Momentum density expectation values

By means of a modification of the TFD method in the near nnclear region for the electron and energy densities, which introduces exact asymptotic properties, radial expectation values and the atomic density at the nucleus are evaluated, comparing fairly closely to the HF results, with a large improvement of the TF estimates. In addition to this, momentum expectation values can be estimated from semiclassical relations. 

Among the average properties which play a special role in the study of quantum fermionic systems are the radial expectation value (r ), the momentum expectation value (j9 ) and the atomic density at the nucleus p(0) = <5(r)>. These density-dependent quantities are defined by... 

The momentum expectation values (p ) are not directly related to the electron density, but to the wave function via its Fourier transform, the momentum density. However we can make use of a semiclassical relation for a local Fermi gas for estimating these values ... 

We can conclude that the present method of correcting TF calculations provides adequate estimations of expectation values for ground state atoms taking into account the simplicity of the model and it self-consistent nature, where no empirical parameters are used. It provides information about the asymptotic behaviour of quantities such as p(0) and (r 2) that cannot be evaluated with the standard semi classical approach and allow us to estimate momentum expectation values which are not directly related to the density in an exact way. 

In the case of the flux of mass, the result is the normal component of pua. But for the flux of momentum and energy, in general the flux density is not the normal component of a vector or tensor function of (t, x), since it will depend on the extended shapes of if and Y. But in the case of short-range forces and slowly varying p, ua, E, it can be shown to have this form with sufficient approximation. Thus one is led to the familiar pressure tensor and heat flow vector Qa, both as functions of (t, x). It is to be emphasized that the general expression of these quantities involves not only expected values of products of momenta (or velocities), but the effect of intermolecular forces. 

It is a fundamental fact of quantum mechanics, that a spin-independent Hamiltonian will have pure spin eigenstates. For approximate wave functions that do not fulfill this criterion, e.g. those obtained with various unrestricted methods, the expectation value of the square of the total spin angular momentum operator, (5 ), has been used as a measure of the degree of spin contamination. is obviously a two-electron operator and the evaluation of its expectation value thus requires knowledge of the two-electron density matrix. 

Each function in the superposition has been assigned a weight equal to the normalization constant [8.33], This means that each cosine function in the supeiposition has an identical probability contribution to expectation value for momentum (see Justification 8.4). Each cosine function contributes with a probability equal to I /m. Furthermore, each cosine function represents a particle momentum that is proportional to the argument n because (d (n. jr)/djc ) (the differential component of the squared momentum operator) is proportional to n. The following plot [Fig. 8.1(b)] of momentum probability against momentum, as represented by n, is an interesting contrast to the plot of probability density against position. 

The calculations include, as said previously, overlaps, conditional probability distributions of the electron probability densities, and these observables oscillator strengths, quadrupole moments (for states with total angular momentum quantum numbers of 1 or more) and expectation values (pi p2)/( pi i>2 )- (Distributions of this last quantity have also been computed, in preparation for two-electron ionization experiments by electron impact, but are not reported here.) We can proceed to summarize these indicators and then examine them and ask how well each model performs. 

Fortunately, the Zeeman interaction term in Eq. (6.102) can be assembled with the inner product of the vector potential and the momentum in Eq. (6.105) by using the magnetization density operator, m, corresponding to the magnetic dipole moment, m, whose expectation value is the reverse sign of the first energy derivative in terms of magnetic field, as... 

It was noted in Section 5.3 that when the 1-electron density matrix is written in the form (5.3.12) the difference between the a and /3 components allows us to define a resultant spin density, essentially as the excess density of up-spin electrons compared with down-spin. In recent years, with the development of magnetic resonance techniques, this quantity has acquired great importance. To indicate its origin we note tiiat fhe expectation value of the z component of spin angular momentum may be written, using for clarity the explicit form (5.2.12),... 

The charge density P(r) and spin density Qz r) are examples of point properties or sub-observables (Hirschfelder, 1977). Their values are inferred (never directly measured) by reference to an integral such as in (5.2.20) that defines an expectation value. Thus VP r) in (5.2.20) may be interpreted as a potential-energy density , since it is the contribution per unit volume to Ve evaluated at point r and similarly, in (5.9.3), Qz(r) is interpreted as a density of spin angular momentum. 

As we should expect, increasing the amount of p character, by increasing the value of a, will shift the maxima of the momentum distribution, to higher values of p. This fact well illustrates Epstein and Tanner s Hybrid Orbital Principle, which states that increased p-character in an s-p type hybrid orbital results in increased density at high momentum . [11]... 

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