59 research outputs found
Quantal Two-Centre Coulomb Problem treated by means of the Phase-Integral Method I. General Theory
The present paper concerns the derivation of phase-integral quantization
conditions for the two-centre Coulomb problem under the assumption that the two
Coulomb centres are fixed. With this restriction we treat the general
two-centre Coulomb problem according to the phase-integral method, in which one
uses an {\it a priori} unspecified {\it base function}. We consider base
functions containing three unspecified parameters and .
When the absolute value of the magnetic quantum number is not too small, it
is most appropriate to choose . When, on the other hand,
is sufficiently small, it is most appropriate to choose .
Arbitrary-order phase-integral quantization conditions are obtained for these
choices of . The parameters and are determined from the
requirement that the results of the first and the third order of the
phase-integral approximation coincide, which makes the first-order
approximation as good as possible.
In order to make the paper to some extent self-contained, a short review of
the phase-integral method is given in the Appendix.Comment: 23 pages, RevTeX, 4 EPS figures, submitted to J. Math. Phy
Critical view of WKB decay widths
A detailed comparison of the expressions for the decay widths obtained within
the semiclassical WKB approximation using different approaches to the tunneling
problem is performed. The differences between the available improved formulae
for tunneling near the top and the bottom of the barrier are investigated.
Though the simple WKB method gives the right order of magnitude of the decay
widths, a small number of parameters are often fitted. The need to perform the
fitting procedure remaining consistently within the WKB framework is emphasized
in the context of the fission model based calculations. Calculations for the
decay widths of some recently found super heavy nuclei using microscopic
alpha-nucleus potentials are presented to demonstrate the importance of a
consistent WKB calculation. The half-lives are found to be sensitive to the
density dependence of the nucleon-nucleon interaction and the implementation of
the Bohr-Sommerfeld quantization condition inherent in the WKB approach.Comment: 18 pages, Late
Quantal Two-Centre Coulomb Problem treated by means of the Phase-Integral Method II. Quantization Conditions in the Symmetric Case Expressed in Terms of Complete Elliptic Integrals. Numerical Illustration
The contour integrals, occurring in the arbitrary-order phase-integral
quantization conditions given in a previous paper, are in the first- and
third-order approximations expressed in terms of complete elliptic integrals in
the case that the charges of the Coulomb centres are equal. The evaluation of
the integrals is facilitated by the knowledge of quasiclassical dynamics. The
resulting quantization conditions involving complete elliptic integrals are
solved numerically to obtain the energy eigenvalues and the separation
constants of the and states of the hydrogen molecule ion
for various values of the internuclear distance. The accuracy of the formulas
obtained is illustrated by comparison with available numerically exact results.Comment: 19 pages, RevTeX 4, 4 EPS figures, submitted to J. Math. Phy
Improved Approximations for Fermion Pair Production in Inhomogeneous Electric Fields
Reformulating the instantons in a complex plane for tunneling or transmitting
states, we calculate the pair-production rate of charged fermions in a
spatially localized electric field, illustrated by the Sauter electric field
E_0 sech^2 (z/L), and in a temporally localized electric field such as E_0
sech^2 (t/T). The integration of the quadratic part of WKB instanton actions
over the frequency and transverse momentum leads to the pair-production rate
obtained by the worldline instanton method, including the prefactor, of Phys.
Rev. D72, 105004 (2005) and D73, 065028 (2006). It is further shown that the
WKB instanton action plus the next-to-leading order contribution in spinor QED
equals the WKB instanton action in scalar QED, thus justifying why the WKB
instanton in scalar QED can work for the pair production of fermions. Finally
we obtain the pair-production rate in a spatially localized electric field
together with a constant magnetic field in the same direction.Comment: RevTex, 12 pages, two figures; replaced by the version accepted in
Phys. Rev.
Chiral tunneling in single and bilayer graphene
We review chiral (Klein) tunneling in single-layer and bilayer graphene and
present its semiclassical theory, including the Berry phase and the Maslov
index. Peculiarities of the chiral tunneling are naturally explained in terms
of classical phase space. In a one-dimensional geometry we reduced the original
Dirac equation, describing the dynamics of charge carriers in the single layer
graphene, to an effective Schr\"odinger equation with a complex potential. This
allowed us to study tunneling in details and obtain analytic formulas. Our
predictions are compared with numerical results. We have also demonstrated
that, for the case of asymmetric n-p-n junction in single layer graphene, there
is total transmission for normal incidence only, side resonances are
suppressed.Comment: submitted to Proceedings of Nobel Symposium on graphene, May 201
Quantum Mechanics as an Approximation to Classical Mechanics in Hilbert Space
Classical mechanics is formulated in complex Hilbert space with the
introduction of a commutative product of operators, an antisymmetric bracket,
and a quasidensity operator. These are analogues of the star product, the Moyal
bracket, and the Wigner function in the phase space formulation of quantum
mechanics. Classical mechanics can now be viewed as a deformation of quantum
mechanics. The forms of semiquantum approximations to classical mechanics are
indicated.Comment: 10 pages, Latex2e file, references added, minor clarifications mad
On the decay of deformed actinide nuclei
decay through a deformed potential barrier produces significant
mixing of angular momenta when mapped from the nuclear interior to the outside.
Using experimental branching ratios and either semi-classical or
coupled-channels transmission matrices, we have found that there is a set of
internal amplitudes which are essentially constant for all even--even actinide
nuclei. These same amplitudes also give good results for the known anisotropic
particle emission of the favored decays of odd nuclei in the same mass
region.
PACS numbers: 23.60.+e, 24.10.Eq, 27.90.+bComment: 5 pages, latex (revtex style), 2 embedded postscript figures
uuencoded gz-compressed .tar file To appear in Physical Review Letter
Exercises in exact quantization
The formalism of exact 1D quantization is reviewed in detail and applied to
the spectral study of three concrete Schr\"odinger Hamiltonians [-\d^2/\d q^2
+ V(q)]^\pm on the half-line , with a Dirichlet (-) or Neumann (+)
condition at q=0. Emphasis is put on the analytical investigation of the
spectral determinants and spectral zeta functions with respect to singular
perturbation parameters. We first discuss the homogeneous potential
as vs its (solvable) limit (an infinite square well):
useful distinctions are established between regular and singular behaviours of
spectral quantities; various identities among the square-well spectral
functions are unraveled as limits of finite-N properties. The second model is
the quartic anharmonic oscillator: its zero-energy spectral determinants
\det(-\d^2/\d q^2 + q^4 + v q^2)^\pm are explicitly analyzed in detail,
revealing many special values, algebraic identities between Taylor
coefficients, and functional equations of a quartic type coupled to asymptotic
properties of Airy type. The third study addresses the
potentials of even degree: their zero-energy spectral
determinants prove computable in closed form, and the generalized eigenvalue
problems with v as spectral variable admit exact quantization formulae which
are perfect extensions of the harmonic oscillator case (corresponding to N=2);
these results probably reflect the presence of supersymmetric potentials in the
family above.Comment: latex txt.tex, 2 files, 34 pages [SPhT-T00/078]; v2: corrections and
updates as indicated by footnote
Glueballs and the Pomeron
Glueballs are considered to be bound states of constituent gluons.
Relativistic wave equation for two massive gluons interacting by the
funnel-type potential is analyzed. Using two exact asymptotic solutions of the
equation, we derive an interpolating mass formula and calculate glueball masses
in agreement with the lattice data. We obtain the complex non-linear Pomeron
trajectory, , in the whole region of . The real part of the
trajectory corresponds to the soft Pomeron, parameters of which are found from
the fit of recent HERA data.Comment: 6 pages, 1 figure; The X international school-seminar on the actual
problems of microword physics Gomel (Belarus), July 15-16, 200
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