Harmonic oscillator Hamiltonian

In tests using the moving ID Hamiltonian harmonic oscillator, (5.25), a velocity Verlet integrator [24] combined with ttapezoidal integration of W (/.) performed well when compared to the analytic solution. An interesting analysis of how... 

The vibrational part of the molecular wave function may be expanded in the basis consisting of products of the eigenfunctions of two 2D harmonic oscillators with the Hamiltonians ffj = 7 -I- 1 /2/coiPa atid 7/p = 7p - - 1 /2fcppp,... 

Now, consider the general case of a V2 multiply excited degenerate vibrational level where V2 > 2, which is dealt with by solving the Schrddinger equation for the isotropic 2D harmonic oscillator with the Hamiltonian assuming the fonn [95]... 

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]. 

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

There are just a few Hamiltonians for which the path integration can be carried out exactly, and the best known case is the driven harmonic oscillator [Feynman and Hibbs, 1965 ... 

Proceeding now to the instanton treatment of the Hamiltonian (5.24) we observe that the spectrum of quasienergies differs from that of the unperturbed harmonic oscillator, f Q) = 0, only by a shift independent of n [Bas et al. 1971],... 

The Hamiltonian operator H for the harmonic oscillator is given in equation (4.12). The quantity c) is then determined as follows... 

The Hamiltonian operator for the unperturbed harmonic oscillator is given by equation (4.12) and its eigenvalues and eigenfunctions are shown in equations (4.30) and (4.41). The perturbation H is... 

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. 

To further illustrate the theory, we apply Jarzynski s identity to the analytically solvable example of a ID moving harmonic oscillator with Hamiltonian... 

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 Eq. (3.1), //<, is a three- or four- dimensional harmonic oscillator Hamiltonian... 

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 ... 

In the general case R denotes a set of coordinates, and Ui(R) and Uf (R) are potential energy surfaces with a high dimension. However, the essential features can be understood from the simplest case, which is that of a diatomic molecule that loses one electron. Then Ui(R) is the potential energy curve for the ground state of the molecule, and Uf(R) that of the ion (see Fig. 19.2). If the ion is stable, which will be true for outer-sphere electron-transfer reactions, Uf(R) has a stable minimum, and its general shape will be similar to that of Ui(R). We can then apply the harmonic approximation to both states, so that the nuclear Hamiltonians Hi and Hf that correspond to Ui and Uf are sums of harmonic oscillator terms. To simplify the mathematics further, we make two additional assumptions ... 

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