Quantum harmonic oscillator Hamiltonians

Some Properties of Coherent States Expansion of the Coherent State on the Eigenvectors of the Quantum Harmonic Oscillator Hamiltonian... 

The eigenvalue equations of the quantum harmonic oscillators Hamiltonians Hpree and H° given by Eqs. (21) and (15) are, respectively,... 

N.l EXPANSION OF THE COHERENT STATE ON THE EIGENVECTORS OF THE QUANTUM HARMONIC OSCILLATOR HAMILTONIAN... 

As a consequence, the expansion (N.2) of the coherent state on the eigenkets of the quantum harmonic oscillator Hamiltonian, gives... 

Then, owing to the orthonormality (n m) = 8 m of the eigenstates of the quantum harmonic oscillator Hamiltonian, the above expression reduces to... 

Owing to Eq. (N.6), this last result shows that the action of the translation operator on the ground state of the quantum harmonic oscillator Hamiltonian, generates a coherent state ... 

In 1999 Somaroo et al. [21] reported the first NMR implementation of a quantum simulation experiment a truncated harmonic oscillator. This is a classical problem, with many applications in physics. The quantum harmonic oscillator Hamiltonian is described by ... 

The ground-state effective Hamiltonian is diagonal with eigenvalues ha n + 5], whereas the excited state one is that of a driven quantum harmonic oscillator that must lead to coherent states. 

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 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... 

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. 

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... 

It may be observed that the two Hamiltonians (269) are those of quantum harmonic oscillators, whereas Hamiltonian (270) is that of a driven damped quantum harmonic oscillator, and Hamiltonians (271) are those of driven undamped quantum harmonic oscillators perturbed by the Davydov coupling... 

Consider the effective Hamiltonian (47) of a driven quantum harmonic oscillator. Since it is not diagonal, it may be suitable to diagonalize it with the aid of a canonical transformation that will affect it or its equivalent form (50), but not that of (46) or its equivalent expression (49), which is yet to be diagonal [13]. 

Consider the effective Hamiltonian (50) of the H-bond bridge when the fast mode is in its first excited state. It is a driven quantum harmonic oscillator, that is,... 

Here, H is the Hamiltonian of the quantum harmonic oscillator coupled to the thermal bath, which is that given by Eq. (1.1). 

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,... 

Hamiltonians (P.26) are those of the undriven quantum harmonic oscillator describing the H-bond bridge moieties a and b. Hamiltonian (P.27) is that of the driven quantum harmonic oscillators describing the a H-bond bridge moiety. Finally, Hamiltonians (P.29) are dealing with the coupling of the H-bond bridge with the thermal bath, whereas Hamiltonian (P.28) is that of the thermal bath. 

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

To simplify our notation, we will suppress in what follows the polarization vector Cf, that is, the vector k will be taken to denote both wavevector and polarization. The time evolution of a mode k of frequency ct>k is determined by a harmonic oscillator Hamiltonian, and its quantum state—by the... 

Evaluate fm H f ) if (a) H is the harmonic-oscillator Hamiltonian operator and f and f are harmonic-oscillator stationary-state wave functions with vibrational quantum numbers m and n (b) H is the particle-in-a-box H and / and f are particle-in-a-box energy eigenfunctions with quantum numbers m and n. 

When the quantum mechanical Hamiltonian for vibration is constructed from Eq. (22.4-16) there are 3 — 5 or 3n — 6 terms, each one of which is a harmonic oscillator Hamiltonian operator. The variables can be separated, and the vibrational Schrodinger equation is solved by a vibrational wave function that is a product of 3 - 5 or 3n — 6 factors ... 

The quantum mechanical hamiltonian for a one-dimensional harmonic oscillator is given by... 

Instead of the quantity given by Eq. (15), the quantity given by Eq. (10) was treated as the activation energy of the process in the earlier papers on the quantum mechanical theory of electron transfer reactions. This difference between the results of the quantum mechanical theory of radiationless transitions and those obtained by the methods of nonequilibrium thermodynamics has also been noted in Ref. 9. The results of the quantum mechanical theory were obtained in the harmonic oscillator model, and Eqs. (9) and (10) are valid only if the vibrations of the oscillators are classical and their frequencies are unchanged in the course of the electron transition (i.e., (o k = w[). It might seem that, in this case, the energy of the transition and the free energy of the transition are equal to each other. However, we have to remember that for the solvent, the oscillators are the effective ones and the parameters of the system Hamiltonian related to the dielectric properties of the medium depend on the temperature. Therefore, the problem of the relationship between the results obtained by the two methods mentioned above deserves to be discussed. 

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