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Nimrod Moiseyev - One of the best experts on this subject based on the ideXlab platform.

  • complex density probability in non hermitian quantum mechanics interpretation and a formula for resonant tunneling probability amplitude
    Physical Review A, 2001
    Co-Authors: Hadas Barkay, Nimrod Moiseyev
    Abstract:

    Non-Hermitian quantum mechanics has been developed to study the dynamics of nuclear, atomic, and molecular sys- tems that can be prepared in metastable finite-lifetime states ~so-called resonance states !@ 1,2# for the study of delocaliza- tion phenomenon which is relevant in different fields, such as bacteria populations, vortex spinning in superconductors, and for the study of the stability conditions of the solutions of hydrodynamical problems @3,4#. In cases where resonance phenomena are studied, the Hamiltonians are non-Hermitian due to the specific boundary conditions that are imposed on the solutions of the Schrodinger equation. The asymptotic solutions should be exponential divergent wave functions ~known as Siegert functions!. In other cases @3,4# the Hamil- tonian is non-Hermitian due to the inclusion of a non- Hermitian operator such as a vii, while the boundary con- ditions of ''conventional'' Hermitian quantum mechanics are kept. There is a way to unify the two types of non-Hermitian quantum problems. Upon complex scaling, i.e., x !x exp(iu), the exponentially divergent metastable reso- nance eigenfunctions become square integrable and thereby become part of the Generalized Hilbert Space @2#. Therefore, the resonances are the eigenfunctions of a complex scaled non-Hermitian Hamiltonian with the same boundary condi- tions as in the conventional Hermitian quantum mechanics. Let us denote the complex non-Hermitian Hamiltonian by H ˆ . A matrix representation of H ˆ ~denoted by H) is obtained by using a finite number of orthogonal functions as a basis set. Since the usual boundary conditions are applied the basis functions can be square integrable or periodic functions. The right and left eigenfunctions of H ˆ, which are defined asC j R and C j , are associated with the right and left eigenvectors of H:

  • State-of-State Transition Probabilities and Control of Laser-Induced Dynamical Processes by The (T, T’) Method
    Multiparticle Quantum Scattering With Applications to Nuclear Atomic and Molecular Physics, 1997
    Co-Authors: Nimrod Moiseyev
    Abstract:

    The time evolution operator for general time-dependent (not necessarily time-periodic) Hamiltonians is given by the (t, t’) method as, \(\hat{u}(x,t\prime ,t) = \exp \{ - \tfrac{i}{\hbar }({{\hat{p}}_{{t\prime }}} + H(x,t\prime )(t - {{t}_{0}}))\}\), where \({{\hat{p}}_{{t\prime }}} \equiv (\hbar /i)\partial /\partial t\prime\) and t’ serves as an additional coordinate. Therefore the inner product is defined in the Generalized Hilbert Space of x and t’. The physical solution is given by ψ(x,t’, t)’ t’-t where ψ(x, t’,t) = u(x, t', t)Ψo.

  • the solution of the time dependent schrodinger equation by the t t method theory computational algorithm and applications
    Journal of Chemical Physics, 1993
    Co-Authors: Uri Peskin, Nimrod Moiseyev
    Abstract:

    A new powerful computational method is introduced for the solution of the time dependent Schrodinger equation with time‐dependent Hamiltonians (not necessarily time‐periodic). The method is based on the use of the Floquet‐type operator in an extended Hilbert Space, which was introduced by H. Sambe [Phys. Rev. A 7, 2203 (1973)] for time periodic Hamiltonians, and was extended by J. Howland [Math Ann. 207, 315 (1974)] for general time dependent Hamiltonians. The new proposed computational algorithm avoids the need to introduce the time ordering operator when the time‐dependent Schrodinger equation is integrated. Therefore it enables one to obtain the solution of the time‐dependent Schrodinger equation by using computational techniques that were originally developed for cases where the Hamiltonian is time independent. A time‐independent expression for state‐to‐state transition probabilities is derived by using the analytical time dependence of the time evolution operator in the Generalized Hilbert Space. Illustrative numerical examples for complex scaled time periodic model Hamiltonians are given.

  • The solution of the time‐dependent Schrödinger equation by the (t,t’) method: Theory, computational algorithm and applications
    The Journal of Chemical Physics, 1993
    Co-Authors: Uri Peskin, Nimrod Moiseyev
    Abstract:

    A new powerful computational method is introduced for the solution of the time dependent Schrodinger equation with time‐dependent Hamiltonians (not necessarily time‐periodic). The method is based on the use of the Floquet‐type operator in an extended Hilbert Space, which was introduced by H. Sambe [Phys. Rev. A 7, 2203 (1973)] for time periodic Hamiltonians, and was extended by J. Howland [Math Ann. 207, 315 (1974)] for general time dependent Hamiltonians. The new proposed computational algorithm avoids the need to introduce the time ordering operator when the time‐dependent Schrodinger equation is integrated. Therefore it enables one to obtain the solution of the time‐dependent Schrodinger equation by using computational techniques that were originally developed for cases where the Hamiltonian is time independent. A time‐independent expression for state‐to‐state transition probabilities is derived by using the analytical time dependence of the time evolution operator in the Generalized Hilbert Space. Illustrative numerical examples for complex scaled time periodic model Hamiltonians are given.

Uri Peskin - One of the best experts on this subject based on the ideXlab platform.

  • the solution of the time dependent schrodinger equation by the t t method theory computational algorithm and applications
    Journal of Chemical Physics, 1993
    Co-Authors: Uri Peskin, Nimrod Moiseyev
    Abstract:

    A new powerful computational method is introduced for the solution of the time dependent Schrodinger equation with time‐dependent Hamiltonians (not necessarily time‐periodic). The method is based on the use of the Floquet‐type operator in an extended Hilbert Space, which was introduced by H. Sambe [Phys. Rev. A 7, 2203 (1973)] for time periodic Hamiltonians, and was extended by J. Howland [Math Ann. 207, 315 (1974)] for general time dependent Hamiltonians. The new proposed computational algorithm avoids the need to introduce the time ordering operator when the time‐dependent Schrodinger equation is integrated. Therefore it enables one to obtain the solution of the time‐dependent Schrodinger equation by using computational techniques that were originally developed for cases where the Hamiltonian is time independent. A time‐independent expression for state‐to‐state transition probabilities is derived by using the analytical time dependence of the time evolution operator in the Generalized Hilbert Space. Illustrative numerical examples for complex scaled time periodic model Hamiltonians are given.

  • The solution of the time‐dependent Schrödinger equation by the (t,t’) method: Theory, computational algorithm and applications
    The Journal of Chemical Physics, 1993
    Co-Authors: Uri Peskin, Nimrod Moiseyev
    Abstract:

    A new powerful computational method is introduced for the solution of the time dependent Schrodinger equation with time‐dependent Hamiltonians (not necessarily time‐periodic). The method is based on the use of the Floquet‐type operator in an extended Hilbert Space, which was introduced by H. Sambe [Phys. Rev. A 7, 2203 (1973)] for time periodic Hamiltonians, and was extended by J. Howland [Math Ann. 207, 315 (1974)] for general time dependent Hamiltonians. The new proposed computational algorithm avoids the need to introduce the time ordering operator when the time‐dependent Schrodinger equation is integrated. Therefore it enables one to obtain the solution of the time‐dependent Schrodinger equation by using computational techniques that were originally developed for cases where the Hamiltonian is time independent. A time‐independent expression for state‐to‐state transition probabilities is derived by using the analytical time dependence of the time evolution operator in the Generalized Hilbert Space. Illustrative numerical examples for complex scaled time periodic model Hamiltonians are given.

Hadas Barkay - One of the best experts on this subject based on the ideXlab platform.

  • complex density probability in non hermitian quantum mechanics interpretation and a formula for resonant tunneling probability amplitude
    Physical Review A, 2001
    Co-Authors: Hadas Barkay, Nimrod Moiseyev
    Abstract:

    Non-Hermitian quantum mechanics has been developed to study the dynamics of nuclear, atomic, and molecular sys- tems that can be prepared in metastable finite-lifetime states ~so-called resonance states !@ 1,2# for the study of delocaliza- tion phenomenon which is relevant in different fields, such as bacteria populations, vortex spinning in superconductors, and for the study of the stability conditions of the solutions of hydrodynamical problems @3,4#. In cases where resonance phenomena are studied, the Hamiltonians are non-Hermitian due to the specific boundary conditions that are imposed on the solutions of the Schrodinger equation. The asymptotic solutions should be exponential divergent wave functions ~known as Siegert functions!. In other cases @3,4# the Hamil- tonian is non-Hermitian due to the inclusion of a non- Hermitian operator such as a vii, while the boundary con- ditions of ''conventional'' Hermitian quantum mechanics are kept. There is a way to unify the two types of non-Hermitian quantum problems. Upon complex scaling, i.e., x !x exp(iu), the exponentially divergent metastable reso- nance eigenfunctions become square integrable and thereby become part of the Generalized Hilbert Space @2#. Therefore, the resonances are the eigenfunctions of a complex scaled non-Hermitian Hamiltonian with the same boundary condi- tions as in the conventional Hermitian quantum mechanics. Let us denote the complex non-Hermitian Hamiltonian by H ˆ . A matrix representation of H ˆ ~denoted by H) is obtained by using a finite number of orthogonal functions as a basis set. Since the usual boundary conditions are applied the basis functions can be square integrable or periodic functions. The right and left eigenfunctions of H ˆ, which are defined asC j R and C j , are associated with the right and left eigenvectors of H:

M. Roushan - One of the best experts on this subject based on the ideXlab platform.

Emile Grgin - One of the best experts on this subject based on the ideXlab platform.

  • Relativistic ring extension of the field of complex numbers
    Physics Letters B, 1998
    Co-Authors: Emile Grgin
    Abstract:

    Abstract Over the past sixty years many attempts have been made at modifying quantum mechanics by extending the field of complex numbers, but none has led to a structural unification of quantum mechanics with relativity. A solution nevertheless exists. It is based on a ring, new to mathematics, which we call the quantal ring. It is manifestly Lorentz covariant, CPT invariant, contains the field of complex numbers as a substructure, and is free of properties that have no physical interpretation. While the systematic derivation of this ring from a comparative abstract analysis of classical and quantum mechanics, as well as the proof of its uniqueness and the construction of a Generalized Hilbert Space over it, are being finalized for publication, we derive it in this note by a simple alternative construction.