Trace, quantum mechanical

Note that equation (A3.11.1881 includes a quantum mechanical trace, which implies a sum over states. The states used for this evaluation are arbitrary as long as they form a complete set and many choices have been considered in recent work. Much of this work has been based on wavepackets [46] or grid point basis frmctions [47]. 

In a mixed quantum-classical calculation the trace operation in the Heisenberg representation is replaced by a quantum-mechanical trace (tTq) over the quantum degrees of freedom and a classical trace (i.e., a phase-space integral over the initial positions xq and momenta Po) over the classical degrees of freedom. This yields... 

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 ... 

Stationary scattering states were used in the derivation of Eq. (5.98). Quantum mechanical traces are, however, independent of the representation in which they are carried out, so that there is no longer any explicit reference to these states, and any other orthonormal set of functions can be used in the trace. The quantum mechanical traces can then, e.g., be evaluated in a coordinate basis. 

The form of the expressions in Eqs (5.98) and (5.114) is closely related to the classical expressions for the rate constant given in Section 5.1. The quantum mechanical trace becomes in classical statistical mechanics an integral over phase space [9] and the Heisenberg operators become the corresponding classical (time-dependent) functions of coordinates and momenta [8]. Thus, Eq. (5.78) is the classical version of Eq. (5.114). Furthermore, note that Eq. (5.98) is related to Eq. (5.49), i.e., the relevant classical (one-way) flux through Ro, at a given time, becomes S(R - Ro)(p/p)9(p/p), exactly as in Eq. (5.49). 

Again, the quantum mechanical expressions can be written in a form that is analogous to the classical expressions for the rate constant given in Section 5.1, remembering that a classical phase-space integral is equivalent to a quantum mechanical trace [9], and classical functions of coordinates and momenta are equivalent to the corresponding quantum mechanical operators. 

We now discuss two ways of taking a classical limit by replacing the quantum mechanical trace operation by an integral over the classical phase space, leading to a classical correlation function... 

The most naive approach to quantizing the TST expression for N(E), Eq. (15), is as follows the phase-space average becomes a quantum mechanical trace, ... 

The quantum mechanical trace in equation (17) is most readily carried out for separable Hamiltonians. This became feasible for collections of harmonic oscillators with the advent in 1973 of the discrete convolution algorithm of Beyer and Swinehart, which was generalized to the case of arbitrary separable Hamiltonians by Stein and Rabinovitch. These direct-count methods are exceedingly fast and exact for separable Hamiltonians. Direct counts can also be implemented for energy levels derived from perturbative expansions of the Hamiltonian. 

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