Partial Wigner transform

Performing a partial Wigner transformation with respect to the nuclear variables, the molecular Hamiltonian can be written as... 

By performing a partial Wigner transform with respect to the coordinates of the environment, we obtain a classical-like phase space representation of those degrees of freedom. The subsystem coordinate operators are left untransformed, thus, retaining the operator character of the density matrix and Hamiltonian in the subsystem Hilbert space [4]. In order to take the partial Wigner transform of Eq. (1) explicitly, we express the Liouville-von Neumann equation in the Q representation,... 

The operator 4 = Vp -Vr —Vr -Vp is the negative of the Poisson bracket operator, and the subscript W indicates the partial Wigner transform. The partial Wigner transform of the total Hamiltonian is written as,... 

Shi and Geva [15] have also derived the QCLE in the adiabatic basis starting from the full path integral expression for the quantum mechanical problem. In this representation the derivation starts with the partial Wigner transform of the environmental degrees of freedom in contrast to what is done... 

Given this correspondence between the matrix elements of a partially Wigner transformed operator in the subsystem and mapping bases, we can express the quantum-classical Liouville equation in the continuous mapping coordinates [53]. The first step in this calculation is to introduce an n-dimensional coordinate space representation of the mapping basis,... 

We consider the same reaction model used in previous studies as a simple model for a proton transfer reaction. [31,57,79] The subsystem consists of a two-level quantum system bilinearly coupled to a quartic oscillator and the bath consists of v — 1 = 300 harmonic oscillators bilinearly coupled to the non-linear oscillator but not directly to the two-level quantum system. In the subsystem representation, the partially Wigner transformed Hamiltonian for this system is,... 

The last line defines the mixed quantum-classical Liouville operator C. The W subscripts denote a partial Wigner transform of an operator or density matrix. The phase space variables of the bath are (R,P) and the partial Wigner transform of the total hamiltonian is given by,... 

In the last equality here we have introduced the partial Wigner transforms of the density matrix and operator. The prime on the trace indicates a trace over the subsystem degrees of freedom. All information on the quantum initial distribution is contained in pw R,P, 0). In the evaluation of this expression we assume that the time evolution of Bw R, P,t) is given by Eq. (4). This... 

The partial Wigner transform of the initial density matrix element (2nd line of Eq.(24)) is... 

To exploit the classical nature of the heavier particles, it is convenient to introduce the partial Wigner transform [10] for the density matrix,... 

The last expression allows us to write the partial Wigner transform of the hamiltonian... 

Performing the partial Wigner transform in eq.(l) and applying the rule for the partial Wigner transform of the product of two operators [10]... 

We begin our discussion with a survey of the quantum dynamics and linear response theory expressions for quantum transport coefficients. Since we wish to make a link to a partial classical description, the use of Wigner transforms... 

As discussed in the Introduction, our interest is in quantum-classical systems where the environmental degrees of freedom can be treated classically. The above formulation of quantum dynamics and quantum statistical mechanics in the Wigner representation suggests that we consider another formulation of quantum mechanics based on a partial Wigner representation where only the degrees of freedom in the subsystem are Wigner transformed [4]. We now sketch how this program can be carried out. 

The partial Wigner transformation [4] of the density matrix with respect to the subset of Q coordinates is... 

The partial Wigner transform of a product of operators satisfies the associative product rule,... 

Quantum mechanics in the partial Wigner representation is exact and the partially Wigner transformed quantum bracket satisfies the Jacobi identity,... 

If the partial Wigner transform of the exact canonical quantum equilibrium density in (49) is expressed in the adiabatic basis and is expanded to linear order in ft, we obtain the same result as in (50), indicating that the quantum-classical expression is exact to this order. 

THE COUPLED QUANTUM-CLASSICAL DESCRIPTION 2.1. Partial Wigner transform... 

The subscript W refers to this partial Wigner transform, N is the eoordinate space dimension of the bath and X = R, P). In this partial Wigner representation, the Hamiltonian of the system takes the form Hw R,P) = P /2M + y-/2m+ V q,R). If the subsystem DOF are represented using the states of an adiabatic basis, a P), which are the solutions of hw R) I R)=Ea R) I where hw K)=p /2m+ V q,R) is the Hamiltonian for the subsystem with fixed eoordinates R of the bath, the density matrix elements are p i -, 0 = ( I Pw( 01 )- From the solution of the quantum Liouville equation given some initial state of the entire quantum system, the reduced density matrix elements of the quantum subsystem of interest can be obtained by integrating over the bath variables, p f t) = dX p X,t), in order to find the populations and off-diagonal elements (coherences) of the density matrix. 

We can write this quantum mechanical expression for the rate coefficient in another form by taking the partial Wigner transform over the bath DOF. Using the rule for the Wigner transform of the trace of two operators, the rate coefficient in eqn (10.3) can be written as ... 

The quantum-classical Liouville equation (QCLE) provides an approximate but accurate description of a quantum subsystem coupled in an arbitrary manner to a bath that can be described by classical dynamics in the absence of coupling to the quantum subsystem. The QCLE describes the time evolution of the partially Wigner transformed density matrix of the system p R,P,t) discussed above, and is given by ... 

The master equation (62) can be transformed to a c-number partial differential equation. Three kinds of equations can be derived from (62) (1) an equation for the Wigner function (sym) related to symmetric (Weyl) ordering of... 

---