Quantum harmonic oscillator modes

This relation may be interpreted as the mean-square amplitude of a quantum harmonic oscillator 3 o ) = 2mco) h coth( /iLorentzian distribution of the system s normal modes. In the absence of friction (2.27) describes thermally activated as well as tunneling processes when < 1, or fhcoo > 1, respectively. At first glance it may seem surprising... 

Now, it may be of interest to look at the connection between the autocorrelation functions appearing in the standard and the adiabatic approaches. Clearly, it is the representation I of the adiabatic approach which is the most narrowing to that of the standard one [see Eqs. (43) and (17)] because both are involving the diagonalization of the matricial representation of Hamiltonians, within the product base built up from the bases of the quantum harmonic oscillators corresponding to the separate slow and fast modes. However, among the... 

The effective Hamiltonian /7 °f, related to the ground states 0 ) and [0]) of the fast and bending modes, is the Hamiltonian of a quantum harmonic oscillator characterizing the slow mode ... 

In Fig. 2 we compare results using e = 0.4 for the two mixed quantum-classical methods outlined in this chapter with exact results obtained from MCTDH wavepacket dynamics calculations. To make a reliable comparison the approximate finite temperature calculations were performed at very low temperatures (/ = 25), though a product of ground state wave functions for the independent harmonic oscillator modes could have been used to make the initial conditions identical to those used in the MCTDH calculations. 

H° and HFree are, respectively, the Hamiltonians of the fast and slow modes viewed as quantum harmonic oscillators, whereas Hint is the anharmonic coupling between the two modes, which are given by Eqs. (15), (21), and (22). Besides, He is the Hamiltonian of the thermal bath, while Hint is the Hamiltonian of the interaction of the H-bond bridge with the thermal bath. 

Now, perform the trace over the eigenstates of the slow mode quantum harmonic oscillator involved in the ACF (114). This leads, after neglecting the zero-point energy of H-bond bridge oscillator, to... 

In this last equation, the right-hand side matrix elements are those of the IP time evolution operator of the driven damped quantum harmonic oscillator describing the H-bond bridge when the fast mode is in its first excited state ... 

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,... 

Bosons of the quantum harmonic oscillator describing the high frequency mode with[b,bt] = 1. 

Effective Hamiltonian of the two H-bond bridge quantum harmonic oscillator in representation II when the two moieties fast modes are in their eigenstates... 

The particular solution for the quantum harmonic oscillator is the first of a set of solutions in fact, we have defined the ground state of the oscillator and the minimum energy that any vibrational mode can have the zero-point energy. In the quantum picture, the atoms can never come to rest because the bond vibration energy can never be lower than that given by Equation (A6.27). 

Here s denotes the coordinate(s) of the quantum particle and Xj are the coordinates of the harmonic oscillator modes which are linearly coupled to the system and which constitute the bath. For simplicity the coupling functions are assumed linear in the system coordinate as well, although extension of the procedure that follows to a more general coupling form of the type f j s)Xj is straightforward. 

In the harmonic approximation the functions Xi and Xf are products of harmonic oscillator functions. We therefore specify the initial state by a set of quantum numbers n — (ni, ri2,..., n/v), and those for the final state by m = (mi,m2,..., tun)- So the nuclear wavefunctions are henceforth denoted by Xi,n and Xf,m- Equation (19.21) tells us how to calculate the rate of transition from one particular initial quantum mode n to a final quantum state m. This is more than we want to know. All we are interested in is the total rate from any initial state to any final state. The ensemble of reactants is in thermal equilibrium therefore... 

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