Set Of Quantum Numbers Research Articles

Amplitudes for scattering of electrons by hydrogenic and alkali-like atomic systems

Amplitudes for scattering of electrons by hydrogenic and alkali-like atomic systems

Quantum numbers for the problem of two coulomb centers

A complete set of commuting operators and the corresponding complete set of quantum numbers are found for the motion of an electron in the field of two Coulomb centers. The symmetry group of the H 2 + ion is found.

Do there exist new light neutral mesons?

The properties of the hypothetical neutral mesons (L) are considered for which the decays into other particles via strong and weak interaction are forbidden. It is shown that some L-mesons may have long lifetime\(\left( {\tau \sim \left( {10^{ - 5} \div 10^{ - 8} } \right){\text{s}}} \right)\) because the decays into 3 or 4ν-quanta are their only possible decay modes. Besides, there is a region of masses and sets of quantum numbers of L-mesons for which the L→ππγ decay is predominant. In fact, the electromagnetic decays of particles produced in strong interactions have not yet been studied experimentally.

Zero-Energy Intercepts of theρandRTrajectories, thepnCharge-Exchange Scattering, and the Difference inppandnpTotal Cross Sections

It is shown that the existing high-energy data for the $\mathrm{np}$ charge-exchange differential cross section in the forward direction and the difference between $\mathrm{np}$ and $\mathrm{pp}$ total cross sections can be simulataneously explained in terms of two Regge trajectories. The two trajectories are the $\ensuremath{\rho}$ and the $R$ (using the notation of Pignotti in the preceding paper). The $R$ has parity and $G$ parity opposite to $\ensuremath{\rho}$ but the same isotopic spin. These two sets of quantum numbers in the crossed channel are the only possible ones with nonvanishing contribution to the processes in question. The conjecture of real analyticity of the generalized coupling constants, together with the existing data, requires the existence of an $R$ trajectory. In fitting the experimental data the intercepts are ${\ensuremath{\alpha}}_{\ensuremath{\rho}}(0)=0.57\ifmmode\pm\else\textpm\fi{}0.1$ and ${\ensuremath{\alpha}}_{R}(0)=0.31\ifmmode\pm\else\textpm\fi{}0.05$.

Studies of polaron motion: Part III: The Hall mobility of the small polaron

The one-dimensional molecular crystal model of polaron motion, developed in parts I and II, is suitably generalized to consider the existence of a Hall effect. As in II, the treatment is confined to the case for which the electronic-overlap term of the total Hamiltonian is a small perturbation. In zeroth order—i.e., in the absence of this term—the electron is localized at a given site. The vibrational state of the system is specified by a set of quantum numbers, N k , giving the degree of excitation of each vibrational mode. The existence of a nonvanishing electronic bandwidth then gives rise to transitions to neighboring sites. Of principal interest in the present paper is the high temperature regime ( T > T t, as defined in II) where polaron motion is predominantly by means of random jumps between neighboring sites. Although the lowest order jump rate is adequate in considering the polaron drift mobility, higher order processes, involving the occupation of (at least) three sites, must be taken into account in treating the Hall effect. In particular, it is demonstrated the relative probability of the electron, initially located on one of the three sites, hopping to one or the other of the remaining two sites, is modified by a contribution which, both in sign and magnitude, is linearly proportional to the applied magnetic field. This effect is shown to arise from the interference between the amplitude for the direct transition from the initial to the final site, and the amplitude for an indirect, second order transition, involving intermediate occupancy of the third site. The (magnetic) field induced components of the jump rates, corresponding to the above processes, are first calculated by a classical occurrence-probability approach which treats the lattice vibrational coordinates as given functions of time. The second approach presents a full quantum mechanical calculation of the jump rates. The results of this treatment agree with those of the occurrence-probability approach in the classical limit ( T ⪢ Θ De bye). The order of magnitude of the calculated Hall Coefficients are found to be greater than or comparable to the “normal” result ( R = −1 nec ) depending on whether or not the three sites involved in the transition are mutually nearest neighbors. A final note corrects an error in a previous paper by one of the authors (T. H.) concerning the sign of the Hall effect in impurity conduction.

Static Approximation and Bounds on Single-Channel Phase Shifts

A single-channel scattering process with the set of quantum numbers $C$ is completely characterized by a phase shift, ${\ensuremath{\eta}}_{C}$. A common approximation in the determination of ${\ensuremath{\eta}}_{C}$ for the scattering of a particle by a compound system is to assume that, apart from the possibility of an exchange of the incident particle with an identical target particle, the target is unaffected by the incident particle. The incident particle is then scattered by the static potential generated by the target in its ground state. The phase shift determined in this approximation, to be called ${{\ensuremath{\eta}}_{C}}^{P}$, can be calculated for a number of scattering processes. Let $H$ represent the Hamiltonian of the entire system, incident particle plus target, let ${E}_{T0}$ be the ground-state energy of the target, and let ${{{E}^{\ensuremath{'}}}_{C}}^{Q}$ be the smallest energy for which ${{{E}^{\ensuremath{'}}}_{C}}^{Q}+{E}_{T0}\ensuremath{-}H$ is a negative definite operator in the space in which the given quantum numbers are $C$ and in which the ground state of the target is projected out. Utilizing the generalized optical potential formalism due to Feshbach and others, it can then be shown that ${\ensuremath{\eta}}_{C}g{{\ensuremath{\eta}}_{C}}^{P}$ if the incident energy is less than ${{{E}^{\ensuremath{'}}}_{C}}^{Q}$. (The bound is probably valid for higher energies, perhaps for all energies for which the process remains a single-channel process. If so, however, the difference between ${\ensuremath{\eta}}_{C}$ and ${{\ensuremath{\eta}}_{C}}^{P}$ for these higher energies will generally be large, of the order of a multiple of $\ensuremath{\pi}$, and the bound will not be immediately useful. We will, therefore, be concerned primarily with incident energies ${E}^{\ensuremath{'}}$ less than ${{{E}^{\ensuremath{'}}}_{C}}^{Q}$.) Furthermore, as one allows for more and more virtual excitation of the target system, the approximate phase shift is guaranteed to improve if ${E}^{\ensuremath{'}}$ is less than ${{{E}^{\ensuremath{'}}}_{C}}^{Q}$, and ${{{E}^{\ensuremath{'}}}_{C}}^{Q}$ will itself increase. Applications are given to the scattering of electrons and of positrons by hydrogen atoms.

Studies of polaron motion: Part II. The “small” polaron

The one-dimensional molecular-crystal model of polaron motion, described in the preceding paper, is here analyzed for the case in which the electronic-overlap term of the total Hamiltonian is a small perturbation. In zeroth order —i.e., in the absence of this term—the electron is localized at a given site, p. The vibrational state of the system is specified by a set of quantum-numbers, N k , giving the degree of excitation of each vibration-mode; the latter differ from the conventional modes in that in each of them, the equilibrium displacement, about which the system oscillates, depends upon the location of the electron. The presence of a nonvanishing electronic-overlap term gives rise to transitions in which the electron jumps to a neighboring site ( p → p ± 1), and in which either all of the N k remain unaltered (“diagonal” transitions) or in which some of them change by ±1 (“nondiagonal” transitions). The two types of transitions play fundamentally different roles. At sufficiently low temperatures, the diagonal transitions are dominant. They give rise to the formation of Bloch-type bands whose widths (see Eq. 37) are each given by the product of the electronic-overlap integral, and a vibrational overlap-integral, the latter being an exponentially falling function of the N k (and, hence, of temperature). In this low-temperature domain, the role of the nondiagonal transitions is essentially one of scattering. In the absence of other scattering mechanisms, such as impurity scattering, they determine the lifetimes of the polaron-band states and, hence, the mean free path for typical transport quantities, such as electron diffusivity. With rising temperature, the probability of the off-diagonal transitions goes up exponentially. This feature, together with the above-mentioned drop in bandwidth, results, e.g., in an exponentially diminishing diffusivity. Eventually, a temperature, T t ∼ 1 2 the Debye θ, is reached at which the energy uncertainty, h ̷ gt , associated with the finite lifetime of the states, is equal to the bandwidth. At this point, the Bloch states lose their individual characteristics (in particular, those which depend upon electronic wave number); the bands may then be considered as “washed out”. For temperatures > T t , electron motion is predominantly a diffusion process. The elementary steps of this process consist of the random-jumps between neighboring sites associated with the nondiagonal transitions. In conformance with this picture, the electron diffusivity is, apart from a numerical factor, the product of the square of the lattice distance and the total non-diagonal transition probability, and is therefore an exponentially rising function of temperature. The limit, J max, of the magnitude of the electronic overlap term, beyond which the perturbation treatment of the present paper becomes inapplicable, is investigated. For representative values of the parameters entering into the theory, J max ∼ 0.12 ev and 0.035 ev for the extreme cases of (a) width of the ground-state polaron-band and (b) high-temperature site-jump probabilities (these numbers correspond to electronic bandwidths of 0.24 ev and 0.07 ev, respectively). For electronic bandwidths in excess of these limits, a treatment based on the adiabatic approach is required; preliminary results of such a treatment are given for the above two cases.