Jacob Katriel | Technion Israel Institute of Technology (original) (raw)

Papers by Jacob Katriel

Journal of Computational and Applied Mathematics, 2010

Position and momentum information measures are evaluated for the ground state of the relativistic... more Position and momentum information measures are evaluated for the ground state of the relativistic hydrogen-like atoms. Consequences of the fact that the radial momentum operator is not self-adjoint are explicitly studied, exhibiting fundamental shortcomings of the conventional uncertainty measures in terms of the radial position and momentum variances. The Shannon and Rényi entropies, the Fisher information measure, as well as several related information measures, are considered as viable alternatives. Detailed results on the onset of relativistic effects for low nuclear charges, and on the extreme relativistic limit, are presented. The relativistic position density decays exponentially at large r, but is singular at the origin. Correspondingly, the momentum density decays as an inverse power of p. Both features yield divergent Rényi entropies away from a finite vicinity of the Shannon entropy. While the position space information measures can be evaluated analytically for both the nonrelativistic and the relativistic hydrogen atom, this is not the case for the relativistic momentum space. Some of the results allow interesting insight into the significance of recently evaluated Dirac-Fock vs. Hartree-Fock complexity measures for many-electron neutral atoms.

Solid State Communications, Nov 1, 1984

Abstract The equation s =-σBσ[βσ|▿sH|]▿sH/|▿sH| is shown to be the magnetization equation for the... more Abstract The equation s =-σBσ[βσ|▿sH|]▿sH/|▿sH| is shown to be the magnetization equation for the arbitrary anisotropic infinite-range spin Hamiltonian, H(s). The known equations for the general anisotropic-Heisenberg, and for arbitrary isotropic and axially symmetric spin Hamiltonians are shown to be special cases of this equation.

Physical review, Nov 1, 1983

A generalized mean-field theory, incorporating the "classical" mean-field theories but having a r... more A generalized mean-field theory, incorporating the "classical" mean-field theories but having a richer structure, is formulated. Thermodynamic consistency is established for this generalized theory. As an application of the formalism, a treatment of coupled orientational and translational order-disorder transitions is presented, resulting in self-consistency equations requiring no numerical integration but exhibiting the essential features of the isotropic-nematic-smectic transitions in liquid crystals.

arXiv (Cornell University), Sep 11, 1995

The fundumental invariant of the Hecke algebra Hn(q) is the q-deformed class-sum of transposition... more The fundumental invariant of the Hecke algebra Hn(q) is the q-deformed class-sum of transpositions of the symmetric group Sn. Irreducible representations of Hn(q), for generic q, are shown to be completely characterized by the corresponding eigenvalues of Cn alone. For Sn more and more invariants are necessary as n inereases. It is pointed out that the q-deformed classical quadratic Casimir of SU (N) plays an analogous role. It is indicated why and how this should be a general phenomenon associated with q-deformation of classical algebras. Apart from this remarkable conceptual aspect Cn can provide powerful and elegant techniques for computations. This is illustrated by using the sequence C 2 , C 3 , • • • , Cn to compute the characters of Hn(q).

arXiv (Cornell University), Feb 27, 1995

The traces of the Murphy operators of the Hecke algebra H n (q), and of products of sets of Murph... more The traces of the Murphy operators of the Hecke algebra H n (q), and of products of sets of Murphy operators with non-consecutive indices, can be evaluated by a straightforward recursive procedure. These traces are shown to determine all the reduced traces in this algebra, which, in turn, determine all other traces. To illustrate the procedure we obtain the set of reduced traces for H 7 (q)-the lowest order Hecke algebra whose character table has not hitherto been reported. This is preceded by the presentation of an explicit algorithm for the reduction of the trace of an arbitrary element of the Hecke algebra into a linear combination of traces of elements consisting of appropriately defined disjoint cycles; and of a proof, presented in order to make the present article reasonably self-contained, that a reduced trace depends only on the set of lengths of the disjoint cycles that it consists of.

Physics Letters, Sep 1, 1976

Abstract Two electron radiative transition probabilities from the L and M shells into a doubly va... more Abstract Two electron radiative transition probabilities from the L and M shells into a doubly vacant K shell are computed non-relativistically, introducing angular correlation in the simplest conceivable way. The relative cross-sections for the different possible processes are in agreement with recently reported experimental values for Fe and Ni and predict interesting changes in the spectra of heavier systems.

Annals of Physics, Oct 1, 2018

The relative ordering of energy levels is investigated for bound two-electron systems with potent... more The relative ordering of energy levels is investigated for bound two-electron systems with potentials of the form V (r 1 , r 2 , r 12) = Z v(r 1) + v(r 2) − v(r 12). Given the two one-body binding potentials v (1) (r) and v (2) (r), it is argued that if f(r) ≡ v (1) (r) − v (2) (r) is positive and monotonically decreasing upon increasing r then the corresponding eigenvalues of the two-electron Hamiltonians *Manuscript Click here to view linked References where λ is the screening parameter. For each of the λ-dependent potentials we compare one-and two-electron spectra corresponding to distinct values of λ. This is followed by pairwise comparison of distinct potentials.

Physical review, Feb 9, 2018

The one-and two-electron Yukawa atoms, also referred to as the Debye-Hückel or screened Coulomb a... more The one-and two-electron Yukawa atoms, also referred to as the Debye-Hückel or screened Coulomb atoms, have been topics of considerable interest both for intrinsic reasons and because of their relevance to terrestrial and astrophysical plasmas. At sufficiently high screening the one-electron Yukawa atom ceases to be bound. Some calculations appeared to suggest that as the screening increases in the ground state of the two-electron Yukawa atom (in which both the one-particle attraction and the interparticle repulsion are screened) the two electrons are detached simultaneously, at the same screening constant at which the one-electron atom becomes unbound. Our results rule this scenario out, offering an alternative that is not less interesting. In particular, it is found that for Z < 1 a mild amount of screening actually increases the binding energy of the second electron. At the nuclear charge Z c ≈ 0.911028. . ., at which the bare Coulomb two-electron atom becomes unbound, and even over a range of lower nuclear charges, an appropriate amount of screening gives rise to a bound two-electron system.

Zeitschrift Für Physik A Hadrons And Nuclei, Sep 1, 1994

An algorithm is presented for the construction of single-and multi-cluster harmonic oscillator wa... more An algorithm is presented for the construction of single-and multi-cluster harmonic oscillator wave functions that are coupled into well-defined irreducible representa

Springer eBooks, Apr 7, 2008

An algorithm for the construction of non-spurious harmonic oscillator wave functions with arbitra... more An algorithm for the construction of non-spurious harmonic oscillator wave functions with arbitrary permutational symmetry is formulated by adapting a procedure recently developed for building general single shell wave functions. The harmonic oscillator functions, expressed in Jacobi coordinates, are calculated recursively using a new type of harmonic oscillator coefficients of fractional parentage. These coefficients are the eigenvectors of the two-cycle class operator of the permutation group in the appropriate basis. The matrix elements of the class operators are evaluated by using a specific version of the harmonic oscillator brackets. The significance of this procedure to atomic, molecular and nuclear physics is pointed out. The presently proposed method is expected to be computationally very efticient.

International Journal of Quantum Chemistry, 2000

As is well known, multiparticle states formed by identical particles with an elementary spin σ ca... more As is well known, multiparticle states formed by identical particles with an elementary spin σ can only belong to irreducible representations of the symmetric group labeled by Young diagrams with at most 2σ+1 rows. Thus, spin ½ particles only give rise to multiparticle states labeled ...

Nuclear Instruments and Methods, Oct 1, 1970

Abstract A computational method for the separation of overlapping decaying gamma rays peaks is pr... more Abstract A computational method for the separation of overlapping decaying gamma rays peaks is proposed. The method, which takes advantage of the decay, overcomes some of the drawbacks of the standard peeling method.

American Journal of Physics, Mar 1, 1971

Journal of physics, Sep 21, 1995

Explicit expressions for the Temperley-Lieb-Martin algebras, i.e., the quotients of the Hecke alg... more Explicit expressions for the Temperley-Lieb-Martin algebras, i.e., the quotients of the Hecke algebra that admit only representations corresponding to Young diagrams with a given maximum number of columns (or rows), are obtained, making explicit use of the Hecke algebra representation theory. Similar techniques are used to construct the algebras whose representations do not contain rectangular subdiagrams of a given size.

European Journal of Combinatorics, Nov 1, 1995

The structure-constant [(P)]n" [(1)t'(2) t2"'" (n)t"].llflFl-*l(2:2-~z...(u)t.+l...fn)t.~, that c... more The structure-constant [(P)]n" [(1)t'(2) t2"'" (n)t"].llflFl-*l(2:2-~z...(u)t.+l...fn)t.~, that corresponds to the merging of the p cycles (1)k1(2) k2... (u-1)k"-L (~.~Z.: kl =p) into a single cycle of length u = ~.~:.~ ikt is derived. The result is used to evaluate the structure constant [(p)],k[tO)ttCr:z...CU_l)(p_l)+l)tv..i., where k is the minimal power of [(P)]n generating the class-sum [(1)t~(py 2... ((i-1)(p-1) + 1) I'.. .]. (i.e. ~-~'-1 (i-1)/, = k).

Physics Letters, Aug 1, 1975

A generalized perturbation theory, embedding the Brfflouin-Wigner and the Rayleigh-SchrOdinger pr... more A generalized perturbation theory, embedding the Brfflouin-Wigner and the Rayleigh-SchrOdinger procedures as special cases, is proposed. It enables the formulation of the expansion for the helium sequence so that the whole correction is incorporated in the second order term.

Proceedings of SPIE, Jan 8, 1987

ABSTRACT

Theoretica chimica acta, 1977

Journal of The Optical Society of America B-optical Physics, Oct 1, 1987

The multiphoton squeezed states defined in this paper are generalizations of the conventional coh... more The multiphoton squeezed states defined in this paper are generalizations of the conventional coherent (Glauber) and squeezed (Yuen) states previously discussed by many authors. We define multiphoton generalizations of the latter by a unified class of states that includes the Holstein-Primakoff realizations of SU(2) and SU(1, 1) as well as the standard harmonic oscillator coherent states (Weyl-Heisenberg group) and squeezed states in a general framework that allows also non-Hermitian realizations. We determine the squeezing properties of these states in a unified formalism and study numerically their dependence on the parameter classifying the states.

Chemical Physics, Feb 1, 1976

The feusibility of coherent multiphoron propn_=tion effects soch as two-photon self-induced trans... more The feusibility of coherent multiphoron propn_=tion effects soch as two-photon self-induced transparency is examined by calcularinp the coherent trnnsirion probabilities for a multiphoton process. For a two-photon excitation of a three-level system, periodic probability functions are obtained which increusc smoothly as the intermediate state approaches resonance with the radiation field. The results for a coherent multiphoton excitation of a multilevel system arc an extension of RPbi's "strong signal theory" Tar a one-photon excikrion of 3 two-level system.

Journal of Computational and Applied Mathematics, 2010

Position and momentum information measures are evaluated for the ground state of the relativistic... more Position and momentum information measures are evaluated for the ground state of the relativistic hydrogen-like atoms. Consequences of the fact that the radial momentum operator is not self-adjoint are explicitly studied, exhibiting fundamental shortcomings of the conventional uncertainty measures in terms of the radial position and momentum variances. The Shannon and Rényi entropies, the Fisher information measure, as well as several related information measures, are considered as viable alternatives. Detailed results on the onset of relativistic effects for low nuclear charges, and on the extreme relativistic limit, are presented. The relativistic position density decays exponentially at large r, but is singular at the origin. Correspondingly, the momentum density decays as an inverse power of p. Both features yield divergent Rényi entropies away from a finite vicinity of the Shannon entropy. While the position space information measures can be evaluated analytically for both the nonrelativistic and the relativistic hydrogen atom, this is not the case for the relativistic momentum space. Some of the results allow interesting insight into the significance of recently evaluated Dirac-Fock vs. Hartree-Fock complexity measures for many-electron neutral atoms.

Solid State Communications, Nov 1, 1984

Abstract The equation s =-σBσ[βσ|▿sH|]▿sH/|▿sH| is shown to be the magnetization equation for the... more Abstract The equation s =-σBσ[βσ|▿sH|]▿sH/|▿sH| is shown to be the magnetization equation for the arbitrary anisotropic infinite-range spin Hamiltonian, H(s). The known equations for the general anisotropic-Heisenberg, and for arbitrary isotropic and axially symmetric spin Hamiltonians are shown to be special cases of this equation.

Physical review, Nov 1, 1983

A generalized mean-field theory, incorporating the "classical" mean-field theories but having a r... more A generalized mean-field theory, incorporating the "classical" mean-field theories but having a richer structure, is formulated. Thermodynamic consistency is established for this generalized theory. As an application of the formalism, a treatment of coupled orientational and translational order-disorder transitions is presented, resulting in self-consistency equations requiring no numerical integration but exhibiting the essential features of the isotropic-nematic-smectic transitions in liquid crystals.

arXiv (Cornell University), Sep 11, 1995

The fundumental invariant of the Hecke algebra Hn(q) is the q-deformed class-sum of transposition... more The fundumental invariant of the Hecke algebra Hn(q) is the q-deformed class-sum of transpositions of the symmetric group Sn. Irreducible representations of Hn(q), for generic q, are shown to be completely characterized by the corresponding eigenvalues of Cn alone. For Sn more and more invariants are necessary as n inereases. It is pointed out that the q-deformed classical quadratic Casimir of SU (N) plays an analogous role. It is indicated why and how this should be a general phenomenon associated with q-deformation of classical algebras. Apart from this remarkable conceptual aspect Cn can provide powerful and elegant techniques for computations. This is illustrated by using the sequence C 2 , C 3 , • • • , Cn to compute the characters of Hn(q).

arXiv (Cornell University), Feb 27, 1995

The traces of the Murphy operators of the Hecke algebra H n (q), and of products of sets of Murph... more The traces of the Murphy operators of the Hecke algebra H n (q), and of products of sets of Murphy operators with non-consecutive indices, can be evaluated by a straightforward recursive procedure. These traces are shown to determine all the reduced traces in this algebra, which, in turn, determine all other traces. To illustrate the procedure we obtain the set of reduced traces for H 7 (q)-the lowest order Hecke algebra whose character table has not hitherto been reported. This is preceded by the presentation of an explicit algorithm for the reduction of the trace of an arbitrary element of the Hecke algebra into a linear combination of traces of elements consisting of appropriately defined disjoint cycles; and of a proof, presented in order to make the present article reasonably self-contained, that a reduced trace depends only on the set of lengths of the disjoint cycles that it consists of.

Physics Letters, Sep 1, 1976

Abstract Two electron radiative transition probabilities from the L and M shells into a doubly va... more Abstract Two electron radiative transition probabilities from the L and M shells into a doubly vacant K shell are computed non-relativistically, introducing angular correlation in the simplest conceivable way. The relative cross-sections for the different possible processes are in agreement with recently reported experimental values for Fe and Ni and predict interesting changes in the spectra of heavier systems.

Annals of Physics, Oct 1, 2018

The relative ordering of energy levels is investigated for bound two-electron systems with potent... more The relative ordering of energy levels is investigated for bound two-electron systems with potentials of the form V (r 1 , r 2 , r 12) = Z v(r 1) + v(r 2) − v(r 12). Given the two one-body binding potentials v (1) (r) and v (2) (r), it is argued that if f(r) ≡ v (1) (r) − v (2) (r) is positive and monotonically decreasing upon increasing r then the corresponding eigenvalues of the two-electron Hamiltonians *Manuscript Click here to view linked References where λ is the screening parameter. For each of the λ-dependent potentials we compare one-and two-electron spectra corresponding to distinct values of λ. This is followed by pairwise comparison of distinct potentials.

Physical review, Feb 9, 2018

The one-and two-electron Yukawa atoms, also referred to as the Debye-Hückel or screened Coulomb a... more The one-and two-electron Yukawa atoms, also referred to as the Debye-Hückel or screened Coulomb atoms, have been topics of considerable interest both for intrinsic reasons and because of their relevance to terrestrial and astrophysical plasmas. At sufficiently high screening the one-electron Yukawa atom ceases to be bound. Some calculations appeared to suggest that as the screening increases in the ground state of the two-electron Yukawa atom (in which both the one-particle attraction and the interparticle repulsion are screened) the two electrons are detached simultaneously, at the same screening constant at which the one-electron atom becomes unbound. Our results rule this scenario out, offering an alternative that is not less interesting. In particular, it is found that for Z < 1 a mild amount of screening actually increases the binding energy of the second electron. At the nuclear charge Z c ≈ 0.911028. . ., at which the bare Coulomb two-electron atom becomes unbound, and even over a range of lower nuclear charges, an appropriate amount of screening gives rise to a bound two-electron system.

Zeitschrift Für Physik A Hadrons And Nuclei, Sep 1, 1994

An algorithm is presented for the construction of single-and multi-cluster harmonic oscillator wa... more An algorithm is presented for the construction of single-and multi-cluster harmonic oscillator wave functions that are coupled into well-defined irreducible representa

Springer eBooks, Apr 7, 2008

An algorithm for the construction of non-spurious harmonic oscillator wave functions with arbitra... more An algorithm for the construction of non-spurious harmonic oscillator wave functions with arbitrary permutational symmetry is formulated by adapting a procedure recently developed for building general single shell wave functions. The harmonic oscillator functions, expressed in Jacobi coordinates, are calculated recursively using a new type of harmonic oscillator coefficients of fractional parentage. These coefficients are the eigenvectors of the two-cycle class operator of the permutation group in the appropriate basis. The matrix elements of the class operators are evaluated by using a specific version of the harmonic oscillator brackets. The significance of this procedure to atomic, molecular and nuclear physics is pointed out. The presently proposed method is expected to be computationally very efticient.

International Journal of Quantum Chemistry, 2000

As is well known, multiparticle states formed by identical particles with an elementary spin σ ca... more As is well known, multiparticle states formed by identical particles with an elementary spin σ can only belong to irreducible representations of the symmetric group labeled by Young diagrams with at most 2σ+1 rows. Thus, spin ½ particles only give rise to multiparticle states labeled ...

Nuclear Instruments and Methods, Oct 1, 1970

Abstract A computational method for the separation of overlapping decaying gamma rays peaks is pr... more Abstract A computational method for the separation of overlapping decaying gamma rays peaks is proposed. The method, which takes advantage of the decay, overcomes some of the drawbacks of the standard peeling method.

American Journal of Physics, Mar 1, 1971

Journal of physics, Sep 21, 1995

Explicit expressions for the Temperley-Lieb-Martin algebras, i.e., the quotients of the Hecke alg... more Explicit expressions for the Temperley-Lieb-Martin algebras, i.e., the quotients of the Hecke algebra that admit only representations corresponding to Young diagrams with a given maximum number of columns (or rows), are obtained, making explicit use of the Hecke algebra representation theory. Similar techniques are used to construct the algebras whose representations do not contain rectangular subdiagrams of a given size.

European Journal of Combinatorics, Nov 1, 1995

The structure-constant [(P)]n" [(1)t'(2) t2"'" (n)t"].llflFl-*l(2:2-~z...(u)t.+l...fn)t.~, that c... more The structure-constant [(P)]n" [(1)t'(2) t2"'" (n)t"].llflFl-*l(2:2-~z...(u)t.+l...fn)t.~, that corresponds to the merging of the p cycles (1)k1(2) k2... (u-1)k"-L (~.~Z.: kl =p) into a single cycle of length u = ~.~:.~ ikt is derived. The result is used to evaluate the structure constant [(p)],k[tO)ttCr:z...CU_l)(p_l)+l)tv..i., where k is the minimal power of [(P)]n generating the class-sum [(1)t~(py 2... ((i-1)(p-1) + 1) I'.. .]. (i.e. ~-~'-1 (i-1)/, = k).

Physics Letters, Aug 1, 1975

A generalized perturbation theory, embedding the Brfflouin-Wigner and the Rayleigh-SchrOdinger pr... more A generalized perturbation theory, embedding the Brfflouin-Wigner and the Rayleigh-SchrOdinger procedures as special cases, is proposed. It enables the formulation of the expansion for the helium sequence so that the whole correction is incorporated in the second order term.

Proceedings of SPIE, Jan 8, 1987

ABSTRACT

Theoretica chimica acta, 1977

Journal of The Optical Society of America B-optical Physics, Oct 1, 1987

The multiphoton squeezed states defined in this paper are generalizations of the conventional coh... more The multiphoton squeezed states defined in this paper are generalizations of the conventional coherent (Glauber) and squeezed (Yuen) states previously discussed by many authors. We define multiphoton generalizations of the latter by a unified class of states that includes the Holstein-Primakoff realizations of SU(2) and SU(1, 1) as well as the standard harmonic oscillator coherent states (Weyl-Heisenberg group) and squeezed states in a general framework that allows also non-Hermitian realizations. We determine the squeezing properties of these states in a unified formalism and study numerically their dependence on the parameter classifying the states.

Chemical Physics, Feb 1, 1976

The feusibility of coherent multiphoron propn_=tion effects soch as two-photon self-induced trans... more The feusibility of coherent multiphoron propn_=tion effects soch as two-photon self-induced transparency is examined by calcularinp the coherent trnnsirion probabilities for a multiphoton process. For a two-photon excitation of a three-level system, periodic probability functions are obtained which increusc smoothly as the intermediate state approaches resonance with the radiation field. The results for a coherent multiphoton excitation of a multilevel system arc an extension of RPbi's "strong signal theory" Tar a one-photon excikrion of 3 two-level system.