Quantum flux operator

We note that the flux is a vector and the expression in Eq. (F.52) is therefore the th component of the quantum flux operator. The quantum flux of probability through a surface given by S(q) = 0 for a system in the quantum state ib) may therefore be determined as the dot product of the quantum flux and the normalized gradient vector VS, and integrated over the entire surface. 

Eq. (351) can be transformed to Eq. (359). Further identifying Ns with 2ti p( ), Eq. (346) becomes identical with Eq. (361). Hence, under certain circumstances the quantized ARRKM theory is equivalent to the rigorous quantum reaction rate theory. A number of remarks are in order. First, assumption (a) is automatically satisfied by definition. Second, assumption (b) implies that Fw in the quantized ARRKM theory be the direct analog of the quantum flux operator in the flux-flux autocorrelation formalism. Third, assumption (c) requires that the action of the operator 0jy(V5 v) at any particular time, say at time zero, is equivalent to the action of the projector P i) at time infinity. Regarding 0vi (V5 v) as the analog... 

The quantum flux operator F measures the probability current density. The latter satisfies the continuity equation resulting from the invariance of the norm of the wave packet in the coordinate basis. For a stationary wave function, the probability density is independent of time and the flux is constant across any fixed hypersurface. In reaction dynamics the flux operator is most generally defined in terms of a dividing surface 0 which... 

We particularly focused on a conically intersecting manifold of two electronic states and described the quantum flux operator formalism within a time-dependent WP approach to calculate the initial state-specific and energy resolved reaction probabilities. The flux operator is represented in the two-state diabatic as well as adiabatic representation. While the... 

In equation (1), F is the quantum flux operator, most conveniently expressed for a curvilinear reaction coordinate s =... 

A consistent quantal TST (QTST) has been worked out by Miller and coworkers [Miller 1974 Miller et al. 1983 Tromp and Miller 1986 Voth et al. 1989a]. In quantum mechanics the classical flux X is replaced by the symmetrized flux operator... 

QTST is predicated on this approach. The exact expression 50 is seen to be a quantum mechanical trace of a product of two operators. It is well known, that such a trace can be recast exactly as a phase space integration of the product of the Wigner representations of the two operators. The Wigner phase space representation of the projection operator limt-joo %) for the parabolic barrier potential is h(p + mwtq). Computing the Wigner phase space representation of the symmetrized thermal flux operator involves only imaginary time matrix elements. As shown by Poliak and Liao, the QTST expression for the rate is then ... 

This derived expression satisfies conditions a-d mentioned above and based on numerical computatiotf 6-2 seems to bound the exact result from above. It is similar but not identical to Wigner s original guess. The quantum phase space function which appears in Eq. 52 is that of the symmetrized thermal flux operator, instead of the quantum density. 

After having described the expression for the rate constant within the framework of classical mechanics, we turn now to the quantum mechanical version. We consider first the definition of a flux operator in quantum mechanics.2 To that end, the flux density operator (for a single particle of mass to) is defined by... 

In the following we present the axioms or basic postulates of quantum mechanics and accompany them by their classical counterparts in the Hamiltonian formalism. We begin the presentation with a brief summary of some of the mathematical background essential for the developments in the following. It is by no means a comprehensive presentation, and the reader is supposed to have some basic knowledge about quantum mechanics that may be obtained from any of the many introductory textbooks in quantum mechanics. The focus here is on results of particular relevance to the subjects of this book. We consider, for example, a derivation of a formal expression for the flux density operator in quantum mechanics and its coordinate representation. A systematic way of generating any representation of any combination of operators is set up, and is of immediate usage for the time autocorrelation function of the flux operator used to determine the rate constants of a chemical process. 

In Section 5.2, we used an expression for the flux operator. In most quantum mechanics textbooks expressions for the probability current density, or probability flux density, are given in terms of the wave function in the coordinate representation. We need an expression for the flux density operator without reference to any particular representation, and since it is rarely found in the textbooks, let us in the following derive this expression. 

As we have shown above the classical flux-flux autocorrelation function Cp can be explicitly analyzed within the framework of the CNF theory. The following natural question arises Can one obtain the time dependence of the quantum flux-flux autocorrelation function Cp using the methods of the QNF theory Of course, the QNF technique provides one with an approximation of the original Hamiltonian operator. This approximation is only accurate in the vicinity of the equilibrium saddle point in phase space, so one should not expect a perfect agreement between the QNF flux-flux autocorrelation function and the exact one to hold up to infinitely long times. Instead, the QNF theory will provide an approximation of the exact flux-flux autocorrelation function in a certain time interval whose length will depend on the effective Planck s constant among other parameters. 

To evaluate a flux correlation function only the eigenstates of the thermal flux operator have to be propagated. Since the number of relevant eigenstates is rather small, this is a manageable task even for larger systems. An interesting analogy in the classical and quantum description can be found if one identifies in the above equation a dynamical factor fr /t). In the limit... 

Further detailed discussions and applications to collinear chemical reactions are given in Reference [225]. Miller and co-workers have formulated the quantum instanton theory, starting from the expression of rate constant in terms of the flux operators, and applied it to various practical processes. Those who are interested in that should refer to References [226,227], The semiclassical instanton theory explained above can be conveniently generalized to the case of nonadiabatic chemical reaction. This is discussed in the next subsection. 

Equation (30) is quite a beguiling expression. For example, in the classical expression for N(E), equation (12), there is a statistical factor 5 E — H), the flux factor F, and a dynamical factor X- A similar structure exist in the quantum expres.sion, equation (26), where the dynamical factor is the projection operator Xf The manipulations following equation (27), however, lead to the result, equation (30), which appears to have no dynamical information, i.e., only the statistical operator S(E — H) and flux operator F are involved in equation (30). This is an example of the fact that dynamics and statistics are inseparably intertwined in quantum mechanics, e.g., a wave-function describes the dynamical motion of the particles and also their statistics. 

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