Representation momentum

The potential energy part is diagonal in the coordinate representation, and we drop the hat indicating an operator henceforth. The kinetic energy part may be evaluated by transfonning to the momentum representation and carrying out a Fourier transform. The result is... 

Error . Writing the potential, V(x), in the momentum representation is not quite as straightforward. The relationship between position and momentum is realized in their... 

This eommutation relation is easily verified in the eoordinate representation leaving x untouehed (x = x ) and using the above definition for p. In the momentum representation... 

We next turn our attention to the problem of finding the position operator q, of which the localized state (t) is an eigenfunction with eigenvalue y. That this operator is not given by q = Vk in the momentum representation becomes dear upon noting that the operator... 

Figure 8.1 Schematic structure of the Hamiltonian matrix for a collision system in an external field expressed in the total angular momentum representation.

This comes about by making the primitive semiclassical approximation in a momentum representation and then Fourier transforming this to obtain (x2 G(E) xi). We found, however, that the direct use of Eq. (3.5) gives quite unstable and erratic results (vide infra). Excellent results are obtained, though, if Eq. (3.5) is subjected to a modified Filinov transformation (MFT) of the type used by Makri and Miller [21] this replaces Eq. (3.5) by... 

So far we considered the density operators in coordinate representation. In many cases, especially for homogeneous systems, it is more convenient to use the momentum representation... 

The essential information about transport properties in many-particle systems is given by the single-particle density matrix or by the singleparticle Wigner distribution. The equations of motion (1.18) and (1.23) for these important quantities are called kinetic equations. For the further consideration we write the latter equation in the momentum representation ... 

Using the relation, the binary density matrix in momentum representation may be expressed in terms of scattering wave functions. 

This formula exactly coincides with the one obtained by integrating the Bethe s cross section (4.13) over q, provided we take qmin = (o0Jttv [when deriving (4.33) we took qmin = relation between q and b, namely, small q corresponds to large b, and vice versa qmax b f a l. Fano150 has made the transformation from momentum representation to a representation in terms of the impact parameter b in the Bethe s formula (4.13) and has obtained an expression for the differential cross section coinciding with (4.44). 

To take advantage from the pseudo-angular momentum representation we shall employ the technique of the irreducible tensor operators as suggested in Ref. [10]. One can easily establish the following interrelations between the matrices Orr and the orbital angular momentum operators ... 

The projection integrals (Yjo y>7) can be interpreted as the (discrete) angular momentum representation of the initial bending wavefunction in the electronic ground state. Employing the semiclassical limit for the spherical harmonics,... 

After about 50 fs (t) has reached the asymptotic region where the torque is essentially zero and the distribution does not alter any further, i.e., the dissociation is over. Figure 10.6 illustrates rotational excitation in the angular momentum representation, whereas Figure 10.2(b) manifests rotational excitation in the coordinate picture. 

The GF of d-electrons takes into account Coulomb interaction on a site, therefore it has a self-energy X j P and obeys the Dyson equation. It can be written in momentum representation as ... 

Iterations in the limit of two first orders determine the terminal part A = Aq + Ai +. ... In the momentum representation one has... 

As an example, suppose for an A-electron system that energy E is approximated by an orbital functional [ , ], which depends on one-electron orbital wave functions , and on occupation numbers n, through a variational A-electron trial wave function A momentum displacement is generated by U = cxp(- 7r D), where D = r - In (he momentum representation of the orbital wave functions,... 

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