A diagonalisation procedure of the Foldy-Wouthuysen type is considered for single-electron Dirac Hamiltonian as a basis many-body applications. modified suggested. In equation completely decoupled into two equations Pauli type. The expressions positive- and negative-energy projection operators in transformed are then trivial and, by performing inverse transformation, functions obtained. Applying similar transformation to two-electron Dirac-Coulomb leads diagonalised part non-diagonal...

The higher derivatives of motion are rarely discussed in the teaching classical mechanics rigid bodies; nevertheless, we experience effect not only acceleration, but also jerk and snap. In this paper will discuss third order displacement with respect to time, using trampolines theme park roller coasters illustrate concept. We effects on human body different types jerk, snap derivatives, how they can be used physics education further enhance learning thus understanding concepts.

Abstract We present a case study of conversation between ourselves and an artificial intelligence-based chatbot ChatGPT. asked the to respond basic physics question that will be familiar most teachers: ‘ A teddy bear is thrown into air. What its acceleration in highest point?’ The chatbot’s responses, while linguistically quite advanced, were unreliable their correctness often full contradictions. then attempted engage Socratic dialogue with resolve errors contradictions, but little success....

The specific mass shift in the electron affinity between $^{35}\mathrm{Cl}$ and $^{37}\mathrm{Cl}$ has been determined by tunable-laser photodetachment spectroscopy to be -0.51(14) GHz. isotope was observed as a difference onset of process for two isotopes. In addition, Cl found 29 138.59(22) ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$, giving factor 2 improvement accuracy over earlier measurements. Many-body calculations including lowest-order correlation effects demonstrate sensitivity show...

The $F=3$ to $F=2$ hyperfine transitions in the $1s$ ground state of two isotopes ${}^{185}{\mathrm{Re}}^{74+}$ and ${}^{187}{\mathrm{Re}}^{74+}$ were measured be $(4560.5\ifmmode\pm\else\textpm\fi{}3)\AA{}$ $(4516.9\ifmmode\pm\else\textpm\fi{}3)\AA{},$ respectively, using emission spectroscopy an electron beam ion trap. After applying appropriate corrections for nuclear charge distribution QED effects, a Bohr-Weisskopf effect \ensuremath{\varepsilon}=2.23(9)% 2.30(9)% are found...

The hyperfine splitting of the $1s$ ground state hydrogenlike Tl has been measured for two stable isotopes using emission spectroscopy in SuperEBIT electron-beam ion trap, giving $3858.22\ifmmode\pm\else\textpm\fi{}0.30 \AA{}$ ${}^{203}{\mathrm{Tl}}^{80+}$ and $3821.84\ifmmode\pm\else\textpm\fi{}0.34 ${}^{205}{\mathrm{Tl}}^{80+}$ with a wavelength difference $\ensuremath{\Delta}\ensuremath{\lambda}=36.38\ifmmode\pm\else\textpm\fi{}0.35 \AA{}.$ This is consistent estimates based on anomaly...

A relativistic coupled-cluster calculation including single and double excitations has been performed for the ground-state energy of beryllium. Comparison with corresponding nonrelativistic calculations shows effect relativity on pair correlation which, together corrections to Hartree-Fock energy, can be combined most accurate available. Many-body radiative were estimated by multiplying results hydrogenlike Be ratio between Dirac-Hartree-Fock hydrogenic electron densities at nucleus. The...

Isotope shifts and hyperfine structure in the 369.4-nm 6s-6${\mathit{p}}_{1/2}$ resonance line of single-valence-electron system ${\mathrm{Yb}}^{+}$ have been determined with an accuracy about 1 MHz by Doppler-free saturated absorption spectroscopy a sputtered vapor. Ab initio many-body perturbation theory calculations coupled-cluster approach were then used to evaluate electronic field shift factor, F=-14.9(2) GHz ${\mathrm{fm}}^{\mathrm{\ensuremath{-}}2}$, estimate specific mass (SMS)...

The electronic part of a possible $P$- and $T$-nonconserving neutral-weak-current-induced electric dipole moment in atomic Xe Hg was calculated with many-body perturbation theory. After the effects this interaction on all core orbitals were treated self-consistently, thereby including single-particle to orders but no correlation effects, results ${\ensuremath{\mu}}_{e}=(5.2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}23}e \mathrm{m}){C}_{T}{\ensuremath{\sigma}}_{N}$...

Rapid changes in forces and the resulting acceleration, jerk higher order derivatives can have undesired consequences beyond effect of themselves. Jerk cause injuries humans racing animals induce fatigue cracks metals other materials, which may ultimately lead to structure failures. This is a reason that it used within standards for limits states. Examples use include amusement rides lifts. Despite its many science engineering applications, rarely discussed university textbooks remains...

The Breit interaction has been treated together with the Coulomb in a self-consistent way, leading to modified occupied orbitals (Dirac-Fock-Breit orbitals), which are analysed. arguments against inclusion of higher orders discussed, as well advantages doing so. new then used calculate electric dipole moment (EDM) Tl and found lead 2% reduction EDM enhancement factor.

The isotope shifts in the resonance lines 4${\mathit{s}}_{1/2}$\ensuremath{\rightarrow}4${\mathit{p}}_{1/2,3/2}$ Ca ii have been measured for isotopes $^{40\mathrm{\ensuremath{-}}48}\mathrm{Ca}$ by fast-ion-beam collinear laser spectroscopy. Atomic many-body perturbation theory was then used to calculate electronic factor field shift, giving F=-285(3) MHz/${\mathrm{fm}}^{2}$. estimate of uncertainty F is based on agreement at level 1% 4s and 4p hyperfine structures obtained using same wave...

We compare the theoretical and experimental energies for ground state of beryllium atom investigate possible sources small discrepancy about 60 \ensuremath{\mu}hartrees found by Bunge [Phys. Rev. A 14, 1965 (1976); 17, 486(E) (1978)]. Indications that correlation energy may be underestimated in Bunge's work have been confirmed a recent, very extensive multiconfigurational Hartree-Fock (MCHF) calculation. emphasize critical part comparison between theory experiment is sum first second...

In a recent article [Phys. Rev. A 30, 703 (1984)] Hardekopf and Sucher solve the relativistic wave equation in momentum space for hydrogenlike systems. They find that by surrounding hydrogenic Hamiltonian with projection operators free-particle positive-energy states, ground-state energy is lowered. this Comment, we investigate problem some detail conclude application of unsuitable will, fact, introduce negative-energy states to interest rather than remove them. The general considerations...

Abstract Changing acceleration and forces are part of the excitement a roller coaster ride. According to Newton’s second law, <?CDATA $\mathbf{F} = m\mathbf{a}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi mathvariant="bold">F</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>m</mml:mi> mathvariant="bold">a</mml:mi> </mml:math> , every our body must be exposed force accelerate. Since bodies not symmetric, direction matters, accounted for by...

The need for a reassessment of nuclear magnetic dipole moments is prompted by recent experiments on the ground-state hyperfine structure in highly charged hydrogenlike systems which are sufficiently sensitive to probe QED effects. This work gives an overview nuclei interest, i.e., ${}^{165}\mathrm{Ho},$ ${}^{185,187}\mathrm{Re},$ ${}^{203,205}\mathrm{Tl},$ ${}^{207}\mathrm{Pb},$ and ${}^{209}\mathrm{Bi}.$ It found that present uncertainties moment limit interpretation accurate experimental...

The ratio of the atomic electric dipole moment (EDM) to a possible electron EDM has been calculated for ground states Cs, Xe and Hg, including all orders diagrams that involve only single excitations, while neglecting correlation effects, which require simultaneous excitation two or more electrons. alkali atoms are known have large enhancement factors as test procedure, calculation was performed Cs state. lowest experimental limit an EDM, so far established state Xe, is closed-shell system,...

In a recent paper [Phys. Rev. A 37, 1087 (1988)] Gorceix and Indelicato study the difference between multiconfiguration Dirac-Fock (MCDF) expectation value of Lorentz Coulomb gauge forms for Breit interaction. After slight rearrangement their expressions, we have found that can be written as commutator with full Hamiltonian. This is nonvanishing due to projection operators onto positive energy states implicit in MCDF approach gives contributions O(${Z}^{2}$${\ensuremath{\alpha}}^{2}$). It...

2014 L'élément de matrice transition dipolaire électrique violant la parité pour 6s ~ 7s Cs a été évalué en utilisant théorie des perturbations le problème à plusieurs corps.Les diagrammes ne con- tenant que excitations une particule, ont évalués tous les ordres par procédure auto-cohérente, qui donne équivalence automatique entre formes longueur et vélocité l' opérateur électrique.Les effets corrélation, exigent l'excitation simultanée deux particules n'ont pas inclus.Le résultat final,...

A procedure is suggested for evaluating matrix elements of an operator between wavefunctions in the coupled-cluster form. The use exponential Ansatz leads to compact expressions also elements. Algorithms are developed summing all effects one-particle clusters and certain chains two-particle (containing well-known random-phase approximation as a subset). treatment perturbations single valence states investigated detail. As examples oscillator strength 2s → 2p transition Li well hyperfine...

Presents calculations of the parity non-conserving (PNC) electric dipole transitions 6s to 7s in Cs and 6p1/2 7p1/2 6p3/2 Tl, made possible by weak interaction between electrons nucleons. In addition effects that can be accounted for an independent-particle description, authors have calculated those second-order correlation described as modifications valence orbitals approximate Brueckner orbitals, including terms where PNC correction occurs on intermediate well external orbitals. Similar...

The hyperfine structure of the s, p and d states alkaline-earth ions have been evaluated using many-body theory. Polarisation effects were included to all orders correlation estimated by comparison with isoelectronic alkali atoms. In addition magnetic dipole constant A for states, nuclear quadrupole moment barium some radium isotopes estimated.

The hyperfine structure of the $4s$, $4p$, and $3d$ states in ${\mathrm{Ca}}^{+}$ has been evaluated with use many-body perturbation theory. Core-polarization correlation effects have included by solving inhomogeneous one- two-particle differential equations. By these equations iteratively, treated also to higher orders. Certain accounted for modifying valence orbitals approximate "Brueckner orbitals." Relativistic were estimated on basis relativistic calculations including core...