Total momentum density

On the other hand, summation of Eqs. (6.10) and (6.11) yields the equation for the total momentum density,... 

It is widely known that total momentum densities for atoms are not always monoton-ically decreasing [10], In fact the degree of non-monotonicity is dependent on the degree of p-population in an atom. This fact is visible as well in the shape of spa hybrids in momentum space. 

Specifically, when all components are mobile, the total momentum density (i.e., that of the fluid and particles) is conserved hence, a mode whose lifetime diverges as g 0 is to be expected. ... 

The one-particle distribution function fp specifies both the total particle density p and momentum density pu, where u is the average fluid velocity. 

Total electric charge operator, 542 Total energy operator, 506,542 Total momentum operator, 506,542 Total number density operator, 452... 

An important phenomenon is the Mott effect. At given temperature T and total momentum P, the binding energy of the deuteron bound state vanishes at the density n 1"" (P. T) due to the Pauli blocking. As a consequence, the... 

The modification of the three and four-particle system due to the medium can be considered in cluster-mean field approximation. Describing the medium in quasi-particle approximation, a medium-modified Faddeev equation can be derived which was already solved for the case of three-particle bound states in [9], as well as for the case of four-particle bound states in [10]. Similar to the two-particle case, due to the Pauli blocking the bound state disappears at a given temperature and total momentum at the corresponding Mott density. 

Figure 3. The dependence of the pairing gap in the LOFF phase on the density asymmetry and the total momentum of the condensate [12].

Eq. (5.34). However, it is possible to construct approximate wavefunctions that lead to electron momentum densities that do not have inversion symmetry. Within the Born-Oppenheimer approximation, the total electronic system must be at rest the at-rest condition... 

Figure 5.4. The radial electron density D(r) (left) and radial momentum density I p) (right) for I S Be li contribution (dotted), 2s contribution (dashed), and total (soUd).

Figure 5.5. Types of electron momentum densities no(p) >n atoms. Solid lines are used for the total density, whereas dotted lines and crosses indicate the contribution from the outermost s and p orbitals, respectively. Top left a type I density for the potassium atom. Top right a typical type II density for the argon atom. Bottom left a typical type III density for the silver atom. Bottom right a closeup of rio(p) for the silver atom showing the minimum and secondary maximum. Adapted from Thakkar [29].

Figure 2.1. Schematic picture illustrating the local probe character in XES for N2 adsorbed on a Ni surface. From the total charge density (gray envelope) valence electrons with p-angular momentum (contour lines) decay into the N Is core hole. From Ref. [3].

We measure the response of an elongated BEC to a two-photon Bragg pulse. If the duration of the pulse is long, the total momentum transferred to the condensate exhibits a nontrivial behavior which reflects the structure of the underlying Bogoliubov spectrum (see Fig. 1). It is thus possible to perform a spectroscopic analysis in which axial phonons with a different number of radial nodes are resolved [Steinhauer 2003], The local density approximation is shown to fail in this regime, while the observed data agree well with the... 

Microscale fluid turbulence is, by deflnition, present only when the continuous fluid phase is present. The coefficients Bpv describe the interaction of the particle phase with the continuous phase. In contrast, Bpvf models rapid fluctuations in the fluid velocity seen by the particle that are not included in the mesoscale drag term Ap. In the mesoscale particle momentum balance, the term that generates Bpv will depend on the fluid-phase mass density and, hence, will be null when the fluid material density (pf) is null. In any case, Bpv models momentum transfer to/from the particle phase in fluid-particle systems for which the total momentum is conserved (see discussion leading to Eq. (5.17)). 

Kaijser and Smith [17] have presented analytic forms for many of the Slater-type orbitals and to Gaussian-type orbitals in both spherical harmonic and Cartesian form. The total momentum-space electron density p(p) is given by... 

The Fourier transform (Eq. (1)) preserves direction, in the sense that one can refer to components of the total momentum in any particular direction. For example, one can distinguish components along any Cartesian axis, or paral-lel/perpendicular to a bond or plane. A further consequence of Eq. (1) is that the momentum density p p) possesses the same symmetry elements as its r-space... 

Fig. 19.3. Types of electron momentum densities Uip) — Hgip) in atoms. Solid lines are used for the total density. Left a Type 1 density for the beryllium atom the contribution from the 2s orbital is indistinguishable from the total density. Right a typical Type 11 density for the neon atom the 2s and 2p contributions are shown as dashed and dotted lines, respectively.

In a binary mixture there are six conserved variables (a) the energy, (b) linear momentum (three components), (c) the solute concentration, and (d) the total fluid density. There is some freedom in specifying the composition of the fluid. For convenience we choose the variables used by Landau and Lifshitz (1960), and specify the composition of the fluid by giving the mass fraction of solute, that is c — Mi/A/. The hydrodynamic state of the binary mixture is then specified by giving the local values of the mass density mp, the mass fraction c, the temperature T, and the local velocity u. 

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