Liouville-von Neumann

Altematively, in the case of incoherent (e.g. statistical) initial conditions, the density matrix operator P(t) I 1>(0) (v(01 at time t can be obtained as the solution of the Liouville-von Neumann equation ... 

The density operator r(<) is a Hermitian and positive function of time, and satisfies the generalized Liouville-von Neumann (LvN) equation(47, 45)... 

A formulation of electronic rearrangement in quantum molecular dynamics has been based on the Liouville-von Neumann equation for the density matrix. Introducing an eikonal representation, it naturally leads to a general treatment where Hamiltonian equations for nuclear motions are coupled to the electronic density matrix equations, in a formally exact theory. Expectation values of molecular operators can be obtained from integrations over initial conditions. 

Ignoring relaxation, the spin evolution is governed by the Liouville-von-Neumann equation... 

Prior to an effective Hamiltonian analysis it is, in order to get this converging to the lowest orders, typical to remove the dominant rf irradiation from the description by transforming the internal Hamiltonian into the interaction frame of the rf irradiation. This procedure is well established and also used in the most simple description of NMR experiments by transforming the Hamiltonian into the rotating frame of the Zeeman interaction (the so-called Zeeman interaction frame). In the Zeeman interaction frame the time-modulations of the rf terms are removed and the internal Hamiltonian is truncated to form the secular high-field approximated Hamiltonian - all facilitating solution of the Liouville-von-Neumann equation in (1) and (2). The transformation into the rf interaction frame is given by... 

Abstract. We review the recent development of quantum dynamics for nonequilibrium phase transitions. To describe the detailed dynamical processes of nonequilibrium phase transitions, the Liouville-von Neumann method is applied to quenched second order phase transitions. Domain growth and topological defect formation is discussed in the second order phase transitions. Thermofield dynamics is extended to nonequilibrium phase transitions. Finally, we discuss the physical implications of nonequilibrium processes such as decoherence of order parameter and thermalization. 

Keywords Nonequilibrium phase transitions, Liouville-von Neumann approach, domain growth, topological defect formation. 

To describe nonequilibrium phase transitions, there have been developed many methods such as the closed-time path integral by Schwinger and Keldysh (J. Schwinger et.al., 1961), the Hartree-Fock or mean field method (A. Ringwald, 1987), and the l/lV-expansion method (F. Cooper et.al., 1997 2000). In this talk, we shall employ the so-called Liouville-von Neumann (LvN) method to describe nonequilibrium phase transitions (S.P. Kim et.al., 2000 2002 2001 S.P. Kim et.al., 2003). The LvN method is a canonical method that first finds invariant operators for the quantum LvN equation and then solves exactly the... 

We applied the Liouville-von Neumann (LvN) method, a canonical method, to nonequilibrium quantum phase transitions. The essential idea of the LvN method is first to solve the LvN equation and then to find exact wave functionals of time-dependent quantum systems. The LvN method has several advantages that it can easily incorporate thermal theory in terms of density operators and that it can also be extended to thermofield dynamics (TFD) by using the time-dependent creation and annihilation operators, invariant operators. Combined with the oscillator representation, the LvN method provides the Fock space of a Hartree-Fock type quadratic part of the Hamiltonian, and further allows to improve wave functionals systematically either by the Green function or perturbation technique. In this sense the LvN method goes beyond the Hartree-Fock approximation. 

By analogy the propagation of a density matrix, which corresponds to the solution of the Liouville-von Neumann equation 231... 

The most successful strategy for approximating the Liouville-von Neumann propagator is to interpolate the operator with polynomial operators. To this end, Newton and Faber polynomials have been suggested to globally approximate the propagator,126,127,225,232-234 as in Eq. [95]. For short-time propagation, short-iterative Arnoldi,235 dual Lanczos,236 and Chebyshev... 

Dependent Liouville-von Neumann Equation Dissipative Evolution. 

Another important hierarchy of equations is obtained by applying the (MCM) to the matrix representation of the Liouville-von Neumann Equation LVNE) [12,13]. In this way the p-order Contracted Liouville-von Neumann Elquation (p-CLVNE) is obtained [4]. It will be shown here that The structure of a particular p-CSE, that involves the higher order CSE s for a given state, can be replaced by an equivalent set of equations, 1-CSE and 1-CLVNE, but for the whole spectrum, i. e. involving all the states. 

In this paragraph, we consider the equation obtained by applying the same MCM to both sides of the matrix representation of the Liouville-von Neumann... 

The evolution of the molecule is described by the generalized Liouville-von Neumann equation [35, 36]... 

These challenges can be dealt with the powerful mathematical tools of quantum chemistry, as advocated by Per-Olov Lowdin.[l, 2, 3, 4] In our studies, linear algebras with matrices,[4] partitioning techniques,[3] operators and superoperators in Liouville space, and the Liouville-von Neumann... 

It is convenient to start with an isolated system which includes the object and medium, with hamiltonian H. Its state is given by the density operator f (t) which satisfies the Liouville-von Neumann (L-vN) equation of motion... 

The powerful mathematical tools of linear algebra and superoperators in Li-ouville space can be used to proceed from the identification of molecular phenomena, to modelling and calculation of physical properties to interpret or predict experimental results. The present overview of our work shows a possible approach to the dissipative dynamics of a many-atom system undergoing localized electronic transitions. The density operator and its Liouville-von Neumann equation play a central role in its mathematical treatments. 

In an NMR experiment, the energy of the lattice is practically constant (the lattice has a large heat capacity). It is therefore assumed that the lattice is always in a state of thermodynamic equilibrium. Thus, it is possible to use a semi-classical description of its interactions with the spin system. Within this approach, the Liouville-von Neumann equation of motion for a spin system is given by ... 

To describe consistently cotunneling, level broadening and higher-order (in tunneling) processes, more sophisticated methods to calculate the reduced density matrix were developed, based on the Liouville - von Neumann equation [186-193] or real-time diagrammatic technique [194-201]. Different approaches were reviewed recently in Ref. [202]. 

We consider also the other method, based on the equations of motion for operators. From Liouville - von Neumann equation we find (all c-operators are Heisenberg operators in the formula below, (t) is omitted for shortness)... 

Having established the Hamiltonian, the next step is to derive an equation of motion for the density matrix. One can start with the Liouville-von Neumann equation for the complete system-plus-bath density matrix a(t)... 

A molecule M plus its bath B in an external field can be described as a total system with a Hamiltonian H = Hm + Hb + H m n I (f) which may depend on time if the total system is subject to an external electromagnetic field, as indicated. Given this, the density operator r(t) for the system satisfies the Liouville-von Neumann (L-vN) equation,... 

---