Bound-state Perturbation Theory Research Articles

Many-Electron QED with Redefined Vacuum Approach

The redefined vacuum approach, which is frequently employed in the many-body perturbation theory, proved to be a powerful tool for formula derivation. Here, we elaborate this approach within the bound-state QED perturbation theory. In addition to general formulation, we consider the particular example of a single particle (electron or vacancy) excitation with respect to the redefined vacuum. Starting with simple one-electron QED diagrams, we deduce first- and second-order many-electron contributions: screened self-energy, screened vacuum polarization, one-photon exchange, and two-photon exchange. The redefined vacuum approach provides a straightforward and streamlined derivation and facilitates its application to any electronic configuration. Moreover, based on the gauge invariance of the one-electron diagrams, we can identify various gauge-invariant subsets within derived many-electron QED contributions.

Analytic Perturbation Theory for Bound States in the Transfer Matrix Spectrum of Weakly Correlated Lattice Ferromagnetic Spin Systems

We consider general d-dimensional lattice ferromagnetic spin systems with nearest neighbor interactions in the high temperature region (β≪1). Each model is characterized by a single site apriori spin distribution taken to be even. We also take the parameter α=〈s4〉−3〈s2〉2>0, i.e. in the region which we call Gaussian subjugation, where 〈sk〉 denotes the kth moment of the apriori distribution. Associated with the model is a lattice quantum field theory known to contain a particle of asymptotic mass −lnβ and a bound state below the two-particle threshold. We develop a β analytic perturbation theory for the binding energy of this bound state. As a key ingredient in obtaining our result we show that the Fourier transform of the two-point function is a meromorphic function, with a simple pole, in a suitable complex spectral parameter and the coefficients of its Laurent expansion are analytic in β.

AN APPROACH TO POTENTIAL SCATTERING

Recently developed time-independent bound-state perturbation theory is extended to treat the scattering domain. The changes in the partial wave phase shifts are derived explicitly and the results are compared with those of other methods.

Operator ordering in quantum radiative processes

In this work we reexamine quantum electrodynamics of atomic electrons in the Coulomb gauge in the dipole approximation and calculate the shift of atomic energy levels in the context of Dalibard, Dupont-Roc and Cohen-Tannoudji formalism by considering the variation rates of physical observable. We then analyze the physical interpretation of the ordering of operators in the dipole approximation interaction Hamiltonian in terms of field fluctuations and self-reaction of atomic electrons, discussing the arbitrariness in the statistical functions in second-order bound-state perturbation theory.

The quantum probability equation: II. Scattering state perturbation theory

The time-independent bound-state perturbation theory derived for the quantum probability equation (Milward G C and Wilkin C 2000 J. Phys. A: Math. Gen. 33 7353) is extended to treat the scattering domain. Formulae for the changes in the partial wave phase shifts are derived to third order in the perturbing potential. The results are compared with those obtained for analytically soluble problems.

The quantum probability equation: I. Bound state perturbation theory

The partial-wave Schrodinger equation with real boundary conditions is recast as an equation for the probability density. When a small additional potential is included, the changes in the bound-state energy eigenvalues are obtained, up to third order in the perturbation, purely in terms of the perturbing potential and the unperturbed probability density. Although the approach is different, our results are equivalent to those derived by Bender (Bender C M 1978 Advanced Mathematical Methods for Scientists and Engineers (New York: McGraw-Hill) p 330). Knowledge of neither the unperturbed energy spectrum nor the wavefunctions of excited states is required. Evaluations of the second-order energy shift are given for some soluble S-wave problems.

Analytic treatment of positronium spin splittings in light-front QED

We study the QED bound-state problem in a light-front hamiltonian approach. Starting with a bare cutoff QED Hamiltonian, $H_{_{B}}$, with matrix elements between free states of drastically different energies removed, we perform a similarity transformation that removes the matrix elements between free states with energy differences between the bare cutoff, $\Lambda$, and effective cutoff, $\lam$ ($\lam < \Lam$). This generates effective interactions in the renormalized Hamiltonian, $H_{_{R}}$. These effective interactions are derived to order $\alpha$ in this work, with $\alpha \ll 1$. $H_{_{R}}$ is renormalized by requiring it to satisfy coupling coherence. A nonrelativistic limit of the theory is taken, and the resulting Hamiltonian is studied using bound-state perturbation theory (BSPT). The effective cutoff, $\lam^2$, is fixed, and the limit, $0 \longleftarrow m^2 \alpha^2\ll \lam^2 \ll m^2 \alpha \longrightarrow \infty$, is taken. This upper bound on $\lam^2$ places the effects of low-energy (energy transfer below $\lam$) emission in the effective interactions in the $| e {\overline e} > $ sector. This lower bound on $\lam^2$ insures that the nonperturbative scale of interest is not removed by the similarity transformation. As an explicit example of the general formalism introduced, we show that the Hamiltonian renormalized to $O(\alpha)$ reproduces the exact spectrum of spin splittings, with degeneracies dictated by rotational symmetry, for the ground state through $O(\alpha^4)$. The entire calculation is performed analytically, and gives the well known singlet-triplet ground state spin splitting of positronium, $7/6 \alpha^2 Ryd$. We discuss remaining corrections other than the spin splittings and how they can be treated in calculating the spectrum with higher precision.

Note on restoring manifest rotational symmetry in hyperfine and fine structure in light-front QED.

We study the part of the renormalized, cutoff QED light-front Hamiltonian that does not change particle number. The Hamiltonian contains interactions that must be treated in second-order bound state perturbation theory to obtain hyperfine structure. We show that a simple unitary transformation leads directly to the familiar Breit-Fermi spin-spin and tensor interactions, which can be treated in degenerate first-order bound-state perturbation theory, thus simplifying analytic light-front QED calculations. To the order in momenta we need to consider, this transformation is equivalent to a Melosh rotation. We also study how the similarity transformation affects spin-orbit interactions.

Octet-decuplet baryon mass splittings from self-consistent one-loop perturbation theory

The bag model of confined relativistic quarks in chiral-invariant interaction with scalar, pseudoscalar, vector, and pseudovector mesons, as well as gluons, is used to calculate the masses and wave functions of the spin-1/2 baryon octet and spin-3/2 decuplet, using selfconsistent Brillouin-Wigner bound state perturbation theory. Chiral symmetry breaking is invoked with the sigma model. SU (6) and SU (3) symmetries are broken by the experimental meson spectrum, and a strange quark mass. Mass corrections are calculated to one loop order, limited to the baryons of the octet and decuplet and the lowest lying mesons. Encouraging results are obtained, especially for theΔ — N and theΣ — Λ splittings. Convergence and stability have not been demonstrated, but are evidently improved by the self-consistency requirement. An initial parameter tuning gives a fit to all the octet and decuplet masses within ≦0.02 GeV, at the price of choosing the bag radius, the non-strange baryon input bag mass, and the strange quark mass. Even these small discrepancies can be dramatically reduced by fine-tuning the vector meson coupling and including an instanton contribution peculiar to theΛ.

Volume-diffraction analysis using bound-state perturbation theory

One-dimensional volume diffraction may be analyzed in a manner similar to the problem of an electron subject to a periodic potential. The quantity analogous to the diffraction Nath parameter ρ is then the ratio ΔW0/V, where V is the strength of the electron potential and ΔW0 the spacing between the free-electron-energy eigenstates. The crossover between the Bragg and Raman–Nath diffraction regimes is marked by a violation of the (1/ρ)≪1 condition, and corresponds to the requirement that (V/ΔW0)≪1 in order to preclude electron eigenstate mixing. Bound-state perturbation theory as developed for use in quantum mechanics may be applied to the diffraction problem to obtain a (1/ρ) expansion for the intensities of diffracted grating orders.

Use of the Schrodinger-Coulomb Green function in the evaluation of the quadratic Stark effect

The Schrodinger-Coulomb Green function is discussed as a tool for evaluating terms in bound-state perturbation theory. As an example the energy shift of the ground state of hydrogen in a uniform electric field is considered. The Green function is used in an evaluation of the second-order perturbation theory expression for this energy shift. For comparison a direct numerical approach to the evaluation of this energy shift is also described. The Green function approach is simpler and more elegant, and leads to an exact result.

Bound-state perturbation theory and annihilation effects in positronium

Bound-state perturbation theory and annihilation effects in positronium

Generalized bound-state perturbation theory

Generalized bound-state perturbation theory

Anomalous contributions to the positronium hyperfine splitting and the radiation gauge

By recasting the Bethe-Salpeter equation in a more general form, the natural ensuing bound-state perturbation theory accounts for the anomalous ${\ensuremath{\alpha}}^{6}\mathrm{ln}{\ensuremath{\alpha}}^{\ensuremath{-}1}$ contribution of the fourth-order vacuum polarization to the ground state of the positronium hyperfine structure as a binding correction of a two-photon diagram. Viewed as such, this contribution is not anomalous. Another two-photon diagram, in which such a binding correction makes it a candidate for an ${\ensuremath{\alpha}}^{6}\mathrm{ln}{\ensuremath{\alpha}}^{\ensuremath{-}1}$ contribution, is found. By using the Bethe-Salpeter equation recast in the more general form and the perturbation theory associated with it, the formal connection with the external field approximation is given.

Relativistic bound states: a mass formula for vector mesons

The problem of interaction in relativistic quantum mechanics remains certainly one of the most attractive in particle physics. Interesting approaches have been proposed by DmAC (~), BAKA~IJIAN and THOMAS (2) and FOLDY (3). More recently some hope was born with the development of theories at infinite momentum (4), using the nonrelativistic structure of the Poincar4 group; but, if the success of such an approach is evident in the current algebra scheme, its use for describing interacting relativistic systems in the general case seems still uncertain. The success of the theory of quarks in hadron physics is one of the motivations for the interest of a relativistic description of bound states. The concept of quarks prisoner of hadrons they build up has led physicists to introduce appealing notions such as colour, gluon theory and more recently bag theories. To these attempts one must add the symmetric harmonic-oscillator quark model (5) for baryons in which quarks constituting baryons are assumed linked by harmonic-oscillator forces (**): first presented in the nonrelativistic scheme (7), its generalization to the relativistic case (s) does not appear so easy. Note at this point a recent paper (9) in which a model for relativistic bound-state perturbation theory is presented. However, it appears very important to have a relativistic t reatment of quarks. Let us recall the transformation from current to constituent quarks obtained by MELOSH (10) in a relativistic free quark model: this transformation can be simply interpreted as a Wigner's rotation (~), and so could not appear in a nonrelativistic t reatment .

On the Stark effect of the plane rotator

The Stark effect of the plane rotator is a pedagogically appealing problem for undergraduate courses on quantum mechanics and, accordingly, has appeared in a number of textbooks as an application of bound-state perturbation theory. Despite this fact and the apparent simplicity of the problem, all of the published accounts of it are erroneous in that they employ nondegenerate perturbation theory and present incorrect expressions for the perturbed eigenvalues and eigenstates. In this paper the problem is given a new treatment, based upon a simple analysis of symmetry features and a careful application of degenerate perturbation theory. The major shortcomings of previous discussions are pointed out and correct expressions are derived, to second order, for the perturbed state vectors and energy levels. The lifting of the twofold degeneracy of the mth unperturbed excitation energy in the 2mth order of perturbation theory is demonstrated. Finally, a comparison of the correct perturbation-theoretic results with the exact solution of the problem shows the two to be in identical correspondence.

Model for relativistic bound-state perturbation theory

A model for relativistic bound-state perturbation theory is presented. For the bound-state wave function needed in the perturbation formula, the covariant-harmonic-oscillator wave function is used. As for the perturbing potential, the Feynman propagator corresponding to one-meson exchange is used. This model produces a relativistic derivation of the Breit-Fermi formula, which was recently used by De R\'ujula, Georgi, and Glashow in their study of hadronic masses.

Initial integral equations in bound-state perturbation theory

Initial integral equations in bound-state perturbation theory

Nuclear Quadrupole Resonance in Solid Nitrogen— An Exercise in Quantum Mechanics

The field of nuclear quadrupole resonance (NQR) in molecular solids provides a rich new source of instructive, easily varied, workable problems in quantum mechanics. Examples which cover the subject areas of spin, identical versus distinguishable particles, degenerate bound-state perturbation theory, symmetry properties of operators and wavefunctions, and selection rules are presented in an integrated discussion of a current research problem involving NQR of N (I = 1)14 in solid α-N2 enriched in the N15 isotope (I = 12). In this solid, the nuclear quadrupole moments interact with an axially symmetric electric field gradient (EFG), and, to an excellent approximation, each diatomic molecule can be treated as a separate two-spin system in which the nuclear quadrupole energy levels are perturbed by a coupling between the magnetic moments of the nuclei. Spectra obtained from first-order perturbation theory applied to both N14-N14 and N14-N15 molecules are compared with the results of an exact calculation and with experiment. In addition, it is demonstrated simply that in the presence of an asymmetric EFG the dipolar interaction of a spin-1 nucleus with a spin-12 nucleus is completely quenched to first order, while some splitting remains for the spin-1-spin-1 case.

Variation-Perturbation Treatment of Scattering Problems. I. Potential Scattering by a Central Field

A variation-perturbation treatment of potential scattering by a central field is presented. The first-order correction ${X}_{l}^{(1)}$, to the lth partial wave, which gives the phase shift ${\ensuremath{\eta}}_{l}$, through third order, may be determined from a variation principle or as the solution of an inhomogeneous differential equation. Optimization of the initial approximation by means of the Hulth\'en condition yields a vanishing ${\ensuremath{\eta}}_{l}^{(1)}$ and a bound ${X}_{l}^{(1)}$. Encouraging results are obtained in trial calculations on the exponential and Yukawa potentials treated as perturbed spherical holes. The likelihood of extending this approach to include the most powerful techniques of bound-state perturbation theory is anticipated.