Positive space

It has long been known from statistical mechanical theory that a Bose-Einstein ideal gas, which at low temperatures would show condensation of molecules into die ground translational state (a condensation in momentum space rather than in position space), should show a third-order phase transition at the temperature at which this condensation starts. Nonnal helium ( He) is a Bose-Einstein substance, but is far from ideal at low temperatures, and the very real forces between molecules make the >L-transition to He II very different from that predicted for a Bose-Einstein gas. 

Figure Bl.28.9. Energetic sitiration for an n-type semiconductor (a) before and (b) after contact with an electrolyte solution. The electrochemical potentials of the two systems reach equilibrium by electron exchange at the interface. Transfer of electrons from the semiconductor to the electrolyte leads to a positive space charge layer, W. is the potential drop in the space-charge layer.

Non-thennal plasmas in contact with insulating walls (substrate) have an important property. The plasma with the hot electrons is positively charged relative to the wall (self-bias). A sheath with a positive space charge and an electric field is fonned between the wall and the plasma. The hot electrons travel faster to the wall than the heavy... 

Caustics The above formulae can only be valid as long as Eq. (9) describes a unique map in position space. Indeed, the underlying Hamilton-Jacobi theory is only valid for the time interval [0,T] if at all instances t [0, T] the map (QOi4o) —> Q t, qo,qo) is one-to-one, [6, 19, 1], i.e., as long as trajectories with different initial data do not cross each other in position space (cf. Fig. 1). Consequently, the detection of any caustics in a numerical simulation is only possible if we propagate a trajectory bundle with different initial values. Thus, in pure QCMD, Eq. (11), caustics cannot be detected. 

Fig. 1. Illustration of a caustic. Different trajectories sample the probability distribution. If they cross each other in position space, the transport or probability density is not longer unique and the approximation might break down.

We will refer to this model as to the semiclassical QCMD bundle. Eqs. (7) and (8) would suggest certain initial conditions for /,. However, those would not include any momentum uncertainty, resulting in a wrong disintegration of the probability distribution in g as compared to the full QD. Eor including an initial momentum uncertainty, a Gaussian distribution in position space is used... 

In order to be useful in practice, the effective transport coefficients have to be determined for a porous medium of given morphology. For this purpose, a broad class of methods is available (for an overview, see [191]). A very straightforward approach is to assume a periodic structure of the porous medium and to compute numerically the flow, concentration or temperature field in a unit cell [117]. Two very general and powerful methods are the effective-medium approximation (EMA) and the position-space renormalization group method. 

As a second method to determine effective transport coefficients in porous media, the position-space renormalization group method will be briefly discussed. 

Although in principle a powerful and elegant method, the position-space renormalization group method yields very complex expressions for the renormalized... 

Here V is the volume of the Bom-von Karman region, i.e. that part of position space which is repeated as a result of the fundamental periodic boundary conditions. The integration in (11.1) is carried out over that region, which we denote by BK. 

The following relation and its inverse hold between the elements of the number density matriees in momentum and position space [21] ... 

If the elements of the number density matrix in position space are invariant under all operations of the space group, i.e. if... 

The symmetry properties of the momentum space wave functions can be obtained either from their position space counterparts or more directly from the counterpart of the Hamiltonian in momentum space. 

Irrespective of the type of extended system we are interested in we impose periodic boundary conditions in position space - "the large period" BK. Such conditions imply a discretisation of momentum and reciprocal space 27] which means that integrations are replaced by summations ... 

The equations to be fulfilled by momentum space orbitals contain convolution integrals which give rise to momentum orbitals ( )(p-q) shifted in momentum space. The so-called form factor F and the interaction terms Wij defined in terms of current momentum coordinates are the momentum space counterparts of the core potentials and Coulomb and/or exchange operators in position space. The nuclear field potential transfers a momentum to electron i, while the interelectronic interaction produces a momentum transfer between each pair of electrons in turn. Nevertheless, the total momentum of the whole molecule remains invariant thanks to the contribution of the nuclear momenta [7]. 

The calculation of e in momentum space is analogous to that in position space. Starting with the r-representation, and expressing the quantity F(r)(pi(r) as the inverse Fourier transform of [F(r) (pi(r)]T(p), one easily finds that ... 

The one-electron energy ej has the same expression in the p-representation as in the position space where the different contributions can be expressed as follows ... 

Because of the terms Ir-RAl and Ir-rT explicit solutions to Eq. 3 carmot be obtained in position space. In such cases approximate solutions are usually expressed as truncated linear combinations of basis functions (LCAO expressions). In spite of its successes, the LCAO approximation experiences various difficulties (truncation limits, nature of the basis functions, etc.) hard to estimate and which are not entirely controllable [51]. 

Spin densities determine many properties of radical species, and have an important effect on the chemical reactivity within the family of the most reactive substances containing free radicals. Momentum densities represent an alternative description of a microscopic many-particle system with emphasis placed on aspects different from those in the more conventional position space particle density model. In particular, momentum densities provide a description of molecules that, in some sense, turns the usual position space electron density model inside out , by reversing the relative emphasis of the peripheral and core regions of atomic neighborhoods. 

Figure 5.1 illustrates the Fourier transform (FT) of a simple function, viz., a Gaussian. The relatively sharp Gaussian function with the exponent a = 1 depicted in Figure 5.1a, yields a diffuse Gaussian (in dotted line) in momentum space. A flat Gaussian function in position space with a = 0.1, transforms to a sharp one (cf. Figure 5.1b). Connected by an FT, the wave functions in position and momentum...

The Schrodinger equation in momentum space for a single particle system is obtained on taking an FT of its position space counterpart in the form... 

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