by Spivey considering set partitions. Since its original derivation it has been derived (and generalized) by many authors using different techniques. In particular, Katriel gave a proof using normal ordering in the Weyl algebra. Mező derived the dual of Spivey's identitiy for factorials involving (unsigned) Stirling num- bers of the first kind considering permutations and cycles. The latter identity has also been been derived (and generalized) in different fashions. What is lacking in literature is a proof using normal ordering in the spirit of Katriel's proof of Spivey's identity. In the present work, this gap is filled by providing a new proof of the dual of Spivey's identity using normal ordering in the shift algebra. [ABSTRACT FROM AUTHOR]

The normal ordering of an integral power of the number operator in terms of boson operators is expressed with the help of the Stirling numbers of the second kind. As a 'degenerate version' of this, we consider the normal ordering of a degenerate integral power of the number operator in terms of boson operators, which is represented by means of the degenerate Stirling numbers of the second kind. As an application of this normal ordering, we derive two equations defining the degenerate Stirling numbers of the second kind and a Dobinski-like formula for the degenerate Bell polynomials. [ABSTRACT FROM AUTHOR]

In this paper, the bijection between k-rook placements on a Ferrers board and k-matchings in the associated bipartite graph is extended to a bijection between k-file placements and certain fractional matchings. Using the latter bijection, a new interpretation is given for the normal ordering coefficients in the shift algebra. Several further results concerning normal ordering in the shift algebra are derived. [ABSTRACT FROM AUTHOR]

Abstract The observation of neutrinoless double beta decay would allow to establish lepton number violation and the Majorana nature of neutrinos. The rate of this process in the case of 3-neutrino mixing is controlled by the neutrinoless double beta decay effective Majorana mass | 〈 m 〉 |. For a neutrino mass spectrum with normal ordering, which is favoured over the spectrum with inverted ordering by recent global fits, | 〈 m 〉 | can be significantly suppressed. Taking into account updated data on the neutrino oscillation parameters, we investigate the conditions under which | 〈 m 〉 | in the case of spectrum with normal ordering exceeds 10 − 3 (5 × 10 − 3) eV : | 〈 m 〉 | NO > 10 − 3 (5 × 10 − 3) eV. We analyse first the generic case with unconstrained leptonic CP violation Majorana phases. We show, in particular, that if the sum of neutrino masses is found to satisfy Σ > 0.10 eV , then | 〈 m 〉 | NO > 5 × 10 − 3 eV for any values of the Majorana phases. We consider also cases where the values for these phases are either CP conserving or are in line with predictive schemes combining flavour and generalised CP symmetries. [ABSTRACT FROM AUTHOR]

We derive combinatorial identities for variables satisfying specific systems of commutation relations, in particular elliptic commutation relations. The identities thus obtained extend corresponding ones for q -commuting variables x and y satisfying y x = q x y . In particular, we obtain weight-dependent binomial theorems, functional equations for generalized exponential functions, we derive results for an elliptic derivative of elliptic commuting variables, and finally study weight-dependent extensions of the Weyl algebra which we connect to rook theory. [ABSTRACT FROM AUTHOR]

The recently proposed generalized Wick theorem offers a simple way to normal ordering the squeeze operator, shown by the present notes dedicated to the memory of J. Janszky. [ABSTRACT FROM AUTHOR]

By virtue of integration technique within ordered product of operators and Dirac's representation theory we find a new general formula for normally ordering coordinate-momentum operator functions, that is $f(g\hat {{Q}}+h\hat {P})= :\exp [\textstyle {g^{2}+h^{2} \over 4}\textstyle {{\partial ^{2}} \over {\partial (g\hat {{Q}}+h\hat {P})^{2}}}]f(g\hat {{Q}}+h\hat {P})$ :, where $\hat {Q}$ and $\hat {P}$ are the coordinate operator and momentum operator respectively, the symbol :: denotes normal ordering. Using this formula we can derive a series of new relations about Hermite polynomial and Laguerre polynomial, as well as some new differential relations. [ABSTRACT FROM AUTHOR]

By virtue of the integration technique within ordered product of operators and Dirac’s representation theory we find a new formula for normally ordering coordinate-operator functions, that is f ( Q ˆ ) = : exp ( 1 4 ∂ 2 ∂ Q ˆ 2 ) f ( Q ˆ ) : , where Q ˆ is the coordinate operator, the symbol : : denotes normal ordering. Using this formula we can arrange a given quantum operator function f ( Q ˆ ) in its normal ordering conveniently and fast. Furthermore, we can derive a series of new relations about Hermite polynomial and Laguerre polynomial, as well as some new differential relations. [ABSTRACT FROM AUTHOR]

In this paper, we study the time evolution of the coherent state in a degenerate parametric amplifier by the technique of integration within an ordered product (IWOP) of operators. We investigate some interesting and important physical properties of this state, such as the quantum statistical properties and squeezing effects, both analytically and numerically. Furthermore, the fidelity between this state and the Schrödinger cat state is also given. It is interesting to find that this quantum state has not only squeezing effect, but also rotating effect by appropriately choosing the related parameters. [ABSTRACT FROM AUTHOR]

Dowling showed that the Whitney numbers of the first kind and of the second kind satisfy Stirling number-like relations. Recently, Kim-Kim introduced the degenerate r -Whitney numbers of the first kind and of the second kind, as degenerate versions and further generalizations of the Whitney numbers of both kinds. The normal ordering of an integral power of the number operator in terms of boson operators is expressed with the help of the Stirling numbers of the second kind. In this paper, it is noted that the normal ordering of a certain quantity involving the number operator is expressed in terms of the degenerate r -Whitney numbers of the second kind. We derive some properties, recurrence relations, orthogonality relations and several identities on those numbers from such normal ordering. In addition, we consider the degenerate r -Dowling polynomials as a natural extension of the degenerate r -Whitney numbers of the second kind and investigate their properties. • Introduction of the degenerate r -Whitney numbers of the second kind and of the first kind. • The normal ordering of the quantity (m a + a + r) k , λ in terms of the degenerate r -Whitney numbers of the second kind. • The generating function of r -Dowling polynomials. [ABSTRACT FROM AUTHOR]

We exploit the ideas of spin-statistics theorem, normal-ordering and the key concepts behind the symmetry principles to derive the canonical (anti)commutators for the case of a one -dimensional (1D) supersymmetric (SUSY) harmonic oscillator (HO) without taking the help of the mathematical definition of canonical conjugate momenta with respect to the bosonic and fermionic variables of this toy model for the Hodge theory (where the continuous and discrete symmetries of the theory provide the physical realizations of the de Rham cohomological operators of differential geometry). In our present endeavor, it is the full set of continuous symmetries and their corresponding generators that lead to the derivation of basic (anti)commutators amongst the creation and annihilation operators that appear in the normal mode expansions of the dynamical fermionic and bosonic variables of our present SUSY theory of a HO. These basic brackets are in complete agreement with such kind of brackets that are derived from the standard canonical method of quantization scheme. [ABSTRACT FROM AUTHOR]

Abstract: By virtue of operator ordering technique and the generating function of polynomials, we provide a simple and neat approach to studying operator identities and mathematical formulas regarding to two-variable Hermite polynomials, which differs from the existing mathematical ways. We not only derive some new integration formulas and summation relations about two-variable Hermite polynomial, but also draw a conclusion that two-variable Hermite polynomial excitation of two-mode squeezed vacuum state is a squeezed two-mode number state. This may open a new route of developing mathematics by virtue of the quantum mechanical representations and operator ordering technique. [Copyright &y& Elsevier]

In this paper, we present a purely algebraic construction of the normal factorization of multimode squeezed states and calculate their inner products. This procedure allows one to orthonormalize bases generated by squeezed states. We calculate several correct representations of the normalizing constant for the normal factorization, discuss an analog of the Maslov index for squeezed states, and show that the Jordan decomposition is a useful mathematical tool for problems with degenerate Hamiltonians. As an application of this theory, we consider a nontrivial class of squeezing problems which are solvable in any dimension. [ABSTRACT FROM AUTHOR]

The unitary group adapted state universal multi-reference coupled cluster (UGA-SUMRCC) theory, recently developed by us using a normal-ordered multi-exponential wave operator Ansatz with spin-free excitations in the cluster operators, has the twin advantages of generating a spin-adapted coupled-cluster (CC) function and having a terminating expression of the so-called ‘direct term’ at the quartic power of cluster amplitudes. Not having any valence spectators, it also has the potentiality to describe orbital- and correlation-relaxation effectively. We illustrate this aspect by applying our formalism to study ionised/excited state energies involving core electrons. The high degree of orbital relaxation attendant on removal of a core electron and the consequent correlation relaxation of the ionised state are demonstrated to be captured very effectively with the Hartree–Fock orbitals for the neutral ground state.Three different formalisms have been used: (1) the ionisation potential/excitation energies (IP/EE) are computed as difference between state energies of the concerned states; (2) a quasi-Fock extension of the UGA-SUMRCC theory, which computes IP/EE directly and (3) use of a special model space to describe both the ground and the excited states such that EE is obtained directly. The model spaces for EE are incomplete and we have indicated the necessary modifications needed to have extensive energies. The results obtained by all the three approaches are very similar but they usually outperform the current CC-based allied theories with roughly the same computational effort. [ABSTRACT FROM PUBLISHER]

From the normally ordered form of the density operator of a squeezed coherent state (SCS), we directly derive the compact expression of the SCS's photon-number distribution (PND). Besides the known oscillation characteristics, we find that the PND is a periodic function with a period of π and extremely sensitive to phase. If the squeezing is strong enough, and the compound phase which is relevant to the complex squeezing and displacement parameters are assigned appropriate values, different oscillation behaviours in PND for even and odd photon numbers appear, respectively. [ABSTRACT FROM AUTHOR]

By virtue of the coherent state representation and the operator ordering method we find a new approach for transiting Hermite polynomials to Laguerre polynomials. We also derive the new reciprocal relation of Laguerre polynomials $$\sum\limits_{n = 0} {( - 1)^n \left( {_n^l } \right)L_n (x) = \tfrac{{x^l }} {{l!}}}$$ , and its application in deriving the sum rule of the Wingner function of Fock states is demonstrated. Some new expansion identities about the operator Laguerre polynomial are also derived. This opens a new route of deriving mathematical polynomials formulas by virtute of the quantum mechanical representations and operator ordering technique. [ABSTRACT FROM AUTHOR]

Using the technique of integration within an ordered product (IWOP) of operators we derive a new kind of bipartite entangled state | β, β〉, which is perfectly different from the preceding entangled states in construction. We also present main properties, generation scheme of | β, β〉, superposition of | β, β〉, Wigner function and some applications of | β, β〉 in quantum optics. [ABSTRACT FROM AUTHOR]

By virtue of the technique of integration within an ordered product of operators we present a new approach to obtain operators' normal ordering. We first put operators into their Weyl ordering through the Weyl-Wigner quantization scheme, and then we convert the Weyl ordered operators into normal ordering by virtue of the normally ordered form of the Wigner operator. [ABSTRACT FROM AUTHOR]

Abstract: Non-commutativity is ubiquitous in mathematical modeling of reality and in many cases same algebraic structures are implemented in different situations. Here we consider the canonical commutation relation of quantum theory and discuss a simple urn model of the latter. It is shown that enumeration of urn histories provides a faithful realization of the Heisenberg–Weyl algebra. Drawing on this analogy we demonstrate how the operator normal forms facilitate counting of histories via generating functions, which in turn yields an intuitive combinatorial picture of the ordering procedure itself. [ABSTRACT FROM AUTHOR]

The technique of the normal ordering of non-commuting operators is an important tool in the solution of problems involving creation and annihilation operators in quantum physics, such as in many-body theory or quantum optics. We point out the inconsistencies in previous definitions of the two standard normal ordering procedures for such operators, and show how consistent definitions may be made. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we give a modified approach to the generalized Stirling numbers of the second kind and . These numbers were firstly defined by Carlitz and recently studied extensively by Blasiak, Penson and Solomon. This approach depends on the previous results obtained by Carlitz, Toscano and Cakić. We show that Blasiak’s results can be investigated from Carlitz and Cakić’s results. Some interesting combinatorial identities are obtained. [Copyright &y& Elsevier]

Normally ordered forms of functions of boson operators are important in many contexts in particular concerning Quantum Field Theory and Quantum Optics. Beginning with the seminal work of Katriel (Lett. Nuovo Cimento 10(13):565–567, ), in the last few years, normally ordered forms have been shown to have a rich combinatorial structure, mainly in virtue of a link with the theory of partitions. In this paper, we attempt to enrich this link. By considering linear representations of noncrossing partitions, we define the notion of noncrossing normal ordering. Given the growing interest in noncrossing partitions, because of their many unexpected connections (like, for example, with free probability), noncrossing normal ordering appears to be an intriguing notion. We explicitly give the noncrossing normally ordered form of the functions ( a r ( a † ) s ) n ) and ( a r +( a † ) s ) n , plus various special cases. We are able to establish for the first time bijections between noncrossing contractions of these functions, k-ary trees and sets of lattice paths. [ABSTRACT FROM AUTHOR]

Abstract: Combinatorial aspects of normal ordering arbitrary products of q-bosons are discussed. In particular, the number operator and its expectation values in Fock space are shown to be intimately related to the q-deformed version of the generalized Stirling and Bell numbers introduced recently. [Copyright &y& Elsevier]

We investigate the combinatorics of the general formulas for the powers of the operator h ∂ d , where ∂ is a differential operator on an arbitrary noncommutative ring in which h is central. New formulas for the generalized Stirling numbers are obtained, as well as results on the divisibility by primes of the coefficients which occur in the normally ordered form of h ∂ d. All of the above applies to the theory of formal differential operator rings. [ABSTRACT FROM AUTHOR]

• Quasi-number states (QNS) of the Morse oscillator form infinite basis set. • Morse Hamiltonian matrix in QNS wave functions is tridiagonal. • Vibrational Hamiltonian can be expressed in normal form using QNS ladder operators. • Normal ordering of QNS ladder operators simplifies implementations of vibrational problems. The Morse oscillator is an adequate zero-order model for describing the highly excited vibrational states and large-amplitude vibrational motion. The corresponding Schrödinger equations can be conveniently solved by algebraic methods using the so-called quasi-number states (QNS) resembling the true wave functions of the Morse oscillator. The associated QNS ladder operators are occupation number, creation and annihilation operators K 0 , K + , K − that satisfy the commutation relations K 0 , K ± = ± K ± and K + , K − = − 2 K 0 of the Lie algebra su(1, 1). The efficient applications of the corresponding algebraic techniques within the variational or perturbative approximations require reduction of arbitrary multiplicative combinations of the su(1,1) ladder operators to the linear combinations of the normalized products K 0 a K + b K − c. The closed form solution of this problem for a typical complex form of the operator product is derived and proven mathematically. The validity and the numerical stability of the derived expressions are demonstrated by the systematic numerical tests for the product of six-term multiplicative operators. [ABSTRACT FROM AUTHOR]