Hamiltonian operator three-dimensional

In Table I, 3D stands for three dimensional. The symbol symbol in connection with the bending potentials means that the bending potentials are considered in the lowest order approximation as already realized by Renner [7], the splitting of the adiabatic potentials has a p dependence at small distortions of linearity. With exact fomi of the spin-orbit part of the Hamiltonian we mean the microscopic (i.e., nonphenomenological) many-elecbon counterpart of, for example, The Breit-Pauli two-electron operator [22] (see also [23]). 

Ehrenfest trajectory for three-dimensional D + H2 generated by the RWP method, that is, the modified Hamiltonian operator f H). Dotted curves in (b) correspond to the Ehrenfest trajectory determined by the usual Schrddinger equation. See text for further details. 

The simplest atomic system that we can consider is the hydrogen atom. To obtain the Hamiltonian operator for this three-dimensional system, we must replace the operator d2/dx2 by the partial differential operator... 

A stress that is describable by a single scalar can be identified with a hydrostatic pressure, and this can perhaps be envisioned as the isotropic effect of the (frozen) medium on the globular-like contour of an entrapped protein. Of course, transduction of the strain at the protein surface via the complex network of chemical bonds of the protein 3-D structure will result in a local strain at the metal site that is not isotropic at all. In terms of the spin Hamiltonian the local strain is just another field (or operator) to be added to our small collection of main players, B, S, and I (section 5.1). We assign it the symbol T, and we note that in three-dimensional space, contrast to B, S, and I, which are each three-component vectors. T is a symmetrical tensor with six independent elements ... 

The eigenvalue problem for the Hamiltonian H [Eq. (2.92)] can be solved in closed form whenever H does not contain all the elements but only a subset of them, the invariant or Casimir operators. For three-dimensional problems there are two such situations corresponding to the two chains discussed in the preceding sections. We begin with chain (I). Restricting oneself only to terms up to quadratic in the elements of the algebra, one can write the most general Hamiltonian with dynamic symmetry (I) as... 

In order to apply the representation theory of so(2,1) to physical problems we need to obtain realizations of the so(2, 1) generators in either coordinate or momentum space. For our purposes the realizations in three-dimensional coordinate space are more suitable so we shall only consider them (for N-dimensional realizations, see Cizek and Paldus, 1977, and references therein). First we shall show how to build realizations in terms of the radial distance and momentum operators, r, pr. These realizations are sufficiently general to express the radial parts of the Hamiltonians we shall consider linearly in the so(2,1) generators. Then we shall obtain the corresponding realizations of the so(2,1) unirreps which are bounded from below. The basis functions of the representation space are simply related to associated Laguerre polynomials. For finding the eigenvalue spectra it is not essential to obtain these explicit realizations of the basis functions, since all matrix elements can... 

Inside the box, where the potential energy is zero everywhere, the Hamiltonian is simply the three-dimensional kinetic energy operator and the Schrddinger equation reads... 

A natural extension of the previous discussion is to consider a particle in the three-dimensional analogue of the square well for the one-dimensional box. Since the motions in the three directions are independent from each other, the hamiltonian in Eq. (2.42) (with F = 0) can be written as a sum of operators, one in each variable x, y or z, and ip can be written as a product of three functions, each one being a function of a different variable ip(x,y,z) = X(x)Y(y)Z(z). The problem to solve is then similar to that of the onedimensional box, taken three times. The energy is the sum of three components similar to Eq. (2.60), involving three quantum numbers ... 

The electron Hamiltonian (15) describes the so-called orbital exchange coupling in a three-dimensional (3D) crystal lattice. The Pauli matrices, cr O ), have the same properties as the z-component spin operator with S = As a i) represents not a real spin but orbital motion of electrons, it is called pseudo spin. For the respective solid-state 3D-exchange problem, basic concepts and approximations were well developed in physics of magnetic phase transitions. The key approach is the mean-fleld approximation. Similar to (8), it is based on the assumption that fluctuations, s(i) = terms quadratic in s i) can be neglected. We do not go into details here because the respective solution is well-known and discussed in many basic texts of solid state physics (e.g., see [15]). 

Enhancement of two-photon cross-sections by two-dimensional and three-dimensional arrangements of monomers has been demonstrated with fluorene V-shapes and dendrimeric structures. Such multidimensional structures lead to lower two-photon absorptions than Hnear oUgomers, but they have better one-photon transparencies. An accurate calculation of large exitonic systems is obtained by diagonalizing the Hamiltonian operator on a reduced basis set. 

There are several theoretical approaches that can be used to calculate the dynamics and correlation properties of two atoms interacting with the quantized electromagnetic held. One of the methods is the wavefunction approach in which the dynamics are given in terms of the probability amplitudes [9]. Another approach is the Heisenberg equation method, in which equations of motion for the atomic and held operators are found from the Hamiltonian of a given system [10], The most popular approach is the master equation method, in which the equation of motion is found for the density operator of an atomic system weakly coupled to a system regarded as a reservoir [7,8,41], There are many possible realizations of reservoirs. The typical reservoir to which atomic systems are coupled is the quantized three-dimensional multimode vacuum held. The major advantage of the master equation is that it allows us to consider the evolution of the atoms plus held system entirely in terms of atomic operators. 

We may generalize the result for the particle in the three-dimensional box in the following way. If the Hamiltonian operator can be written as a sum of groups of terms, each of which depends only on one coordinate or one set of coordinates, then the wave function can be written as a product of functions each of which depends only on the one coordinate or the one set of coordinates correspondingly, the total energy is the sum of the energies associated with each coordinate or each set of coordinates. 

In three-dimensional space the positions of the two structureless atoms are specified by six coordinates whidi, after separating the motion of the center of mass, are reduced to three. Introducing the reduced mass fi of the system (/Lt = [mim2/(mim2)] ), we obtain a hamiltonian which is the sum of a kinetic part represented by the Laplacian operator acting in a three-dimensional space plus a potential energy V (r), which is a function only of the distance, r, between the two particles ... 

To summarize, we have seen that it is possible to explain the degeneracy pattern characteristic of a three-dimensional harmonic oscillator by introducing, as a proper symmetry group of the Hamiltonian operator (here referred to as the degeneracy group), a group of unitary transformations in a three-dimensional complex space. 

To reiterate, we prefer to describe the one-dimensional model first because of its mathematical simplicity in comparison to the three-dimensional model. From a strictly historical point of view, the situation is slightly more involved. The vibron model was officially introduced in 1981 by lachello [26]. In his work one can find the fundamental idea of the dynamical symmetry, based on U(4), for realizing an algebraic version of the three-dimensional Hamiltonian operator of a single diatomic molecule. After this work, many other realizations followed (see specific... 

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