ATDHF versus TDHF in a soluble model

The equations of motion for a system of interacting fermions with SU(3) symmetry are established in TDHF and ATDHF approximations. These equations are presented in a form such that the comparison between different approximations can be made in an analytic way. The problem of coupling between large and small amplitude modes is also discussed.     

Eigenchannel calculation for a soluble model

We examine the accuracy of the eigenchannel method for treating nuclear reactions by a comparison with the exact solution of a problem of two coupled channels. No crucial deviations between the exact and the eigenchannel cross sections are observed. We also discuss the questions connected with the non-orthogonality of the sets of radial states used in the eigenchannel treatment, and show how to minimize the resulting errors. Our conclusion is that, in principle, the eigenchannel method is capable of producing the correct shape of nuclear cross sections, without introducing spurious wiggles.     

Approach to statistical equilibrium in a soluble model

The soluble model of an oscillator coupled to a scalar field is used to describe the Brownian motion of an oscillator in a thermal bath. The approach to equilibrium is shown by studying the generating functional that corresponds to a non-equilibrium situation fixed as an initial condition. It is shown how this functional goes over into the functional associated with the equilibrium situation. The solution of the functional equations is discussed.     

Quantum entanglement in a soluble two-electron model atom

We investigate the entanglement properties of bound states in an exactly soluble two-electron model, the Moshinsky atom. We present exact entanglement calculations for the ground, first and second excited states of the system. We find that these states become more entangled when the relative inter-particle interaction becomes stronger. As a general trend, we also observe that the entanglement of the eigenstates tends to increase with the states’ energy. There are, however, “entanglement level-crossings” where the entanglement of a state becomes larger than the entanglement of other states with higher energy. In the limit of weak interaction, we also compute (exactly) the entanglement of higher excited states. Excited states with anti-parallel spins are found to involve a considerable amount of entanglement even for an arbitrarily weak (but non zero) interaction. This minimum amount of entanglement increases monotonically with the state’s energy. Finally, the connection between entanglement and the Hartree-Fock approximation in the Moshinsky model is addressed. The quality of the ground-state Hartree-Fock approximation is shown to deteriorate, and the corresponding correlation energy to grow, as the entanglement of the (exact) ground state increases. The present work goes beyond previous related studies because we fully take into account the identical character of the two constituting particles in the entanglement calculations, and provide analytical, exact results both for the ground and the first few excited states.     

Less singular terms and small x evolution in a soluble model

We calculate the effect of the less singular terms at small x on the evolution of the coefficient function in φ³ theory in six dimensions, which result from a complete solution of the ladder equation. Scale-invariant next-to-leading order contributions are also studied. We show that the small-x approximation does not deliver the dominant contributions.     

Eigenstate expansions in a soluble model of identical particle scattering

Solutions to the Schrödinger equation and the inhomogeneous equation for the case of two identical particles interacting with a center of force are studied. Eigenstate expansions for solving each equation are explicitly introduced and their properties discussed. The case when the interparticle interaction v₁₂ is zero is then examined; this is a completely soluble problem. The eigenstate expansion solutions for the Schrödinger and inhomogeneous equations are used to explore the means by which the correct solution is obtained. Finally, approximate solutions, obtained by truncating the eigenfunction expansions, are introduced. It is seen that both methods lead to the correct amplitude when τ₁₂ = 0, even though the approximate solution to the inhomogeneous equation does not lead, in the end, to an antisymmetric solution.     

One-dimensional diffusion in soluble model potentials

The one-dimensional diffusion in potentials which have a finite number of jumps in their value and in their derivative is investigated. The jump conditions of the eigenfunctions of the corresponding Fokker-Planck-operator are derived. These conditions are then applied to bistable models, to a metastable model and to a periodic potential model. At least for one model in all these three cases all eigenvalues and eigenfunctions are given analytically.     

An exactly soluble model of a shallow double well

Shallow one-dimensional double-well potentials appear in atomic and molecular physics and other fields. Unlike the “deep” wells of macroscopic quantum coherent systems, shallow double wells need not present low-lying two-level systems. We argue that this feature, the absence of a low-lying two-level system in certain shallow double wells, may allow the finding of new test grounds for quantum mechanics in mesoscopic systems. We illustrate the above ideas with a family of shallow double wells obtained from reflectionless potentials through the Darboux–Bäcklund transform.     

A simple and exactly soluble model of a deep two-electron trap

A simple and exactly soluble model is presented for a two-electron capture process accompanied by a large lattice relaxation. The model is a system of two interacting electrons in a deformable lattice with a single donor-type impurity. It is demonstrated that the model has the following two advantages. First, it exhibits a larger lattice relaxation for the binding of the second electron than for the first. Second, it enables us to discuss possible magnetic behaviors of the impurity in the ground state.     

Symmetries and time evolution in discrete phase spaces: a soluble model calculation

Using the flexibility and constructive definition of the Schwinger bases, we developed different mapping procedures to enhance different aspects of the dynamics and of the symmetries of an extended version of the two-level Lipkin model. The classical limits of the dynamics are discussed in connection with the different mappings. Discrete Wigner functions are also calculated.     

Quasi-spin projection in an exactly soluble model

An exactly soluble model for the study of projection techniques within the framework of the Hartree-Fock theory is presented. Properties of the exact solutions are analyzed and projections, both before and after variation, are discussed.     

Quantum orders in an exact soluble model

We find all the exact eigenstates and eigenvalues of a spin-1/2 model on square lattice: H=16g Sum S(y)(i)S(x)(i + empty set x)S(y)(i + empty set x + empty set y)S(x)(i + empty set y). We show that the ground states for g < 0 and g > 0 have different quantum orders described by Z2A and Z2B projective symmetry groups. The phase transition at g = 0 represents a new kind of phase transition that changes quantum orders but not symmetry. Both the Z2A and Z2B states contain Z2 lattice gauge theories at low energies. They have robust topologically degenerate ground states and gapless edge excitations.     

Position and momentum densities. Complementarity at work: refining a quantum model from different data sets

Although Bragg and Compton scattering are well-established techniques, only very few attempts to simultaneously combine information originating from these two experiments have been made so far. This remark also holds for Bragg neutron magnetic combined with X-ray scattering. We propose a quite general procedure to refine a quantum model from different data sets using basic Bayesian probability theory. As an illustration, a qualitative preliminary study to extract chemical information such as charge transfer in ionic-covalent compounds is reported.     

Complementarity and Scientific Rationality

Bohrs interpretation of quantum mechanics has been criticized as incoherent and opportunistic, and based on doubtful philosophical premises. If so Bohrs influence, in the pre-war period of 1927–1939, is the harder to explain, and the acceptance of his approach to quantum mechanics over de Broglies had no reasonable foundation. But Bohrs interpretation changed little from the time of its first appearance, and stood independent of any philosophical presuppositions. The principle of complementarity is itself best read as a conjecture of unusually wide scope, on the nature and future course of explanations in the sciences (and not only the physical sciences). If it must be judged a failure today, it is not because of any internal inconsistency.     

Complementarity via Minimum Error Measurement in a Two-Path Interferometer

We study the fringe visibility and the which-path information(WPI) of a general Mach-Zehnder interferometer with an asymmetric beam splitter(BS). A minimum error measurement in the detector is used to extract the WPI. Both the fringe visibility V and the WPI I_(path) are affected by the initial state of the photon and the second asymmetric BS. The condition in which the WPI takes the maximum is obtained. The complementarity relationship V~2 + I_(path)~2 ≤ 1 is found, and the conditions for equality are also presented.