Boltzmann density operator

Boltzmann Density Operators in Both Representations The Evolution Operator of a Driven Quantum Harmonic Oscillator [59]... 

Here, pTot is the Boltzmann density operator, p, the dipole moment operator, X, a variable having the dimension of p, which is given by... 

Here, Hx0t is the total Hamiltonian of the system. The Boltzmann density operator pTot is... 

Besides, within the Bosons representation, the Boltzmann density operator appearing in Eq. (63), is given in Table VI. 

Now, we will show that the SD (71) that is the Fourier transform of the ACF (68) via Eq. (69), is equivalent to that obtained by Marechal and Witkowski in their pioneering work [18], For this purpose, look at the ACF (68). It is possible to write the translation operator appearing in it, which has to be averaged on the Boltzmann density operator, as a product of two translation operators. According to Eq. (F.4), it is possible to write... 

Quantum Representation II. In quantum representation II and from Table VI, the Hamiltonian and the Boltzmann density operator of the H-bond bridge are given, respectively, by the equations ... 

Here, pj7° is the Boltzmann density operator of the H-bond bridge,... 

By using the eigenvalue equations (229) and then neglecting the zero-point energy of the Boltzmann density operator, this ACF transforms into... 

Again, keeping in mind that the Hamiltonian involved in the Boltzmann density operator has no reason to be non-Hermitean, whereas the Hamiltonians involved in the evolution operators must be non-Hermitean in order to take into account the damping, the ACF appears to be given by the following trace trFermi over the base spanned by the Hamiltonian HFermi ... 

At last, pDav is the Boltzmann density operator given by... 

Besides, observe that the probabilities pk are at thermal equilibrium with the matrix elements of the Boltzmann density operator that is diagonal with respect to k), ... 

The operator A(ka°) is the translation operator that is unitary according to Eq. (E.8). In the following, the B operators will be either the effective Hamiltonians H or the Boltzmann density operators pj7 built up from these Hamiltonians. 

In quantum representation //, the Boltzmann density operator corresponding to the H-bond bridge viewed as a quantum harmonic oscillator may be written, neglecting the zero-point energy, according to... 

Now, the damped time-dependent density operator (1.14) may be viewed as resulting from the following canonical transformation on the Boltzmann density operator, involving the damped translation operator appearing in Eq. (131), that is,... 

Here, pj is the Boltzmann density operator of the H-bond bridge viewed as a quantum harmonic oscillator, pe is the Boltzmann density operator of the thermal bath, and (t) are effective time-evolution operators governing the dynamics of the H-bond bridge depending on the excited-state degree k of the fast mode. They are given by Eq. (110), that is,... 

Here, (3 has its usual statistical meaning, whereas X. is a scalar, and pB is the Boltzmann density operator describing the system. It was been written [Go//(f)]° in place of [Gofl(t)]° in order to make clear the differences in the fashion so we could calculate the ACF. 

Within the adiabatic approximation, and within the quantum representation II, the Boltzmann density operator playing the physical role is... 

This last result shows that the canonical transformation over the Boltzmann density operator and involving the translation operator, gives the density operator pa. Now, one may observe that owing to Eq. (N.12) ... 

Again, note that when the absolute temperature is vanishing, the Boltzmann density operator reduces to that of the ground state of the Hamiltonian of the quantum harmonic oscillator, that is,... 

At this step of factorization of the u1 and iC subspaces, it appears that the traces trsiow must be the same because of the presence of the same Boltzmann density operator p ]u on the right-hand side addition, so that they may be written more simply trSiow -... 

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