scholarly journals Coquaternionic Quantum Dynamics for Two-level Systems

10.14311/1394 ◽  
2011 ◽  
Vol 51 (4) ◽  
Author(s):  
D. C. Brody ◽  
E. M. Graefe
Keyword(s):  
State Space ◽  
Time Evolution ◽  
Quantum Dynamics ◽  
Complex Conjugate ◽  
Conjugate Pair ◽  
Open Orbit ◽  
Energy Eigenvalues ◽  
Complex Conjugate Pair ◽  
Two Level Systems ◽  
Different Characteristics

The dynamical aspects of a spin-1/2 particle in Hermitian coquaternionic quantum theory are investigated. It is shown that the time evolution exhibits three different characteristics, depending on the values of the parameters of the Hamiltonian. When energy eigenvalues are real, the evolution is either isomorphic to that of a complex Hermitian theory on a spherical state space, or else it remains unitary along an open orbit on a hyperbolic state space. When energy eigenvalues form a complex conjugate pair, the orbit of the time evolution closes again even though the state space is hyperbolic.

Nonlinear evolution of the elliptical instability: an example of inertial wave breakdown

1999 ◽  
Vol 396 ◽  
pp. 73-108 ◽  
Author(s):  
D. M. MASON ◽  
R. R. KERSWELL
Keyword(s):  
Finite Amplitude ◽  
Complex Conjugate ◽  
Small Scale ◽  
Conjugate Pair ◽  
Weakly Nonlinear ◽  
Elliptical Instability ◽  
Elliptical Flow ◽  
Complex Conjugate Pair ◽  
Generic Instability ◽  
Secondary Instabilities

A direct numerical simulation is presented of an elliptical instability observed in the laboratory within an elliptically distorted, rapidly rotating, fluid-filled cylinder (Malkus 1989). Generically, the instability manifests itself as the pairwise resonance of two different inertial modes with the underlying elliptical flow. We study in detail the simplest ‘subharmonic’ form of the instability where the waves are a complex conjugate pair and which at weakly supercritical elliptical distortion should ultimately saturate at some finite amplitude (Waleffe 1989; Kerswell 1992). Such states have yet to be experimentally identified since the flow invariably breaks down to small-scale disorder. Evidence is presented here to support the argument that such weakly nonlinear states are never seen because they are either unstable to secondary instabilities at observable amplitudes or neighbouring competitor elliptical instabilities grow to ultimately disrupt them. The former scenario confirms earlier work (Kerswell 1999) which highlights the generic instability of inertial waves even at very small amplitudes. The latter represents a first numerical demonstration of two competing elliptical instabilities co-existing in a bounded system.

scholarly journals A New Signal Representation Using Complex Conjugate Pair Sums

2019 ◽  
Vol 26 (2) ◽  
pp. 252-256 ◽  
Author(s):  
Basheeruddin Shah Shaik ◽  
Vijay Kumar Chakka ◽  
Arikatla Satyanarayana Reddy
Keyword(s):  
Complex Conjugate ◽  
Signal Representation ◽  
Conjugate Pair ◽  
Complex Conjugate Pair

On the onset of instability by oscillatory modes in hydromagnetic Couette flow

1967 ◽  
Vol 301 (1467) ◽  
pp. 451-471 ◽  
Keyword(s):  
Couette Flow ◽  
Hartmann Number ◽  
Outer Wall ◽  
Angular Speed ◽  
Complex Conjugate ◽  
Neutral Stability ◽  
Narrow Gap ◽  
Conjugate Pair ◽  
Average Value ◽  
Complex Conjugate Pair

The transition of the onset of instability from stationary modes to oscillatory modes for an incompressible, conducting Couette flow between two coaxial, perfectly conducting, non-permeable, rotating cylinders under the influence of an axially applied magnetic field is considered. Results for three cases are reported. These pertain to flow between (1) a rotating inner wall with a stationary outer wall, (2) counterrotating walls, and (3) corotating walls. It is found that, for high values of the Hartmann number, there may exist some range of convective wavenumbers for which neither of the two lowest modes of axisymmetrical disturbances will become stationary. Within this range, the neutral stability curve is determined by a complex-conjugate pair of oscillatory axisymmetrical modes of equal stability. The oscillatory modes may, in fact, become more critical than the stationary modes. It is demonstrated that the approximation of replacing the angular speed by its average value, combined with the assumption of a narrow gap between the cylindrical walls, eliminates the oscillatory axisymmetrical modes.

Poles and Zeros of a Bandpass Filter Obtained by Transformation of a Lowpass Filter

1979 ◽  
Vol 16 (1) ◽  
pp. 39-41
Author(s):  
S. C. Duttaroy
Keyword(s):  
Bandpass Filter ◽  
Damping Ratio ◽  
Complex Conjugate ◽  
Lowpass Filter ◽  
Transformed Roots ◽  
Conjugate Pair ◽  
Complex Conjugate Pair ◽  
Poles And Zeros ◽  
Explicit Expressions

The standard lowpass to bandpass transformation is shown to transform a complex conjugate pair of roots (poles or zeros) to two such pairs having equal damping ratio. Explicit expressions are given for the locations of the transformed roots; these should be useful in active R.C. bandpass realizations.

scholarly journals On Complex Conjugate Pair Sums and Complex Conjugate Subspaces

2019 ◽  
Vol 26 (9) ◽  
pp. 1403-1407
Author(s):  
Shaik Basheeruddin Shah ◽  
Vijay Kumar Chakka ◽  
Arikatla Satyanarayana Reddy
Keyword(s):  
Complex Conjugate ◽  
Conjugate Pair ◽  
Complex Conjugate Pair

scholarly journals PT symmetry as a necessary and sufficient condition for unitary time evolution

2013 ◽  
Vol 371 (1989) ◽  
pp. 20120060 ◽  
Author(s):  
Philip D. Mannheim
Keyword(s):  
Time Evolution ◽  
Path Integral ◽  
Secular Equation ◽  
Hermitian Operator ◽  
Complex Conjugate ◽  
Energy Eigenvalues ◽  
Integral Measure ◽  
Klein Gordon ◽  
Necessary And Sufficient ◽  
Root Operator

While Hermiticity of a time-independent Hamiltonian leads to unitary time evolution, in and of itself, the requirement of Hermiticity is only sufficient for unitary time evolution. In this paper, we provide conditions that are both necessary and sufficient. We show that symmetry of a time-independent Hamiltonian, or equivalently, reality of the secular equation that determines its eigenvalues, is both necessary and sufficient for unitary time evolution. For any -symmetric Hamiltonian H , there always exists an operator V that relates H to its Hermitian adjoint according to V HV −1 = H † . When the energy spectrum of H is complete, Hilbert space norms 〈 ψ 1 | V | ψ 2 〉 constructed with this V are always preserved in time. With the energy eigenvalues of a real secular equation being either real or appearing in complex conjugate pairs, we thus establish the unitarity of time evolution in both cases. We also establish the unitarity of time evolution for Hamiltonians whose energy spectra are not complete. We show that when the energy eigenvalues of a Hamiltonian are real and complete, the operator V is a positive Hermitian operator, which has an associated square root operator that can be used to bring the Hamiltonian to a Hermitian form. We show that systems with -symmetric Hamiltonians obey causality. We note that Hermitian theories are ordinarily associated with a path integral quantization prescription in which the path integral measure is real, while in contrast, non-Hermitian but -symmetric theories are ordinarily associated with path integrals in which the measure needs to be complex, but in which the Euclidean time continuation of the path integral is nonetheless real. Just as the second-order Klein–Gordon theory is stabilized against transitions to negative frequencies because its Hamiltonian is positive-definite, through symmetry, the fourth-order derivative Pais–Uhlenbeck theory can equally be stabilized.

QUARK-GLUON VERTEX AND STRUCTURE OF MESONS AND DIQUARKS

1994 ◽  
Vol 09 (38) ◽  
pp. 3551-3563 ◽  
Author(s):  
S.J. STAINSBY ◽  
R.T. CAHILL
Keyword(s):  
Form Factors ◽  
Quark Propagator ◽  
The Other ◽  
Analytic Structure ◽  
Complex Conjugate ◽  
Branch Points ◽  
Conjugate Pair ◽  
Gluon Vertex ◽  
Scalar Diquark ◽  
Complex Conjugate Pair

Two quark propagators with different analytic structure are employed in Bethe-Salpeter type equations for the pion and scalar diquark form factors. One of the quark propagators has been calculated with the inclusion of a trivial (bare) quark-gluon vertex and, as a consequence, contains a complex conjugate pair of logarithmic branch points. The other quark propagator is obtained using a non-trivial (dressed) vertex ansatz and is entire, with an essential singularity at infinity. The effects of these different quark propagators on the BSE solutions are compared.

Density matrix and purity evolution in dissipative two-level systems: II. Relaxation

2021 ◽  
Author(s):  
Sambarta Chatterjee ◽  
Nancy Makri
Keyword(s):  
Density Matrix ◽  
Time Evolution ◽  
Reduced Density Matrix ◽  
Level System ◽  
Two Level Systems ◽  
Reduced Density ◽  
Harmonic Bath

We investigate the time evolution of the reduced density matrix (RDM) and its purity in the dynamics of a two-level system coupled to a dissipative harmonic bath, when the system is initially placed in one of its eigenstates.

scholarly journals Hidden Nambu mechanics II: Quantum/semiclassical dynamics

2019 ◽  
Vol 2019 (12) ◽  
Author(s):  
Atsushi Horikoshi
Keyword(s):  
Time Evolution ◽  
Quantum Dynamics ◽  
Degrees Of Freedom ◽  
Hamiltonian Dynamics ◽  
Extended Phase ◽  
Nambu Mechanics ◽  
Mechanical Structure ◽  
Wave Packet Dynamics ◽  
Semiclassical Dynamics ◽  
Metastable Potential

Abstract Nambu mechanics is a generalized Hamiltonian dynamics characterized by an extended phase space and multiple Hamiltonians. In a previous paper [Prog. Theor. Exp. Phys. 2013, 073A01 (2013)] we revealed that the Nambu mechanical structure is hidden in Hamiltonian dynamics, that is, the classical time evolution of variables including redundant degrees of freedom can be formulated as Nambu mechanics. In the present paper we show that the Nambu mechanical structure is also hidden in some quantum or semiclassical dynamics, that is, in some cases the quantum or semiclassical time evolution of expectation values of quantum mechanical operators, including composite operators, can be formulated as Nambu mechanics. We present a procedure to find hidden Nambu structures in quantum/semiclassical systems of one degree of freedom, and give two examples: the exact quantum dynamics of a harmonic oscillator, and semiclassical wave packet dynamics. Our formalism can be extended to many-degrees-of-freedom systems; however, there is a serious difficulty in this case due to interactions between degrees of freedom. To illustrate our formalism we present two sets of numerical results on semiclassical dynamics: from a one-dimensional metastable potential model and a simplified Henon–Heiles model of two interacting oscillators.

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