Wigner transform

Another way to study the quantum dynamics of a system is to consider quantum phase space representations, that can be compared directly with classical results. Although there is no unique way to define a phase space representation of quantum mechanics, the most popular are the Wigner (Wigner, 1932) and Husimi (Husimi, 1940) functions. The Wigner transform... 

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

More complex integral transforms have been studied, e.g. the two-dimensional Wigner transform, but so far without success. Even the best-quality data are too sparse and noisy to give reliable information when transformed in this way. 

The concentration of the Wigner transforms of the eigenfunctions on the quantized tori provides the counterpart in classically integrable systems to the scarring phenomena associated with unstable periodic orbits. 

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,... 

Using the definition of the Wigner transform [5] of the density matrix,... 

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,... 

In the next section we describe how the QCL equation may be expressed in any basis that spans the subsystem Hilbert space. Here we observe that the subsystem may also be Wigner transformed to obtain a phase-space-like representation of the subsystem variables as well as those of the environment. Taking the Wigner transform of Eq. (8) over the subsystem, we obtain the quantum-classical Wigner-Liouville equation [24],... 

This equation gives the dynamics of the quantum-classical system in terms of phase space variables (R, P) for the bath and the Wigner transform variables (r,p) for the quantum subsystem. This equation cannot be simulated easily but can be used when a representation in a discrete basis is not appropriate. It is easy to recover a classical description of the entire system by expanding the potential energy terms in a Taylor series to linear order in r. Such classical approximations, in conjunction with quantum equilibrium sampling, are often used to estimate quantum correlation functions and expectation values. Classical evolution in this full Wigner representation is exact for harmonic systems since the Taylor expansion truncates. 

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,... 

Carrying out the this change of representation on the quantum-classical Liouville equation and using the product rule formula for the Wigner transform... 

Here we used the fact that the matrix element of an operator A can be expressed in terms of its Wigner transform Aw X) as... 

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... 

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]... 

To demonstrate this equality, we show that the Fourier-Wigner transform preserves the norm. The action of p on / can be determined by means of eq.(15), which yields... 

Using the Fourier-Wigner transform, we can calculate the explicit expressions for position and momentum operators, corresponding to eqs.(10). The transformation is applied to the functions X,/>( ) and hDjip(0 t° obtain the results... 

Thus, a generic hermitian operator can be expressed in terms of the representation of the group Hn. The existence of this expansion is demonstrated in Appendix 1. There, we also prove that the functions B(g) have to be obtained as the inverse Fourier transform of the functions Bw (q,p) of the phase-space variables (q,p) = qi,. . ., qn,Pi, , Pn), which are associated to the quantum operators, B, via the Wigner transform [10]. Since we are considering hermitian operators of the form B X, IiD J, the coefficients in eq.(22) are... 

Another crucial ingredient to derive classical mechanics, is the Fourier-Wigner transform, defined in eq.(16). Thereby, we introduced the vector space C /j 2 (M2n) spanned by the functions f(q, p), on which position X3 and momentum hD3 operators (as usual, j = 1,..., n is a vector index in Rn) act as... 

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