We discuss the possibility of zeros in the nonrelativistic radiative continuum-continuum matrix element for electron-atom (inverse) bremsstrahlung. As demonstrated earlier for upward transitions from bound states, the occurrence of different signs for the free-free matrix element in limiting cases, plus the requirement of continuity, implies the existence of zeros. Using knowledge of the sign of the dipole matrix element in the soft- and hard-photon limits with one continuum electron energy held fixed, we show that zeros can occur in the
s-
p matrix element. We discuss the connection of our results to elastic scattering and to Ramsauer-Townsend minima. We consider the observability of zeros in this
(s-
p) matrix element manifested as minima in the cross sections.
The radial electric dipole moments of bound-bound, bound-free, and free-free transitions of high-lying
(n>15) Rydberg states and low-lying continuum states $(\ensuremath{\epsilon}<400$ ${\mathrm{cm}}^{\ensuremath{-}1})$ are calculated for hydrogen and the alkali-metal atoms. All dipole matrix elements for the transitions between low-
l (s,
p,
d, and
f) states of H, Li, and Rb are presented. Results of similar calculations for the other alkali-metal atoms (Na, K, and Cs) are summarized. These numerical results, using quantum defect theory, are compared with semiclassical approximations. The dipole matrix elements cross zero at certain combinations of initial and final states. The matrix elements at these so-called Cooper minima in Li, K, Rb, and Cs are presented in detail.
The authors have carried out exhaustive calculations of monopole, dipole, and quadrupole transition radial matrix elements as functions of the effective principal quantum numbers of the initial and final electronic state. The general features of the results of these calculations have been displayed in three-dimensional graphs for the general cases. The locations in (n*,n{prime}*)-space of the zeroes of these matrix elements have been found and appear to follow a generic behavior. This behavior has been fitted to a simple analytic form and the parameters of the fits are given for monopole, dipole and quadrupole transitions involving l = 0, 1, 2, and 3 states. These results are expected to be of great value to experimentalists working with atomic Rydberg states, especially of alkali-like systems.
Heisenberg's form of the correspondence principle for non-relativistic matrix elements has been used to evaluate matrix elements for various potentials. These are compared with available quantum mechanical results to check the general validity of the use of classical mechanics and correspondence principles in the derivation of quantum mechanical expressions. Using a one dimensional harmonic oscillator potential matrix elements of the form (n mod qm mod n+s) and (n mod pm mod n+s) are worked out. A Morse potential is then considered and the matrix elements for position, momentum, and kinetic energy are evaluated. Using a Coulomb potential a similar procedure is used for functions of various position coordinates, from which dipole and quadrupole moments from transitions between n, l states are calculated. The agreement with quantum mechanics is found to be generally good, and in some cases identical results are obtained.
The quantum mechanical position operators, and their products, are not well-defined in systems obeying periodic boundary conditions. Here we extend the work of Resta [Phys. Rev. Lett. 80, 1800 (1998)], who developed a formalism to calculate the electronic polarization as an expectation value of a many-body operator, to include higher multipole moments, e.g., quadrupole and octupole. We define
nth-order multipole operators whose expectation values can be used to calculate the
nth multipole moment when all of the lower moments are vanishing (modulo a quantum). We show that changes in our operators are tied to flows of $n\ensuremath{-}1\mathrm{st}$ multipole currents, and encode the adiabatic evolution of the system in the presence of an $n\ensuremath{-}1\mathrm{st}$ gradient of the electric field. Finally, we test our operators on a set of tight-binding models to show that they correctly determine the phase diagrams of topological quadrupole and octupole models, capture an adiabatic quadrupole pump, and distinguish a bulk quadrupole moment from other mechanisms that generate corner charges.
In view of the possibilities of laser spectroscopy with very short pulses and relatively high frequencies, we examine aspects of the theory of atom-field interactions that are related to violation of the condition of the long-wavelength approximation (LWA) according to which
kr⪡1, where
r is ``of the order of atomic dimensions'' and
k is the magnitude of the wave vector. On- and off-resonance transitions are considered, with
kr being larger than unity due to the large extent of the two wave functions involved in the coupling matrix element. The implementation of the analysis uses bound-free transition matrix elements with the
n=50 and 80 hydrogenic functions as initial states and values of
k up to
27eV, which is sufficient to produce
kr>1, thereby rendering the LWA inoperative. In spite of this, it is shown that, for on-resonance transitions, the results from the use of the well-known multipole operators resulting from the LWA [e.g., the electric dipole approximation (EDA)] agree with those from the exact expressions derived here from the application of the multipolar Hamiltonian. This numerical agreement is proven analytically. As a test of the kind of convergence of the multipole series expansion for small values of
k and large
r, it is shown that the lowest-order ratio of electric dipole to quadrupole matrix elements decreases rapidly within 1.0 atomic unit above threshold. Finally, it is shown that off-resonance couplings lead to differences between the full-interaction operator and the EDA which cannot be neglected. In the extreme case of intrashell couplings for the
n=50 shell, calculation shows that the
50p state is coupled to angular momentum states up to $\ensuremath{\ell}=21$, compared to the electric dipole coupling of $\ensuremath{\Delta}\ensuremath{\ell}=\ifmmode\pm\else\textpm\fi{}1$.