Showing papers on "Ladder operator published in 1987"
TL;DR: In this paper, a new method of algebraisation of quantum mechanical eigenvalue equations is presented, where the dynamical algebra is represented on the space of group matrix elements and the ladder operators of the algebra are obtained from Infeld-Hull-Miller factorisations.
Abstract: A new method of algebraisation of quantum mechanical eigenvalue equations is presented. In this method the dynamical algebra is represented on the space of group matrix elements. The ladder operators of the dynamical algebra are obtained from Infeld-Hull-Miller factorisations. The method is used to study the first Poschl-Teller equation even in the non-symmetric case. The energy spectrum and the exact normalised solutions are obtained in agreement with the results of non-algebraic methods.
72 citations
31 Dec 1987
49 citations
TL;DR: In this paper, a fundamental link between system theory and statistical mechanics has been established by the Kolmogorov entropy K. Since K represents a measure for the internal information creation rate of dynamical systems, it provides an approach to irreversibility.
Abstract: A fundamental link between system theory and statistical mechanics has been found to be established by the Kolmogorov entropy K. By this quantity the temporal evolution of dynamical systems can be classified into regular, chaotic, and stochastic processes. Since K represents a measure for the internal information creation rate of dynamical systems, it provides an approach to irreversibility. The formal relationship to statistical mechanics is derived by means of an operator formalism originally introduced by Prigogine. For a Liouville operator L and an information operator\(\tilde M\) acting on a distribution in phase space, it is shown that i[L,\(\tilde M\)]≡KI (I=identity operator). As a first consequence of this equivalence, a relation is obtained between the chaotic correlation time of a system and Prigogine's concept of a “finite duration of presence.” Finally, the existence of chaos in quantum systems is discussed with respect to the existence of a quantum mechanical time operator.
47 citations
TL;DR: In this paper, the authors conjecture the operator content of the finite-size limit of the spectra of the Ashkin-Teller model with free boundary conditions, and obtain the same operator content from a Hamiltonian with a four-fermion interaction and a U(1) Kac-Moody Sugawara structure.
Abstract: Based on numerical analysis, the authors conjecture the operator content of the finite-size limit of the spectra of the Ashkin-Teller model with free boundary conditions. The same operator content is obtained from a Hamiltonian with a four-fermion interaction and a U(1) Kac-Moody Sugawara structure. For some special values of the coupling constant the model exhibits N=2 superconformal and Zamolodchikov-Fateev invariance. The operator content in these cases is expressed in terms of irreducible representations of the corresponding algebras.
38 citations
TL;DR: In this paper, the algebra of the two-dimensional harmonic oscillator is exploited to obtain matrix elements between eigenstates of the Morse potential, and the latter is mapped into the radial equation of the former problem by means of a change of variable and the use of the angular variable as a dummy variable.
Abstract: The algebra of the two-dimensional harmonic oscillator is exploited to obtain matrix elements between eigenstates of the Morse potential This follows after mapping the latter into the radial equation of the former problem by means of a change of variable and the use of the angular variable as a dummy variable
30 citations
TL;DR: Using the radial ladder operators A+l and A-l defined in solving the hydrogen atom radial equation with the factorisation method, the author developed a new device by which the calculations on the average value of rs, the inner product of the radial wavefunctions and the matrix element of r are simplified and by which some new recurrence relations are derived as mentioned in this paper.
Abstract: Using the radial ladder operators A+l and A-l defined in solving the hydrogen atom radial equation with the factorisation method, the author develops a new device by which the calculations on the average value of rs, the inner product of the radial wavefunctions and the matrix element of r are simplified and by which some new recurrence relations are derived.
24 citations
TL;DR: In this paper, the authors investigated the structure of the linear differential operators whose solutions determine the four-point correlations of primary operators in the SU(2) α-model with Wess-Sumino term and the d = 2 critical statistical systems with central Virasoro charge smaller than one.
23 citations
TL;DR: In this article, it was shown that a perturbed operator with disjoint parts of the spectrum is similar to an operator for which the subspaces constructed from the isolated parts of an unperturbed operator are invariant.
Abstract: The basis for most of the results in this paper is a theorem that a perturbed operator with disjoint parts of the spectrum is similar to an operator for which the subspaces constructed from the isolated parts of the unperturbed operator are invariant. In particular, estimates are obtained for the eigenvalues and projections of the perturbed operators, results about equiconvergence of spectral decompositions are obtained, and convergence questions for the eigenvalues are investigated with the use of projection methods. Bibliography: 15 titles.
21 citations
TL;DR: The time evolution operator for the time-dependent harmonic oscillator H = (1)/(2) {α(t)p2 +β( t)q2} is exactly obtained as the exponential of an anti-Hermitian operator.
Abstract: The time‐evolution operator for the time‐dependent harmonic oscillator H= (1)/(2) {α(t)p2 +β(t)q2} is exactly obtained as the exponential of an anti‐Hermitian operator. The method is based on the equations of motion for the coordinate and momentum operators in the Heisenberg representation. The problem is reduced to solving the classical equations of motion.
15 citations
TL;DR: In this paper, a new approximate differential form of the Boltzmann collision operator for a Rayleigh gas (or Brownian particles) is derived, which retains all the terms of first and second order in the small quantity μ 1 = M/(m+M), M and m being the masses of the light particles and of the heavy particles, respectively.
Abstract: A new approximate differential form of the Boltzmann collision operator for a Rayleigh gas (or Brownian particles) is derived. The calculation procedure retains all the terms of first and second order in the small quantity μ1 = M/(m+M), M and m being the masses of the light particles and of the heavy particles, respectively. The special conditions under which the obtained operator can be compared (and agree) with the different one derived by Wannier for heavy ions in a cold gas, are discussed. Moreover, some properties of the new operator are studied. In particular, it is shown that, contrarily to what happens for the usual first-order approximate (Fokker-Planck) collision operator, the new operator has the Burnett functions as eigenfunctions only in the Maxwell model. Finally, the reliability and accuracy of the new operator are examined and compared with those of the usual Fokker-Planck collision operator. From this discussion, the advantages offered by the new operator in many instances result manifest.
TL;DR: In this paper, it was shown that the spin connection can be obtained by averaging the Levi-Civita connection over the finite group generated by an orthonormal frame of the base manifold.
Abstract: It is shown, how, in the setting of Clifford bundles, the spin connection (or Dirac operator) may be obtained by averaging the Levi-Civita connection (or Kahler-Dirac operator) over the finite group generated by an orthonormal frame of the base manifold.
TL;DR: In this paper, it was shown that the radial momentum wave function for the hydrogen atom can be factorized and a ladder operator P±l is derived that is linear in the position operator r and is nonlinear function of the momentum operator p. The results complement the familiar operator solution for hydrogen atom that is based on factorization of the radial Hamiltonian.
Abstract: The radial form of Hylleraas’ equation for the hydrogen atom, Λl‖El〉=4ℏ4a−2‖El〉 (a=Bohr radius), is considered and it is shown that the operator Λl can be factorized. Hence ladder operators P±l are derived that are linear in the position operator r and are nonlinear functions of the momentum operator p. It is proven that P±l‖El〉 =2ℏ2a−1[1+(l+ (1)/(2) ± (1)/(2) )2 ×(2Mℏ−2a2E)]1/2‖E,l±1〉. In the momentum representation of wave mechanics the solutions to these equations are the radial momentum‐space wavefunctions for the hydrogen atom. Thus a simple method of calculating these wavefunctions is obtained. The results complement the familiar operator solution for the hydrogen atom that is based on factorization of the radial Hamiltonian and yields operators that are linear in p and are nonlinear functions of r.
TL;DR: In this article, a manufacturing process is proposed for calculating matrix elements of families of basic functions between the eigenfunctions Psi mj(x) of factorisable equations, based on the well known property that solutions of factorizable equations are also solutions of a couple of first-order difference-differential equations.
Abstract: A manufacturing process is proposed for calculating matrix elements of families of basic functions Qt(x) between the eigenfunctions Psi mj(x) of factorisable equations. This procedure, well adapted for computer algebra, relies on the well known property that solutions of factorisable equations are also solutions of a couple of first-order difference-differential equations. As a particular case, it is applied to the determination of closed form expressions of the 'curved' hydrogenic pseudoradial integrals which are needed when studying space-curvature effects in atomic structure calculations. Several other applications are pointed out.
01 Jan 1987
TL;DR: For an eigenvalue problem on I0 = (0, 1), the maxima and infima of all eigenvalues when the coefficient p in the operator y″ + py is allowed to vary in the class of all integrable functions with ∫ p+ = B or ∫ P+ = b where we integrate over the interval I0 as discussed by the authors.
Abstract: For an eigenvalue problem on I0 = (0,1), we determine the maxima and infima of all eigenvalues when the coefficient p in the operator –y″ + py is allowed to vary in the class of all integrable functions with ∫ P+ = B or ∫ P+ = B where we integrate over the interval I0.
TL;DR: In this article, the axial exchange current operator is considered within the Skyrma model and it is shown that at large internucleon separations the time component of the AXial exchange-current operator agrees with the well-known axial pion-exchange charge operator originally derived on the basis of soft-pion theorems.
TL;DR: In this paper, a relativistic spinor formalism was used to study a system of charged zero-spin bosons in the presence of a uniform magnetic field using a relatival spinor.
Abstract: A system of charged zero‐spin bosons in the presence of a uniform magnetic field is studied using a relativistic spinor formalism. Equations of motion for the relevant operators are developed. The eigenfunctions of two different sets of commuting operators are obtained by the use of ladder operators and also by direct solution of the Klein–Gordon equation in both coordinate and momentum representations. Matrix elements are calculated which will be required in later consideration of the collisionless conductivity and dielectric tensors for the boson–antiboson plasma in a strong magnetic field (see papers II and III in this series).
TL;DR: In this article, a new approach to the formulation of operator field equations is proposed and applied to the quantum mechanical system in one dimension and obtain explicit and general expressions for the transfer operator and for corresponding matrix elements.
Abstract: We propose a new approach to the formulation of some operator field equations In Section 1 we study the consistency of some particular difference equations and the convergence of the exact solutions In Section 2 we apply these results to the quantum mechanical system in one dimension and obtain explicit and general expressions for the transfer operator and for the corresponding matrix elements
01 Jan 1987
TL;DR: The description of dissipative processes in quantum mechanics in this article is based on the existence of a time operator canonically conjugate to the Liouville operator and the entropy can be defined as a functional of this time operator and a new dissipative semi-group of time evolution can be constructed.
Abstract: The description of dissipative processes in quantum mechanics here presented is based on the existence of a time operator canonically conjugate to the Liouville operator. The entropy can be defined as a functional of this time operator and a new dissipative semi-group of time evolution can be constructed. This formalism seems well adapted to the description of the decay phenomena and the measurement process.
TL;DR: The instability index of a linear operator with point spectrum is defined to be the total multiplicity of its eigenvalues with positive real parts as mentioned in this paper, and it is shown that the original operator can be found in the form of an integro-differential operator which can be realized constructively by solving a special elliptic boundary value problem.
Abstract: The instability index of a linear operator with point spectrum is defined to be the total multiplicity of its eigenvalues with positive real parts. Under certain conditions the computation of the instability index of an unbounded nonsymmetric operator acting in a Hilbert space can be reduced to an analogous problem for a certain selfadjoint operator. It is shown that if the original operator is a differential operator acting in a space of vector-valued functions with a single variable, then the selfadjoint operator corresponding to it can be found in the form of an integro-differential operator which can be realized constructively by solving a special elliptic boundary value problem. Analogues of known theorems of Morse on the connection between the instability index and the number of conjugate points are established for this integro-differential operator.Bibliography: 26 titles.
TL;DR: In this paper, a unitary operator is introduced to transform the aggregate boson operator into the fermion operator that also acts as an operator form of the Young diagram, in case it acts on vacuum.
Abstract: Aggregate creation (or annihilation) operators of one-dimensional spinless bosons are found as operator realization of Schur-functions i.e. Young diagrams. A unitary operator is introduced to transform the aggregate boson operator into the fermion operator that also acts as an operator form of the Young diagram. The fermion operator produces, in case it acts on vacuum, the uncoupled fermion state just describable in Sato's Maya diagram. Further, characteristic of S-function comes out in such quantum-mechanical solvable models, a boson model of the present author and a simplified Tomonaga-Luttinger model. Usefulness of the unitary operator is demonstrated also as a partial restatement of the theory of Kadomtsev-Petviashvili equation.
TL;DR: The general statistical properties of the Zwanzig projection operator are discussed in this paper, where it is shown that the mean value of the memory effects contained in the evolution equations generated by the projection operator is always zero.
Abstract: The general statistical properties of the Zwanzig projection operator are discussed. In particular it is shown that the mean value of the memory effects contained in the evolution equations generated by the projection operator are always zero. The results are applied to the nonlinear three-wave interaction in a plasma. Simple evolution equations for the mean number of photons of the interacting waves are derived.
TL;DR: The markovian description of the time evolution of the coordinate operator Q and the momentum operator P of a quantum-mechanical harmonic oscillator is shown explicitly to preserve the canonical commutation relation [Q ( t ), P ( t )] = i ℏ.
TL;DR: In this paper, the authors considered the factorization of four vector operators, D±(ω) and D± (−ω), which occur in a representationindependent, spectrum generating algebra for the three-dimensional, isotropic harmonic oscillator in an angular momentum basis, and showed that these operators can be factorized in two different ways to yield operators that are linear in either p or r.
Abstract: The factorization of four vector operators, D±(ω) and D±(−ω), which occur in a representation‐independent, spectrum‐generating algebra for the three‐dimensional, isotropic harmonic oscillator in an angular momentum basis, is considered (ω is the angular frequency of the oscillator). The D±(ω) are quantum‐mechanical analogs of the classical vectors (1∓iL×)Fc (ω), where Fc(ω)=−Mωr×L+pL is constant in a frame rotating with angular velocity ωL. It is shown that these four vector operators can be factorized in two different ways to yield operators that, apart from their dependence on a constant of the motion (L2), are linear in either p or r. In this way 20 abstract operators are obtained. The properties of these operators are discussed: (i) Twelve are ladder operators for the quantum numbers l, and l and m, in the eigenkets ‖lm〉 of L2 and Lz. In linearized, differential form six of these operators are ladder operators for the spherical harmonics in the coordinate representation, while the other six are the ...
TL;DR: In this article, the Schroedinger operator in R/sup 3/ with potential equal to the sum of a Coulomb part and a rapidly decreasing part was studied, and the determinant of the regularized scattering operator at zero energy to the characteristics of the discrete spectrum was derived.
Abstract: A study is made of the Schroedinger operator in R/sup 3/ with potential equal to the sum of a Coulomb part and a rapidly decreasing part For this operator, zeroth-order trace formulas are obtained that relate the determinant of the regularized scattering operator at zero energy to the characteristics of the discrete spectrum
TL;DR: In this paper, it was shown that the electric operator generates a degenerate representation of the Virasoro algebra and that it transforms as the fundamental representation under the SU(2) symmetry.
TL;DR: In this article, an explicit finite difference scheme was proposed to solve operator equations of motion in quantum mechanics and in quantum scalar field theory, and it was shown that the finite difference can be used to solve both problems.
Abstract: We propose an explicit finite difference scheme to solve operator equations of motion in quantum mechanics and in a quantum scalar field theory.