Projected evolution superoperators and the density operator: theory and applications to inelastic scattering

1982 ◽  
Vol 60 (10) ◽  
pp. 1371-1386 ◽  
Author(s):  
R. E. Turner ◽  
J. S. Dahler ◽  
R. F. Snider
Keyword(s):  
Inelastic Scattering ◽  
Density Operator ◽  
Operator Method ◽  
Time Dependent ◽  
Interaction Picture ◽  
Von Neumann ◽  
First Order ◽  
Density Operators ◽  
Projection Operator Method ◽  
Time Dependent Solution

The projection operator method of Zwanzig and Feshbach is used to construct the time dependent density operator associated with a binary scattering event. The formula developed to describe this time dependence involves time-ordered cosine and sine projected evolution (memory) superoperators. Both Sehrödinger and interaction picture results are presented. The former is used to demonstrate the equivalence of the time dependent solution of the von Neumann equation and the more familiar, frequency dependent Laplaee transform solution. For two particular classes of projection superoperators projected density operators arc shown to be equivalent to projected wave functions. Except for these two special eases, no projected wave function analogs of projected density operators exist. Along with the decoupled-motions approximation, projected interaction picture density operators arc applied to inelastic scattering events. Simple illustrations arc provided of how this formalism is related to previously established results for two-state processes, namely, the theory of resonant transfer events, the first order Magnus approximation, and the Landau–Zener theory.

scholarly journals Evolution Equation and Transport Coefficients of Swarms in Initial Relaxation Processes

1987 ◽  
Vol 40 (3) ◽  
pp. 367 ◽  
Author(s):  
Keiichi Kondo
Keyword(s):  
Evolution Equation ◽  
Transport Coefficients ◽  
Collision Operator ◽  
Operator Method ◽  
Time Dependent ◽  
Relaxation Processes ◽  
Projection Operator Method ◽  
Model Collision ◽  
The Difference ◽  
Dependent Transport

The problem of a swarm approaching the hydrodynamic regime is studied by using the projection operator method. An evolution equation for the density and the related time-dependent transport coefficient are derived. The effects of the initial condition on the transport characteristics of a swarm are separated from the intrinsic evolution of the swarms, and the difference from the continuity equation with time-dependent transport coefficients introduced by Tagashira et al. (1977, 1978) is discussed. To illustrate this method, calculations on the relaxation model collision operator have been carried out. The results are found to agree with the analysis by Robson (1975).

On the derivation of the Pauli and van Hove master equations

1978 ◽  
Vol 56 (9) ◽  
pp. 1204-1217 ◽  
Author(s):  
K. M. van Vliet
Keyword(s):  
Projection Operator ◽  
Evolution Operator ◽  
Random Phase ◽  
Stochastic Theory ◽  
Operator Method ◽  
Original Method ◽  
Von Neumann ◽  
Projection Operator Method ◽  
Von Neumann Equation ◽  
Pauli Master Equation

We discuss the derivation of the Pauli master equation, based on a repeated random phase assumption, and of van Hove's result, based on an initial random phase assumption. For the former we indicate a derivation which is closer to the general approach of stochastic theory than Pauli's original method. For the van Hove result, we show that the diagonal and nondiagonal parts of the evolution operator of the Schrödinger or von Neumann equation are readily obtained by Zwanzig's projection operator method.

Entanglement in composite systems due to external influences

2018 ◽  
Vol 33 (21) ◽  
pp. 1850128
Author(s):  
D. M. Gitman ◽  
M. S. Meireles ◽  
A. D. Levin ◽  
A. A. Shishmarev ◽  
R. A. Castro
Keyword(s):  
Magnetic Field ◽  
Magnetic Fields ◽  
Density Operator ◽  
Time Dependent ◽  
Von Neumann Entropy ◽  
Entanglement Measure ◽  
Photon Beams ◽  
Von Neumann ◽  
Electron Positron ◽  
In The Beginning

In this paper, we consider two examples of an entanglement in two-qubit systems and an example of entanglement in quantum field theory (QFT). In the beginning, we study the entanglement of two spin states by a magnetic field. A nonzero entanglement appears for interacting spins. When the coupling between the spins is constant, we study the entanglement by several types of time-dependent magnetic fields. In the case of a constant difference between [Formula: see text] components of magnetic fields acting on each spin, we find several time-dependent coupling functions [Formula: see text] that also allow us to analyze analytically and numerically the entanglement measure. Considering two photons moving in an electron medium, we demonstrate that they can be entangled in a controlled way by applying an external magnetic field. The magnetic field affecting electrons of the medium affects photons and, thus, causes an entanglement of the photon beams. The third example is related to the effect of production of electron–positron pairs from the vacuum by a strong external electric field. Here, we have used a general nonperturbative expression for the density operator of the system under consideration. Applying a reduction procedure to this density operator, we construct mixed states of electron and positron subsystems. Calculating the von Neumann entropy of such states, we obtain the loss of information due to the reduction and, at the same time, the entanglement measure of electron and positron subsystems. This entanglement can be considered as an example of an entanglement in QFT.

scholarly journals Hyperpolarization and the physical boundary of Liouville space

2021 ◽  
Vol 2 (1) ◽  
pp. 395-407
Author(s):  
Malcolm H. Levitt ◽  
Christian Bengs
Keyword(s):  
Master Equation ◽  
Density Operator ◽  
Spin Dynamics ◽  
Physical Region ◽  
State Density ◽  
Von Neumann Entropy ◽  
Von Neumann ◽  
Liouville Space ◽  
Density Operators ◽  
Physical Boundary

Abstract. The quantum state of a spin ensemble is described by a density operator, which corresponds to a point in the Liouville space of orthogonal spin operators. Valid density operators are confined to a particular region of Liouville space, which we call the physical region and which is bounded by multidimensional figures called simplexes. Each vertex of a simplex corresponds to a pure-state density operator. We provide examples for spins I=1/2, I=1, I=3/2 and for coupled pairs of spins-1/2. We use the von Neumann entropy as a criterion for hyperpolarization. It is shown that the inhomogeneous master equation for spin dynamics leads to non-physical results in some cases, a problem that may be avoided by using the Lindbladian master equation.

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