Harmonic oscillator Hamiltonian systems

Suppose that Hamiltonian 7/(p, q) is expressed in a region around a saddle point of interest as an expansion in a small parameter e, so that the zero-order Hamiltonian Hq is regular in that region specifically, it is written as a sum of harmonic-oscillator Hamiltonians. Such a zero-order system is a function of action variables J of Hq only, and it does not depend on the conjugate angle variables 0. The higher-order terms of the Hamiltonian are expressed as sums of... 

Consider a system with a time-independent Hamiltonian H that involves parameters. An obvious example is the molecular electronic Hamiltonian (13.5), which depends parametrically on the nuclear coordinates. However, the Hamiltonian of any system contains parameters. For example, in the one-dimensional harmonic-oscillator Hamiltonian operator - f /2m) (f/dx ) + kx, the force constant is a parameter, as is the mass m. Although is a constant, we can consider it as a parameter also. The stationary-state energies E are functions of the same parameters as H. For example, for the harmonic oscillator... 

The Hamiltonian operator of the system is a sum of four harmonic oscillator Hamiltonians ... 

Now consider a system of N one-dimensional harmonic oscillators with the Hamiltonian... 

A mapping is said to be symplectic or canonical if it preserves the differential form dp A dq which defines the symplectic structure in the phase space. Differential forms provide a geometric interpretation of symplectic-ness in terms of conservation of areas which follows from Liouville s theorem [14]. In one-degree-of-freedom example symplecticness is the preservation of oriented area. An example is the harmonic oscillator where the t-flow is just a rigid rotation and the area is preserved. The area-preserving character of the solution operator holds only for Hamiltonian systems. In more then one-degree-of-freedom examples the preservation of area is symplecticness rather than preservation of volume [5]. 

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. 

The Hamiltonian function for a system of bound harmonic oscillators is, in the most general form, a sum of two positively definite quadratic forms composed of the particle momentum vectors and the Cartesian projections of particle displacements about equilibrium positions ... 

This is not as useful as Eq. (19.4) because products of different coordinates appear in the second term. However, the symmetry properties of this term ensure the existence of a coordinate system in which the cross-terms can be eliminated and the nuclear Hamiltonian reduces to a sum of harmonic oscillator terms ... 

The dynamics of the normal mode Hamiltonian is trivial, each stable mode evolves separately as a harmonic oscillator while the imstable mode evolves as a parabolic barrier. To find the time dependence of any function in the system phase space (q,pq) all one needs to do is rewrite the system phase space variables in terms of the normal modes and then average over the relevant thermal distribution. The continuum limit is introduced through use of the spectral density of the normal modes. The relationship between this microscopic view of the evolution... 

The model consists of a system S in interaction with a bath B. The system S is a harmonic oscillator with frequency Q. Its canonical variables are q0, p0 and its Hamiltonian is... 

All of the above methods are easily extended to more degrees of freedom. There are also approaches that rely on a certain form of multidimensional Hamiltonian which should be represented as some system coupled to a bath of harmonic oscillators, as described in Section 2.2. In Chapter 5 we see that integration over these bath... 

A complete treatment of this derivation can be found in Ref. [19]. The first three terms in the kinetic energy operator indicates the presence of a 3D harmonic oscillator, and the final two terms indicate the presence of a 2D rotator (as for the hydrogen atom). A similar conclusion was made by Auberbach et al. [22] where they use a semi-classical quantisation method and the molecule is said to undergo a unimodal distortion and then the semi-classical Hamiltonian is found to be separated into two parts - a harmonic oscillator part with three vibrational coordinates and a rotational part with two rotational coordinates. However, more progress in terms of specifying the wavefunctions of the system can be made by following a different approach. 

One point which has not been addressed in the example of the time-independent harmonic oscillator is the non-perturbative treatment of the time dependence in the system Hamiltonians. Both the TL and the TNL non-Markovian theories employ auxiliary operators or density matrices, respectively, and can be applied in strongly driven systems [29,32]. This point will be shown to be very important in the examples for the molecular wires under the influence of strong laser fields. 

We consider the same reaction model used in previous studies as a simple model for a proton transfer reaction. [31,57,79] The subsystem consists of a two-level quantum system bilinearly coupled to a quartic oscillator and the bath consists of v — 1 = 300 harmonic oscillators bilinearly coupled to the non-linear oscillator but not directly to the two-level quantum system. In the subsystem representation, the partially Wigner transformed Hamiltonian for this system is,... 

In order to obtain the potential energy surfaces associated with chemical reactions we, typically, need the lowest eigenvalue of the electronic Hamiltonian. Unlike systems such as a harmonic oscillator and the hydrogen atom, most problems in quantum mechanics cannot be solved exactly. There are, however, approximate methods that can be used to obtain solutions to almost any degree of accuracy. One such method is the variational method. This method is based on the variational principle, which says... 

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,... 

In order to solve this problem, it is possible to use the Hamiltonian procedure of classical mechanics [8], Hence, the classical Hamiltonian of a system of coupled harmonic oscillators can be written as follows [7] ... 

The simplest example of a classical or quantum dissipative system is a particle evolving in a potential V(x) and coupled linearly to a fluctuating dynamical reservoir or bath. If the bath is only weakly perturbed by the system, it can be considered as linear, described by an ensemble of harmonic oscillators. Starting from the corresponding system-plus-bath Hamiltonian and using some convenient approximations, it is possible to get a description of the dissipative dynamics of the system. 

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