Tunneling of an energy eigenstate through a parabolic barrier viewed from Wigner phase space

We analyze the tunneling of a particle through a repulsive potential resulting from an inverted harmonic oscillator in the quantum mechanical phase space described by the Wigner function. In particular, we solve the partial differential equations in phase space determining the Wigner function of an energy eigenstate of the inverted oscillator. The reflection or transmission coefficients R or T are then given by the total weight of all classical phase-space trajectories corresponding to energies below, or above the top of the barrier given by the Wigner function.     

三维谐振子Wigner函数的星乘解

Wigner函数作为相空间中的一个准概率分布函数,也是密度矩阵的特殊表示形式,具有十分重要的物理意义。首先介绍了Wigner函数的性质及其计算方法,然后利用星本征方程(Moyal方程)计算了三维谐振子的Wigner函数。最后讨论了在相空间中描述声子与电子(或光子)相互作用的方法,并得到了跃迁几率在相空间中所满足的方程。     

Quantum phenomena via complex measure: Holomorphic extension

The complex measure theoretic approach proposed earlier is reviewed and a general version of density matrix as well as conditional density matrix is introduced. The holomorphic extension of the complex measure density (CMD) is identified to be the Wigner distribution function of the conventional quantum mechanical theory. A variety of situations in quantum optical phenomena are discussed within such a holomorphic complex measure theoretic framework. A model of a quantum oscillator in interaction with a bath is analyzed and explicit solution for the CMD of the coordinate as well as the Wigner distribution function is obtained. A brief discussion on the assignment of probability to path history of the test oscillator is provided.     

A new parametrized quantum distribution function and its time development

We discuss a new general class of quantum distribution functions characterized by an arbitrary parameter b. The values b = -1, 0, 1 correspond to the anti-standard (Kirkwood), the Wigner, and the standard distribution functions, respectively. An analytic form of the equation of motion is derived. We conclude that, for time-dependent problems involving a potential which is a function of coordinates only, the Wigner distribution function is the optimum one to use, from a simplicity standpoint.     

Quantum Dynamical Behavior of the Morse Oscillator: the Wigner Function Approach

We explore the quantum dynamical behavior of the Morse oscillator in the phase space using the Wigner function. For an initial wave packet excited with Gaussian probability distribution, we calculate the associated Wigner function and compute its time evolution. By calculating the marginal probabilities, we study the formation of quantum carpets both in the position space and in the momentum space. In addition, in view of these probabilities, we present the time evolution of the position and momentum expectation values. The structure of quantum carpets and the time-evolved expectation values mimic the emergence of quantum revivals and fractional revivals.     

Energy dependence of fission probabilities induced by negative pions in Sn,Au and Bi

Fission probabilities induced by negative pions in Sn, Au and Bi at different energies using variety of nuclear track detectors have been studied. The target-detector assemblies in 4π-geometric configuration were irradiated at the AGS facility of Brookhaven National Laboratory, USA. After etching the exposed detectors at appropriate etching conditions the detectors have been scanned for the tracks of fission fragments produced as a result of interaction of pions with the target nuclei. Based on the track counts, the values of fission cross sections have been measured and fission probabilities have been calculated using the reaction cross-section calculated with the help of the cascade-exciton model code CEM95. The values of fission probability based on experimental fission cross-sections have been compared with the theoretically calculated values of fission probabilities obtained using the CEM95 code. Theoretical values of fission probability have been computed for incident pion energy up to 2500 MeV in Sn, Au and Bi for comparison to fission data for high energy negative pions at 500, 672, 1068, 1665 MeV and 2300 MeV energies for the same targets. The values of fission probability based on the experimental fission cross section and theoretically calculated values of fission probability have been compared. Reasonable agreement has been observed among the experimentally measured and theoretically computed values of fission probabilities. A saturation of probabilities has been observed for Au and Bi at higher energies, but for Sn an increase of probability with the increase of pion beam momentum up to 1665 MeV has been observed.     

Decay times for second-chance fission of 239U studied by crystal blocking

Neutron-induced fission of ²³⁸U has been studied by the crystal-blocking technique for neutron energies just below and above the threshold for second-chance fission. In agreement with earlier measurements, in this energy range the lifetime for first-chance fission is found to be too short to have an observable effect on the blocking dips. Above the threshold, however, an appreciable filling-in of the dips is observed. The results are analyzed in terms of a two-component lifetime distribution and then indicate an average lifetime of a few fsec for second-chance fission at a neutron energy of E_n ≈ 7.2 MeV, in agreement with results from a simple calculation. It is shown that in this analysis it is important to take into account the anisotropy of the fission-fragment distribution and, in particular, the difference between the angular distributions for first- and second-chance fission.     

Wigner Distribution Function for the Time-Dependent Quadratic-Hamiltonian Quantum System using the Lewis–Riesenfeld Invariant Operator

We investigate the Wigner distribution function of the general time-dependent quadratic-Hamiltonian quantum system with the Lewis–Riesenfeld invariant operator method. The Wigner distribution function of the system in Fock state, coherent state, squeezed state, and thermal state are derived. We apply our study to the one-dimensional motion of a Brownian particle and to the driven oscillator with strongly pulsating mass.PACS: 03.65.-w, 03.65.Ca     

The Wigner distribution function

In this letter, the relationship between the characteristic function for two arbitrary noncommuting observables and a generalized Wigner distribution function is established. This distribution function is shown to have no simple interpretation in the sense of probability theory but, in lieu of its special properties, can be used directly for calculating the expectation values of observables.     

Probability representation of the quantum evolution and energy-level equations for optical tomograms

The von Neumann evolution equation for the density matrix and the Moyal equation for the Wigner function are mapped onto the evolution equation for the optical tomogram of the quantum state. The connection with the known evolution equation for the symplectic tomogram of the quantum state is clarified. The stationary states corresponding to quantum energy levels are associated with the probability representation of the von Neumann and Moyal equations written for optical tomograms. The classical Liouville equation for optical tomogram is obtained. An example of the parametric oscillator is considered in detail.     

Wigner function for the Dirac oscillator in spinor space

The Wigner function for the Dirac oscillator in spinor space is studied in this paper. Firstly, since the Dirac equation is described as a matrix equation in phase space, it is necessary to define the Wigner function as a matrix function in spinor space. Secondly, the matrix form of the Wigner function is proven to support the Dirac equation. Thirdly, by solving the Dirac equation, energy levels and the Wigner function for the Dirac oscillator in spinor space are obtained.     

Wigner function for the Dirac oscillator in spinor space

The Wigner function for the Dirac oscillator in spinor space is studied in this paper.Firstly,since the Dirac equation is described as a matrix equation in phase space,it is necessary to define the Wigner function as a matrix function in spinor space.Secondly,the matrix form of the Wigner function is proven to support the Dirac equation.Thirdly,by solving the Dirac equation,energy levels and the Wigner function for the Dirac oscillator in spinor space are obtained.     

Systematics of neutron emission probabilities from delayed neutron precursors

A simple correlation based on the gross theory ofβ-decay is derived between the neutron emission probabilitiesP _n of delayed neutron precursors, theirβ-decay energies and the neutron binding energies of the daughter nuclei. The correlation is shown to be valid for delayed neutron precursors among the fission products. TheP _n-values of several expected but still unidentified neutron precursors are estimated together with their contributions to the delayed neutron groups in thermal-neutron induced fission of²³⁵U. Some aspects of theβ-strength function for transitions into highly excited states are discussed.     

Approximate analytical representation of Wigner distribution function for Gaussian beams passing through ABCD optical systems with hard aperture

Based on the treatment that a rectangular function can be expanded as an approximate sum of complex Gaussian functions with finite numbers, the analytical expression of the Wigner distribution function for a Gaussian beam passing through a paraxial ABCD optical system with hard-edged aperture is obtained. By numerical simulation, it is shown that the effect of the aperture on the Wigner distribution function is prominent. By comparing the analytical results with the numerical integral results, it is shown that this method of expanding hard-edged aperture into Gaussian functions with finite numbers is proper and ascendant. This method could be extended to study the Wigner distribution functions of other light beams passing through a paraxial ABCD optical system with hard-edged aperture.     

Transition fission states for heated nuclei

Induced fission reactions of fissioning compound nuclei that result from the capture of various incident particles (nucleons, γ rays, multiply charged ions) by target nuclei are investigated using the generalized nucleus model and the Wigner random matrix method. The effect produced on the fission widths of the compound nucleus by the competition between the excitation energies of its collective vibrational degrees of freedom that lead to its scission into fission fragments and its rotational and multi-quasiparticle states is analyzed. Bohr’s concept of transition fission states developed for near-barrier nuclear fission is generalized to the induced fission of nuclei with the excitation energies noticeably higher than the fission barriers. The temperature of the fissioning nucleus in the vicinity of the point of its scission into fission fragments is estimated.     

Dissipative Dynamics and the Statistics of Energy States of a Hookean Model for Protein Folding

A generic model of a random polypeptide chain, with discrete torsional degrees of freedom and Hookean spring connecting pairs of hydrophobic residues, reproduces the energy probability distribution of real proteins over a very large range of energies. We show that this system with harmonic interactions, under dissipative dynamics driven by random noise, leads to a distribution of energy states obeying a modified one-dimensional Ornstein–Uhlenbeck process and giving rise to the so-called Wigner distribution. A tunably fine- or coarse-grained sampling of the energy landscape yields a family of distributions for the energies and energy spacings.     

Study of the reaction238U (α, α′f) atE α=480 MeV

The excitation function of the fission probability P _E E _x) for²³⁸U has been measured in the reaction²³⁸U(α, α′ f) at 480 MeV bombarding energy. The reaction mechanism of this reaction is discussed for excitation energies belowB _nf , the threshold for second chance fission, and aboveB _nf up toE _x =37 MeV. In comparing with results from fission induced by photons and by particle transfer reactions the (α, α′f) reaction gives too low values for the fission probabilityP _f at excitation energies well aboveB _nE . The role of the quasi-elastic knock-out process in this reaction is discussed.