Bound-state Coulomb wave functions

The usual Hilbert-space requirement of continuous gradients is not appropriate to Coulombic point-singularities of the potential function u(r) [ 196]. This is illustrated by the cusp behavior of hydrogenic bound-state wave functions, for which the Hamiltonian operator is... 

Fig. 4.10. Electron momentum distributions for neon ( 75oi = 0.79 a.u. and /. 02 = 1.51 a.u.) subject to a linearly polarized monochromatic field with frequency ui = 0.057 a.u. and intensity I = 3.0 x 1014W/cm2, as functions of the electron momentum components parallel to the laser-field polarization. The left and the right panels correspond to the classical and to the quantum-mechanical model, respectively. The upper and lower panels have been computed for a contact and Coulomb-type interaction Vi2, respectively. In panels (a) and (d), and (h) and (e), the second electron is taken to be initially in a Is, and in a 2p state, respectively, whereas in panels (c) and (/) the spatial extension of the bound-state wave function has been neglected. The transverse momenta have been integrated over...

In Fig. 4.8 the effect of the initial-state wave functions is explored, for the case where the crucial electron-electron interaction is the two-body Coulomb interaction (4.14a) and for the case where this interaction is the two-body contact interaction (4.14d), which is not restricted to the position of the ion. In both cases, the form factor includes the function (4.23), which favors momenta such that pi + p2 is large. This is clearly visible for the contact interaction (4.14d) and less so for the Coulomb interaction (4.14a) whose form factor also includes the factor (4.19), which favors pi = 0 (or p2 = 0)- We conclude that (i) the effect of the specific bound state of the second electron is marginal and (ii) that a pure two-body interaction, be it of Coulomb type as in (4.14a) or contact type as in (4.14d), yields a rather poor description of the data. A three-body effective interaction, which only acts if the second electron is positioned at the ion, provides superior results, notably the three-body contact interaction (4.14b), cf. the left-hand panel (d). This points to the significance of the interaction of the electrons with the ion, which so far has not been incorporated into the S-matrix theory beyond the very approximate description via effective three-body interactions such as (4.14b) or (4.14c). 

Lastly we note that the width of the well identified bound states is zero. But if we diagonalize a hamiltonian with a Coulombic tail, where sits an infinity of discrete but loosely bound Rydberg states, we will reproduce in our projected calculation one discrete state which averages these states, and is surely also an admixture of continuum states. Typically cross sections to these states are small. If one is really interested in cross sections to them an adjunct basis, P, may be used which connects them to P through the potential matrix elements. A t-matrix expression which has the desired state as the final entry, but P%, as the approximate state wave function, can be used. For example, if charge transfer is responsible for a small amount of flux loss from the target nucleus it is not necessary to use a two-centered basis the procedure described can be used, e.g.,... 

In 1957, Bardeen, Cooper, and Schrieffer published their theory of superconductivity, known as the BCS theory. It predicts that under certain conditions, the attraction between two conduction electrons due to a succession of phonon interactions can slightly exceed the repulsion that they exert directly on one another due to the Coulomb interaction of their like charges. The two electrons are thus weakly bound together forming a so-called Cooper pair. It is these Cooper pairs that are responsible for superconductivity. In conventional superconductors, these electrons are paired so that their spin and orbital angular momenta cancel. They are described by a wave function, known as an order parameter. In this case the order parameter has symmetry similar to that of the wave function of s electrons and represents a singlet state. 

The separability used here leads to a clear relationship between chemical species and ground state electronic wave functions. Each isomeric species is determined by its own stationary ground state electronic wave function. The latter determines a stationary arrangement of Coulomb sources which is different for the different isomers. The nuclei are then hold around a stationary configuration if eq.(10) has bound solutions. An interconversion between them would require a Franck-Condon process, as it is discussed in Section 4. 

Physical chemistry of the positron and Ps is unique in itself, since the positron possesses its own quantum mechanics, thermodynamics and kinetics. The positron can be treated by the quantum theory of the electron with two important modifications the sign of the Coulomb force and absence of the Pauli exclusion principle with electrons in many electron systems. The positron can form a bound state or scatter when it interacts with electrons or with molecules. The positron wave function can be calculated more accurately than the electron wave function by taking advantage of simplified, no-exchange interaction with electrons. However, positron wave functions in molecular and atomic systems have not been documented in the literature as electrons have. Most researchers perform calculations at certain levels of approximation for specific purposes. Once the positron wave function is calculated, experimental annihilation parameters can be obtained by incorporating the known electron wave functions. This will be discussed in Chapter 2. 

A-z-k) in the lab system, respectively, wave function of TT+TT bound state with the Coulomb potential only squared at the origin with the principal quantum number n and the orbital momentum Z = 0, Pb is the Bohr momentum in 2, dan/dpidp2 is the double inclusive production cross section for tt+tt -pairs from short lived sources without taking into account 7r+7T Coulomb interaction in the hnal state, p and P2 are the 7t+ and 7t momenta in the lab system. The momenta of 7t+ and 7t mesons obey the relation Pi P2 The A2W are produced in states with different principal quantum numbers n and are distributed according to n Wi = 83%, W2 = 10.4%, W3 = 3.1%, W >4 = 3.5%. The probability of A2tt production in K, K, p, p, xp and T mesons decay were calculated in [34]. 

For the case cited above, the ponderomotive energy is approximately 1 eV. For typical short pulse experiments today, this energy can easily be hundreds of electron volts. Therefore the wave function of a photoelectron in an intense laser field does not resemble that of the normal field-free Coulomb state, but is dressed by the field, becoming, in the absence of a binding potential, a Volkov state [5], This complex motion of the photoelectrons in the continuum is very difficult to reproduce in terms of the field-free atomic basis functions, so that we have chosen to define our electron wave functions on a finite difference grid. These numerical wave functions have the flexibility to represent both the bound and continuum states in the laser field accurately. 

However, the numerical treatment of such explicitly time-dependent basis states would be time consuming compared to the treatment of bound states. Thus we search for a further simplification of by investigating the asymptotic behavior of Coulomb wave functions [42]. For rAe tt the radial wave function z/j. / is nearly independent of e and may be considered constant for integration. For et tt the exponential function in equation (18) is nearly independent of e. In both cases in equation (18) may be replaced by... 

In the Z — 1 limit, the kinetic terms involving derivatives remain, the centrifugal terms drop out, and the scaled Coulombic potentials metamorphize into delta functions. This hyperquantum limit is tantamount to — oo in the unsealed wave equation. For electronic structure, the low-D limit is generally less useful than the large-D limit, because only the ground state of a delta-function potential [2,5,9] is bound and that can accommodate only two electrons. However, for... 

---