Quantum harmonic oscillator operators

Here, t/(f) is the reduced time evolution operator of the driven damped quantum harmonic oscillator. Recall that representation II was used in preceding treatments, taking into account the indirect damping of the hydrogen bond. After rearrangements, the autocorrelation function (45) takes the form [8]... 

These operators allow for the description of the quantum harmonic oscillator that is very parsimonious. The quantum harmonic oscillator has evenly spaced eigenstates, and the state of the system may be changed according to... 

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

Thermal Average of the Translation Operator The Driven Damped Quantum Harmonic Oscillator Quantum Representation II ... 

In this expression, according to the theory of the quantum harmonic oscillator, the operator q appearing on the right-hand side, may couple two successive eigenstates /c ) of the Hamiltonian of the harmonic oscillator. Consequently, by ignoring the scalar term p(0,0), which does not couple these states, we may write the dipole moment operator according to... 

InEq. (113), I)] (tfv is the IP time-evolution operator of the driven quantum harmonic oscillator interacting with the thermal bath,... 

Another possibility is to extract the reduced time evolution operator from the analytical solution obtained by Louisell and Walker for the reduced time-dependent density operator of a driven damped quantum harmonic oscillator. 

Appendix D shows that the IP time evolution operator of a driven quantum harmonic oscillator is given by Eq. (D.23), that is,... 

From this viewpoint, which is the most fundamental, the line shape as a whole is the sum of the diagonal matrix elements of the time evolution operator of the driven damped quantum harmonic oscillator in the IP representation with respect to the diagonal part of the Hamiltonian of this oscillator. According to Eq. (120), each diagonal element is a sum of time-dependent terms... 

Figure 16. The effects of the parity operator C2 on the ground and the first excited states of the symmetrized g and u eigenfunctions of the g and u quantum harmonic oscillators involved in the centrosymmetric cyclic dimer. (The subscripts 1 and 2 refer, respectively, to the a and b moieties of the centrosymmetric cyclic dimer).

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... 

APPENDIX D THE EVOLUTION OPERATOR OF A DRIVEN QUANTUM HARMONIC OSCILLATOR [54]... 

As a consequence, because of Eqs. (D.7) and (D.23), the full-time evolution operator (D.2) of the driven quantum harmonic oscillator takes the form ... 

It is shown in Appendix N that the action of the translation operator on the ground state (0) ) of the Hamiltonian of the quantum harmonic oscillator gives a coherent state a ) ... 

Of course, the reduced-density operator of the driven damped quantum harmonic oscillator at time t is the partial trace over the thermal bath of the full density operator ... 

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,... 

By definition, a coherent state a) is the eigenvector of the non-Hermitean lowering operator a of the quantum harmonic oscillator. Thus, the basic equation and its conjugate are, respectively,... 

Now, we may insert in front of a coherent state, the closeness relation on the eigenstates of the number operator at a of the quantum harmonic oscillator (with [a, at] = 1), in the following way ... 

Now, consider the normalized density operator pa of a system of equivalent quantum harmonic oscillators embedded in a thermal bath at temperature T owing to the fact that the average values of the Hamiltonian //, of the coordinate Q and of the conjugate momentum P, of these oscillators (with [Q, P] = ih) are known. The equations governing the statistical entropy S,... 

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,... 

Again, use the fact that the exponential operators commute with the kets and the bras appearing explicitly in this equation. Then use the orthonormality properties dealing with these kets and these bras. At last make explicit the trace involving the eigenstates of the u quantum harmonic oscillator. 

Density operator of the thermal bath. Monomer and g and u density operators of the H-bond bridge quantum harmonic oscillator for situation 0 ). 

Let us consider an example, that of the derivative operator in the orthonormal basis of Harmonic Oscillator functions. The fact that the solutions of the quantum Harmonic... 

Here a and are the usual oscillator creation and annihilation operators with bosonic commutation relations (73), and 0i,..., 1 ,..., 0Af) denotes a harmonic-oscillator eigenstate with a single quantum excitation in the mode n. According to Eq. (80a), the bosonic representation of the Hamiltonian (79) is given by... 

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