BORN–OPPENHEIMER APPROXIMATION FOR THE BROWN–RAVENHALL EQUATION

2001 ◽  
Vol 13 (08) ◽  
pp. 921-951
Author(s):  
VANIA SORDONI
Keyword(s):  
Semiclassical Limit ◽  
Energy Levels ◽  
Mass Ratio ◽  
Nuclear Mass ◽  
Oppenheimer Approximation ◽  
Brown And Ravenhall

In this paper we study the pseudo-relativistic Hamiltonian proposed by Brown and Ravenhall in the semiclassical limit when the mass ratio h2 of electronic to nuclear mass tends to zero. We show that the relativistic contribution of the nuclei on WKB-type expansions of the first energy levels are of order o(h2), as h→0.

scholarly journals The Uehling correction to the energy levels in a pionic atom

2006 ◽  
Vol 84 (2) ◽  
pp. 107-113 ◽  
Author(s):  
S G Karshenboim ◽  
E Yu. Korzinin ◽  
V G Ivanov
Keyword(s):  
Free Electron ◽  
Vacuum Polarization ◽  
Energy Levels ◽  
Analytic Form ◽  
Electron Mass ◽  
Mass Ratio ◽  
Pionic Atom ◽  
Closed Analytic Form

We consider a correction to energy levels in a pionic atom induced by the Uehling potential, i.e., by a free electron vacuum-polarization loop. The calculation is performed for circular states (l = n–1). The result is obtained in a closed analytic form as a function of Zα and the pion-to-electron mass ratio. Certain asymptotics of the result are also presented.PACS Nos.: 12.20.Ds, 36.10.Gv

Investigation of logarithmic contributions in the electron-to-muon mass ratio to the shift of the S energy levels in the muonium atom

2003 ◽  
Vol 66 (5) ◽  
pp. 893-901 ◽  
Author(s):  
N. A. Boikova ◽  
S. V. Kleshchevskaya ◽  
Yu. N. Tyukhtyaev ◽  
R. N. Faustov
Keyword(s):  
Energy Levels ◽  
Mass Ratio ◽  
Muonium Atom ◽  
Muon Mass

Polarizabilities and other properties of the tdµ molecular ion

2001 ◽  
Vol 79 (9) ◽  
pp. 1149-1158
Author(s):  
A K Bhatia ◽  
R J Drachman
Keyword(s):  
Energy Levels ◽  
Molecular Ions ◽  
Muon Catalyzed Fusion ◽  
Oppenheimer Approximation ◽  
Good Energy ◽  
Cusp Conditions ◽  
Fermi Contact ◽  
Muonic System ◽  
Three Body ◽  
Contact Parameters

Wave functions of the Hylleraas type were used earlier to calculate energy levels of muonic systems. Recently, we found in the case of the molecular ions H2+, D2+, and HD+ that it was necessary to include high powers of the internuclear distance in the Hylleraas functions to localize the nuclear motion when treating the ions as three-body systems without invoking the Born–Oppenheimer approximation. We tried the same approach in a muonic system, tdµ– (triton, deuteron, and muon). Improved convergence was obtained for J = 0 and 1 states for shorter expansions when we used this type of generalized Hylleraas function, but as the expansion length increased the high powers were no longer useful. We obtained good energy values for the two lowest J = 0 and 1 states and compared them with the best earlier calculations. Expectation values were obtained for various operators, the Fermi contact parameters, and the permanent quadrupole moment. The cusp conditions were also calculated. The polarizability of the ground state was then calculated using second-order perturbation theory with intermediate J = 1 pseudostates. (It should be possible to measure the polarizability by observing Rydberg states of atoms with tdµ– acting as the nucleus.) In addition, the initial sticking probability (an essential quantity in the analysis of muon catalyzed fusion) was calculated and compared with earlier results. PACS Nos.: 30.00, 36.10-k, 02.70-c

The Bakerian Lecture, 1987. Quantum chaology

1987 ◽  
Vol 413 (1844) ◽  
pp. 183-198 ◽  
Keyword(s):  
Chaotic Systems ◽  
Semiclassical Limit ◽  
Energy Levels ◽  
Classical Motion ◽  
Classical Systems ◽  
Closed Orbits ◽  
Spectral Behaviour ◽  
Universality Classes ◽  
The Mean ◽  
Exponential Instability

Bounded or driven classical systems often exhibit chaos (exponential instability that persists), but their quantum counterparts do not. Nevertheless, there are new régimes of quantum behaviour that emerge in the semiclassical limit and depend on whether the classical orbits are regular or chaotic, and this motivates the following definition. Definition . Quantum chaology is the study of semiclassical, but nonclassical, behaviour characteristic of systems whose classical motion exhibits chaos. This is illustrated by the statistics of energy levels. On scales comparable with the mean level spacing (of order h N for N freedoms), these fall into universality classes: for classically chaotic systems, the statistics are those of random matrices (real symmetric or complex hermitian, depending on the presence or absence of time-reversal symmetry); for classically regular ones, the statistics are Poisson. On larger scales (of order h , i. e. classically small but semiclassically large), universality breaks down. These phenomena are being explained by representing spectra in terms of classical closed orbits: universal spectral behaviour has its origin in very long orbits; non-universal behaviour depends only on short ones.

High-accuracy calculations in the H+2 molecular ion: towards a measurement of mp/me

2007 ◽  
Vol 85 (5) ◽  
pp. 497-507 ◽  
Author(s):  
J Ph. Karr ◽  
F Bielsa ◽  
T Valenzuela ◽  
A Douillet ◽  
L Hilico ◽  
...  
Keyword(s):  
Vibrational Spectroscopy ◽  
Quantum Cascade Laser ◽  
Energy Levels ◽  
High Accuracy ◽  
Electron Mass ◽  
Mass Ratio ◽  
Molecular Ion ◽  
Quantum Cascade ◽  
Two Photon

We report on our recent advances in the calculation of the energy levels of the H+2 molecular ion, including relativistic and radiative corrections. These theoretical efforts are linked to the prospect of obtaining a new determination of the proton to electron mass ratio mp/me through precise vibrational spectroscopy of H+2. We describe the setup of our experiment, aiming at a measurement of the L = 2, υ = 0 → L = 2, υ = 1 two-photon transition at 9.166 μm using a phase-locked quantum cascade laser as excitation source.PACS Nos.: 31.15.Pf, 31.30.Jv, 32.10.Hq

scholarly journals Further investigations with a Wilson Chamber. III.—The accuracy of the angle determination

1932 ◽  
Vol 136 (829) ◽  
pp. 338-348 ◽  
Keyword(s):  
Energy Levels ◽  
Probable Error ◽  
Average Error ◽  
Mass Ratio ◽  
Angle Measurements ◽  
Mass Ratios ◽  
Calculated Mass ◽  
The Difference ◽  
Error Of Measurement

I.—The question of the precision of the determination of the angles of forked tracks is of considerable importance, in particular owing to the possibility of determining nuclear energy levels from Wilson photographs. In an earlier paper some experiments were described in which an artificial track consisting of a bent glass fibre was photographed in different positions. The average error of determination of the angle was found to be 10 minutes of arc. This error was attributed to the lack of perfect adjustment of the camera. That the error of measuring actual tracks could be nearly as small as this was shown by measurements of three collisions in which the difference between the calculated and expected mass ratios was consistent with a probable error of about 10 minutes of arc for the angle measurements. Only such forks were used for these calculations of the mass ratio, for which the three arms appeared unusually straight and for which the test for coplanarity was accurately satisfied. Subsequently two collisions with hydrogen nuclei were described in which the error of the angles was held to be as low as 6 minutes of arc. It was pointed out at the time that many tracks did not, in fact, satisfy these conditions, but sufficient data were not then available for a statistical analysis of the distribution of calculated mass ratios, from which a reliable estimation of the probable error of measurement of an average fork could be made. Since then a great many more photographs have been taken with a larger chamber and an improved camera and such a statistical test is now possible. To test the camera itself five photographs were taken of two black lines ruled on a card. The angles calculated from the photographs were:—

Semiclassical theory of spectral rigidity

1985 ◽  
Vol 400 (1819) ◽  
pp. 229-251 ◽  
Keyword(s):  
Periodic Orbits ◽  
Matrix Theory ◽  
Semiclassical Limit ◽  
Energy Levels ◽  
Gaussian Unitary Ensemble ◽  
A Value ◽  
Straight Line ◽  
Spectral Rigidity ◽  
Riemann Zeta ◽  
Classical Orbits

The spectral rigidity ⊿( L ) of a set of quantal energy levels is the mean square deviation of the spectral staircase from the straight line that best fits it over a range of L mean level spacings. In the semiclassical limit (ℏ→0), formulae are obtained giving ⊿( L ) as a sum over classical periodic orbits. When L ≪ L max , where L max ~ℏ-(N-1) for a system of N freedoms, ⊿( L ) is shown to display the following universal behaviour as a result of properties of very long classical orbits: if the system is classically integrable (all periodic orbits filling tori), ⊿( L )═ 1 / 5 L (as in an uncorrelated (Poisson) eigenvalue sequence); if the system is classically chaotic (all periodic orbits isolated and unstable) and has no symmetry, ⊿( L ) ═ In L /2π 2 + D if 1≪ L ≪ L max (as in the gaussian unitary ensemble of random-matrix theory); if the system is chaotic and has time-reversal symmetry, ⊿( L ) = In L /π 2 + E if 1 ≪ L ≪ L max (as in the gaussian orthogonal ensemble). When L ≫ L max , ⊿( L ) saturates non-universally at a value, determined by short classical orbits, of order ℏ –(N–1) for integrable systems and In (ℏ -1 ) for chaotic systems. These results are obtained by using the periodic-orbit expansion for the spectral density, together with classical sum rules for the intensities of long orbits and a semiclassical sum rule restricting the manner in which their contributions interfere. For two examples ⊿(L) is studied in detail: the rectangular billiard (integrable), and the Riemann zeta function (assuming its zeros to be the eigenvalues of an unknown quantum system whose unknown classical limit is chaotic).

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