Amplitude momentum space

Let < j(k) be the Klein-Gordon amplitude corresponding to a spin zero particle localized at the origin at time t = 0. Since in momentum space the space displacement operator is multiplication by exp (— tk a), the state localized at y at time t = 0 is given by exp (—ik-y) (k). This displaced state by condition (b) above must be orthogonal to (k), i.e. 

We shall call an amplitude ( ), which satisfies Eq. (9-516), a transversal amplitude.20 We can summarize the above statements as follows in momentum space, a one-photon amplitude u(k) is defined on the forward light cone, i.e., for Jc2 = 0, k0 > 0, and satisfies the subsidiary... 

In the Anderson picture the suppression of classical chaotic diffusion is understood as a destructive phase interference phenomenon that limits the spread of the rotor wave function over the available angular momentum space. The localization effect has no classical analogue. It is purely quantum mechanical in origin. The localization of the quantum rotor wave function in the angular momentum space can be demonstrated readily by plotting the absolute squares of the time averaged expansion amplitudes... 

The amplitude (k/kj T qko) may be considered as a probe amplitude for the momentum-space structure amplitude (q/ 0). It is the structure amplitude that is essentially probed by the reaction. Increasingly-refined forms may be used for the probe amplitude. The plane-wave impulse approximation is... 

The spectral function (11.9) in the weak-coupling approximation is proportional to the square of the one-hole structure amplitude (a a(q) 0), averaged over the angles of q. This amplitude is the momentum-space... 

Thus, the adjoint relationship, expressed by the matrix G, is particularly simple. In quantum mechanics the coefficients ak have an important interpretation since they represent the amplitude of the wave function in momentum space. Equations (23) and (24) are direct analogues to the continuous Fourier transformation, which changes a coordinate... 

The theoretical form of the EXAFS as described by Eq. (11) is a sum of damped sinusoidal functions, with frequencies related to the distance of the absorber atom from the backscattering atoms, and an amplitude function which contains information about the number of backscatterers at that distance. This structural information can be best extracted by the Fourier transform technique, which converts data from k or momentum space into R or distance space. The following Fourier transformation of... 

An important progress of the 1970s has been the development of the relativistic OBEP [17]. In this model, the full, relativistic Feynmann amplitudes for the various one-boson-exchanges are used to define the potential. These nonlocal expressions do not pose any numerical problems when used in momentum space . The quantitative deficiencies of the nonrelativistic OBEP disappear immediately when the nonsimplified, relativistic, nonlocal OBE amplitudes are used. 

Recently Elster, Liu and Thaler (ELT) [El 91] proposed a novel method for dealing with the momentum space Coulomb problem, which is, in principle, exact and may be less prone to numerical difficulties than the VP method. Their approach is based on the separation of the optical potential in eq. (3.63) and employs the two-potential formula [Ro 67] to express the full scattering amplitude as a sum of the point Coulomb amplitude and the point Coulomb distorted nuclear amplitude. The latter is obtained by numerically solving an integral equation represented in terms of Coulomb wave function basis states rather than the usual plane wave states. 

A medium is called isotropic and homogeneous when its properties are the same everywhere in space and whatever the direction considered. Within such a medium the propagation velocity does not depend on the wave intensity. In a non-dispersive medium the wave velocity is no longer dependent on the wave frequency, i.e., no energy loss or decrease in amplitude occur during propagation. Let us call p the variation of pressure (p = 0 at equilibrium), and assuming mass and momentum conservation ... 

Mitroy et al. (1984) carried out an extensive configuration-interaction calculation of the structure amplitude (q/ 0) for correlated target and ion states. The long-dashed curve in fig. 11.7(a) shows their momentum distribution multiplied by 2. They found that the dominant contribution came from the pseudo-orbital 3d, calculated by the natural-orbital transformation. Pseudo-orbitals are localised to the same part of space as the occupied 3s and 3p Hartree—Fock orbitals and therefore contribute to the cross section at much higher momenta than the diffuse Hartree—Fock 3d and 4d orbitals. The measurements show that the 4d orbital has a larger weight than is calculated by Mitroy et al, who overestimate the 3d component. 

In the quantum mechanical treatment of this model, the equations of motion in the harmonic approximation become analogous to those for electromagnetic waves in space [2-4]. Thus, each wave is associated with a quantum of vibrational energy hu and a crystal momentum hq. By analogy to the photon for the electromagnetic quantum, the lattice vibrational quantum is called a phonon. The amplitude of the wave reflects the phonon population in the vibrational mode (i.e., the mode with frequency co and... 

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