Isotropic harmonic potential

To analyze the phase dependence on the relative distance and orientation of the dipole moments, we assume that each molecule is in a ground translational state of a one- or two-dimensional optical lattice potential. We use a translational ground state wavefunction of the three-dimensional isotropic harmonic potential for a simple estimate of the mean distance a molecule travels in the ground state of an optical lattice potential. We make a harmonic approximation of the potential Vok > giving the frequency m = k /TT Jm with the corresponding translational ground state wavefunction width a = %lrru , for a molecule of mass m. A typical potential depth a molecule experiences in a lattice is Vo = ti r, where r = is... 

A full and comprehensive treatment of the incoherent scattering of an harmonic oscillator can be found in ref.[9]. We shall just give the results and the application to our systems. For a particle of mass M in an isotropic harmonic potential, one finds ... 

Figure 2.6 The potential V(r) that corresponds to the dynamical symmetry (I). The potential is nonrigid because [cf. Eq. (2.113)] the rotational spacings are comparable to the vibrational ones. Tn the harmonic limit V(r) is the potential of an isotropic harmonic oscillator.

Now, consider another life-time , Ate, which is due to the potential energy. By taking a certain isotropic potential d> (r) - say Lennard-Jones type [15] and assuming the potential to be symmetric [16], there will be a characteristic frequency, uc, associated with the harmonic potential. Thus, Ate, can be written as ... 

For w = 1 or 2 they have the general form of a radial eigenvalue problem arising from some Hamiltonian. In fact, the radial parts of the nonrelativistic hydrogenic Hamiltonian, Klein-Gordon, and second-order iterated Dirac Hamiltonians with 1/r potential can all be expressed in this form for w = 1 and suitable choices of the parameters , rj, x. Similarly, the three-dimensional isotropic harmonic oscillator radial equation has this form for w = 2. 

Consider now one of these variable and its contribution to the potential energy, z(r) = 27rg 2(7Xz(r)2. This is the potential energy of a three-dimensional isotropic harmonic oscillator. The total potential energy, Eq. (16.82) is essentially a sum over such contributions. This additive form indicates that these oscillators are independent of each other. Furthermore, all oscillators are characterized by the same force constant. We now also assume that all masses associated with these oscillators are the same, namely we postulate the existence of a single frequency Ms., sometimes referred to as the Einstein frequency of the solvent polarization fluctuations, and Ws are related as usual by the force constant... 

The great potential of the X-ray data for obtaining motional information has recently led to a molecular dynamics test197 of the standard refinement techniques that assume isotropic and harmonic motion. Since simulations have shown that the atomic fluctuations are highly anisotropic and, in some cases, anharmonic (see Chapt. VI.A.1), it is important to determine the errors introduced in the refinement process by their neglect. A direct experimental estimate of the errors resulting from the assumption of isotropic, harmonic temperature factors is difficult because sufficient data are not yet available for protein crystals. Moreover, any data set includes other errors that would obscure the analysis, and the specific correlation of temperature factors and motion is complicated by the need to account for static disorder in the crystal. As an alternative to an experimental analysis of the errors in the refinement of proteins, a purely theoretical approach has been used.197 The basic idea is to generate X-ray data from a molecular dynamics simulation... 

With the use of these we find for the kinetic and potential energies of the isotropic harmonic oscillator the following expressions ... 

We have attempted to take into account this oscillatory motion of the neighboring atoms, by averaging the potential over this motion using an uncoupled isotropic harmonic oscillator approximation. This determined the cell potential of the central atom. We assumed that the motion of the central atom can be described by the eigenfunction of a harmonic oscillator. 

Solid state physicists are familiar with the free- and nearly free-electron models of simple metals [9]. The essence of those models is the fact that the effective potential seen by the conduction electrons in metals like Na, K, etc., is nearly constant through the volume of the metal. This is so because (a) the ion cores occupy only a small fraction of the atomic volume, and (b) the effective ionic potential is weak. Under these circumstances, a constant potential in the interior of the metal is a good approximation—even better if the metal is liquid. However, electrons cannot escape from the metal spontaneously in fact, the energy needed to extract one electron through the surface is called the work function. This means that the potential rises abruptly at the surface of the metal. If the piece of metal has microscopic dimensions and we assume for simplicity its form to be spherical - like a classical liquid drop, then the effective potential confining the valence electrons will be spherically symmetric, with a form intermediate between an isotropic harmonic oscillator and a square well [10]. These simple model potentials can already give an idea of the reason for the magic numbers the formation of electronic shells. 

More recently, Fisher information has been studied as an intrinsic accuracy measure for concrete atomic models and densities [43, 44] and also for quantum mechanics central potentials [45]. Also, the concept of phase space Fisher information, where position and momentum variables are included, was analyzed for hydrogenlike atoms and the isotropic harmonic oscillator [46]. The net Fisher information measure is found to correlate well with the inverse of the ionization potential and dipole polarizability [44]. 

The diabatic potential is taken to be a two-dimensional isotropic harmonic oscillator... 

The model of N quantum particles confined in a harmonic trap has been widely used to model the Bose-Einstein condensation in trapped dilute gases [57-68], and more recently in trapped Fermi gases [69-73]. A simplified model of the trap may be constmcted assuming a non-interacting system of quanmm particles in an external isotropic harmonic trap, as a homogeneous potential, n = 2, with the following Hamiltonian H = hk = /2m) V - a x +yl +zl), where... 

We consider an electron in a spAiraically symmetric harmonic potential [9]. In polar coordinates, the Hamiltonian of such a three-dimensional isotropic harmonic oscillator (HO) is given by... 

Fig. 5.12. Comparison of the potential dependence in isotropic (a) and anisotropic (b) components to the second harmonic intensity from Ag(l 11) at the indicated incident wavelengths. The solution was 0.25 M Na2S04 at a pH of 3.5. An incident angle of 45° was used. From Ref. 132 and 137.

In our model with the aid of parameter e we continuously pass from a zero bistable potential (magnetically isotropic particle) to a pair of symmetric wells of infinite depth (highly anisotropic particle). For the magnetic case, as for those of Refs. 21 and 22, a crucial circumstance enabling the harmonic suppression is that an antisymmetric contribution (bias) should be present in the potential. On the other hand, the presence of a symmetric contribution turns out to be an... 

The energetics of such atomic motion can be investigated. If the probability density function is a Gaussian function, the potential energy in which the atom vibrates will be isotropic and harmonic and will have a normal Boltzmann distribution over energy levels. This potential energy will have the form ... 

The following sections develop three subjects the classical approximations for the strain/stress in isotropic polycrystals, isotropic polycrystals under hydrostatic pressure and the spherical harmonic analysis to determine the average strain/stress tensors and the intergranular strain/stress in textured samples of any crystal and sample symmetry. Most of the expressions that are obtained for the peak shift have the potential to be implemented in the Rietveld routine, but only a few have been implemented already. 

The discussion has been focussed on the hydrogen atom, but some of the results are applicable also for the confinement by a dihedral angle of any system with a central potential, or simply with rotational symmetry around the z-axis, including isotropic and anisotropic harmonic oscillators. 

Another three-dimensional problem which is soluble in Cartesian coordinates is the three-dimensional harmonic oscillator, a special case of which, the isotropic oscillator, we have treated in Section la by the use of classical mechanics. The more general system consists of a particle bound to the origin by a force whose components along the x, y, and z axes are equal to —kzx, —Jtvy, and — k,z, respectively, where kx, kv, kt are the force constants in the three directions and x, y, z are the components of the displacement along the three axes. The potential energy is thus... 

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