We apply an approximate form of the exact one-particle-irreducible renormalization-group generator to the calculation of the equation of state. Several approaches are explored. (i) Global nonlinear trajectories for the Hamiltonian parameters are exploited to give nonlinear crossover equations of state. Logarithms of the reduced temperature $t\ensuremath{\equiv}\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$ are automatically exponentiated to give power-law behavior. (ii) The form and asymptotic properties of the equation of state are described for those systems whose complete nonlinear trajectories cannot be explicitly obtained. (iii) Nonlinear trajectories for the irreducible Green's functions are solved to give a fully exponentiated equation of state containing no logarithmic terms. (iv) Simple critical (noncrossover) equations of state are obtained by iteration of the generator with linear and quasilinear trajectories. (v) Operator nonlinear equations are solved to give the crossover equation of state for an arbitrary order $\ensuremath{\theta}$ critical system, both in an $\ensuremath{\epsilon}$ expansion and at the borderline dimension. (vi) Combination of the Green's functions are formed into Green's eigenfunctions or operators and their nonlinear trajectories used to calculate fully exponentiated equations of state for order $\ensuremath{\theta}$ Ising systems. Systems considered include the usual Wilson-Fisher Hamiltonian [methods (i)-(iv)], the Sak-model compressible ferromagnet [(i)-(ii)], the $\mathrm{nm}$ -hypercubical model (of which the dilute quenched random ferromagnet forms the $n\ensuremath{\rightarrow}0$ limit) [(ii)], and the isotropic $n$ -component order $\ensuremath{\theta}$ ferromagnet [(iv)-(vi)]. The tricritical ($\ensuremath{\theta}=3$) case is given explicitly as an example of the general formalism of methods (iv)-(vi). Throughout, we use a bare critical propagator which is an arbitrary generalized homogeneous function of the components of the wave vector. This allows us to simultaneously describe ordinary critical systems and anisotropic Lifschitz points as well as certain structural and spin-reorientation phase transitions.We apply an approximate form of the exact one-particle-irreducible renormalization-group generator to the calculation of the equation of state. Several approaches are explored. (i) Global nonlinear trajectories for the Hamiltonian parameters are exploited to give nonlinear crossover equations of state. Logarithms of the reduced temperature $t\ensuremath{\equiv}\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$ are automatically exponentiated to give power-law behavior. (ii) The form and asymptotic properties of the equation of state are described for those systems whose complete nonlinear trajectories cannot be explicitly obtained. (iii) Nonlinear trajectories for the irreducible Green's functions are solved to give a fully exponentiated equation of state containing no logarithmic terms. (iv) Simple critical (noncrossover) equations of state are obtained by iteration of the generator with linear and quasilinear trajectories. (v) Operator nonlinear equations are solved to give the crossover equation of state for an arbitrary order $\ensuremath{\theta}$ critical system, both in an $\ensuremath{\epsilon}$ expansion and at the borderline dimension. (vi) Combination of the Green's functions are formed into Green's eigenfunctions or operators and their nonlinear trajectories used to calculate fully exponentiated equations of state for order $\ensuremath{\theta}$ Ising systems. Systems considered include the usual Wilson-Fisher Hamiltonian [methods (i)-(iv)], the Sak-model compressible ferromagnet [(i)-(ii)], the $\mathrm{nm}$-hypercubical model (of which the dilute quenched random ferromagnet forms the $n\ensuremath{\rightarrow}0$ limit) [(ii)], and the isotropic $n$-component order $\ensuremath{\theta}$ ferromagnet [(iv)-(vi)]. The tricritical ($\ensuremath{\theta}=3$) case is given explicitly as an example of the general formalism of methods (iv)-(vi). Throughout, we use a bare critical propagator which is an arbitrary generalized homogeneous function of the components of the wave vector. This allows us to simultaneously describe ordinary critical systems and anisotropic Lifschitz points as well as certain structural and spin-reorientation phase transitions.

Physical Review AJun 1, 197835 citations

Quantum Trajectories III

Lecture notes in physics. M, MonographsFeb 17, 20091 citation

Quantum Trajectories II

Lecture notes in physics. M, MonographsFeb 17, 20091 citation

Comment on “Laser-assisted formation of antihydrogen”

James S. Cohen

In a recent paper, Whitehead, McCann, and Shimamura [Phys. Rev. A 64, 023401 (2001)] presented classical-trajectory Monte Carlo (CTMC) calculations of the effects of laser fields on the cross section for antihydrogen formation in collisions of antiprotons with positronium. They also use the same method to calculate the photoionization rate of isolated positronium in a strong field. The purpose of this Comment is to point out (i) the criteria for validity of the quasiclassical approximation are not satisfied for all the conditions used; (ii) some of their numerical results do not agree with the values obtained with a corroborated code; (iii) the ambiguity of a cross section when the target can ionize even before the projectile arrives; and (iv) the essential effect of quantum-mechanical tunneling on the strong-field ionization rate. A recently published method allowing tunneling in a CTMC calculation [J. S. Cohen, Phys. Rev. A 64, 043412 (2001)] is used for (iv). It is concluded that the laser effect on formation of antihydrogen may be much greater than that found by Whitehead et al., but the pulse must be better characterized for quantitative predictions.In a recent paper, Whitehead, McCann, and Shimamura [Phys. Rev. A 64, 023401 (2001)] presented classical-trajectory Monte Carlo (CTMC) calculations of the effects of laser fields on the cross section for antihydrogen formation in collisions of antiprotons with positronium. They also use the same method to calculate the photoionization rate of isolated positronium in a strong field. The purpose of this Comment is to point out (i) the criteria for validity of the quasiclassical approximation are not satisfied for all the conditions used; (ii) some of their numerical results do not agree with the values obtained with a corroborated code; (iii) the ambiguity of a cross section when the target can ionize even before the projectile arrives; and (iv) the essential effect of quantum-mechanical tunneling on the strong-field ionization rate. A recently published method allowing tunneling in a CTMC calculation [J. S. Cohen, Phys. Rev. A 64, 043412 (2001)] is used for (iv). It is concluded that the laser effect on formation of antihydrogen may be much greater than that found by Whitehead et al., but the pulse must be better characterized for quantitative predictions.

Physical Review AJan 31, 20036 citations

Reconciling Semiclassical and Bohmian Mechanics: IV. Multisurface Dynamics

Bill Poirier, Gérard Parlant

In previous articles (J. Chem. Phys. 2004, 121, 4501; 2006, 124, 034115; 2006, 124, 034116) a bipolar counter-propagating wave decomposition, Psi = Psi+ + Psi-, was presented for stationary states Psi of the one-dimensional Schrödinger equation, such that the components Psi+/- approach their semiclassical WKB analogs in the large action limit. The corresponding bipolar quantum trajectories are classical-like and well-behaved, even when Psi has many nodes or is wildly oscillatory. In this paper, the method is generalized for multisurface scattering applications and applied to several benchmark problems. A natural connection is established between intersurface transitions and (+ <--> -) transitions.In previous articles (J. Chem. Phys. 2004, 121, 4501; 2006, 124, 034115; 2006, 124, 034116) a bipolar counter-propagating wave decomposition, Psi = Psi+ + Psi-, was presented for stationary states Psi of the one-dimensional Schrödinger equation, such that the components Psi+/- approach their semiclassical WKB analogs in the large action limit. The corresponding bipolar quantum trajectories are classical-like and well-behaved, even when Psi has many nodes or is wildly oscillatory. In this paper, the method is generalized for multisurface scattering applications and applied to several benchmark problems. A natural connection is established between intersurface transitions and (+ <--> -) transitions.

The Journal of Physical Chemistry ASep 22, 200751 citations

Quantum Trajectories I

Lecture notes in physics. M, MonographsJan 1, 19931 citation

\mathrm{nm}

-hypercubical model (of which the dilute quenched random ferromagnet forms the $n\ensuremath{\rightarrow}0$ limit) [(ii)], and the isotropic

n

-component order $\ensuremath{\theta}$ ferromagnet [(iv)-(vi)]. The tricritical ($\ensuremath{\theta}=3$) case is given explicitly as an example of the general formalism of methods (iv)-(vi). Throughout, we use a bare critical propagator which is an arbitrary generalized homogeneous function of the components of the wave vector. This allows us to simultaneously describe ordinary critical systems and anisotropic Lifschitz points as well as certain structural and spin-reorientation phase transitions.