Large-order Perturbation Theory Research Articles

Large-order perturbation theory without a wavefunction for the LoSurdo-Stark effect in hydrogenic atoms

The author obtains exact large-order perturbation corrections to the energy shifts of the LoSurdo-Stark effect in hydrogenic atoms for both non-degenerate and degenerate states. The method consists of applying perturbation theory to a recurrence relation among properly selected moments of the eigenfunction which do not explicitly appear in the calculation. The resulting recurrence relation for the perturbation corrections is suitable for computer algebra calculation of perturbation corrections to the energy. The method is most useful to treat both separable and non-separable problems in any convenient coordinate system. The author uses spherical polar coordinates and calculates the shifts of the first two hydrogenic energy levels as illustrative examples.

Complex Stark eigenvalues for excited states of hydrogenic ions from analytic continuation of real variationally based large-order perturbation theory

Real variationally based large-order perturbation theory (LOPT), implemented via the perturbational-variational Rayleigh-Ritz (PV-RR) matrix formalism, and combined with analytic continuation into the complex plane by shifted-origin (SO) perturbation series, has been applied to compute accurately with Padé approximants the complex Stark eigenvalues (resonances) of numerous excited states of hydrogenic ions over a range of field strengths. The problem of hydrogenic excited-state zeroth-order degeneracy in spherical coordinates is eliminated by the novel procedure of introducing a new SO unperturbed Hamiltonian with nondegenerate eigenfunctions. The SO PV-RR LOPT procedure is not restricted to one-electron systems.

Large order perturbation theory and summation methods in quantum mechanics

Large order perturbation theory and summation methods in quantum mechanics

Large-order perturbation theory of the Zeeman effect in hydrogen from a four-dimensional anisotropic anharmonic oscillator.

The Zeeman Hamiltonian for (spinless) hydrogen in a constant magnetic field is shown to be equivalent to a four-dimensional anisotropic anharmonic oscillator. Using this relation, Rayleigh-Schr\"odinger perturbation series expansions of both systems can be related to each other and analyzed in a unified way. Special emphasis is laid upon analytical estimates of their behavior in large orders of perturbation theory. Employing the path-integral approach, a new large-order formula is derived for the expansion of the ground-state energy of the oscillator system. With use of known Bender-Wu formulas for isotropic anharmonic oscillators, the major part of this calculation becomes straightforward. Combined with the new equivalence, this calculation represents the simplest path-integral derivation of large-order formulas for the Zeeman system.

Complex Stark eigenvalues via analytic continuation of real high-order perturbation series

We have calculated complex Stark eigenvalues for the ground state of the H atom for field strengths 0.025–1.0 au by combining a procedure due to Reinhardt for the analytic continuation of real large-order eigenvalue (polarizability) perturbation series with the perturbational-variational Rayleigh-Ritz (PV-RR) method for generating such series; the analytic continuation is accomplished by shifting the origin of the real series into the complex plane where the divergent series involved are summed by Padé approximants. The method yields results for hydrogenic Stark resonances of good accuracy over a wide range of field strengths with slight computational effort. PV-RR implements large-order perturbation theory with the variational method, thus rendering feasible the extension of the technique to larger systems.

Calculation of high-order electric polarizabilities via the perturbational-variational Rayleigh-Ritz formalism: Application to the hydrogenic Stark effect.

It is demonstrated that perturbation theory combined with the variational principle in the form of the perturbational-variational Rayleigh-Ritz (PV-RR) matrix formalism is a powerful method for applying large-order perturbation theory (LOPT) to the Stark effect for hydrogenic ions. Unlike other more specialized LOPT methods, which do not invoke the variational principle, application of the PV-RR formalism is not restricted to one-electron systems. A paradigm is derived and implemented which permits the precise construction of a Rayleigh-Ritz variational ansatz from which the real Stark ground-state Rayleigh-Schr\"odinger perturbation series for the eigenfunction, eigenvalue, and arbitrary expectation values can be generated via the PV-RR procedure to any prespecified order in the most economical way possible. A novel method of analysis, based on a combination of the Stark virial, Hellmann-Feynman, and remainder theorems, is developed for studying the physical origins of high-order atomic and molecular polarizabilities; the procedure is illustrated by application to the hydrogenic Stark effect. It is concluded on the basis of this work and of similar high-order PV-RR polarizability calculations for ${\mathrm{H}}_{2}$${\mathrm{}}^{+}$ that the prospects for extending the PV-RR formalism to accurate polarizability studies of more complex systems are promising.

High-order non-born-oppenheimer electric polarizabilities for excited states of H +2 via the perturbational-variational rayleigh-ritz formalism

Large-order perturbation theory in the form of the perturbational-variational Rayleigh-Ritz (PV RR) formalism has been applied to excited energy levels in the Stark effect for H + 2,yielding non-Born-Oppenheimer electric polarizabilities through 16th, 8th, 4th, and 2nd order, respectively, for the first four excited non-rotational states; these non-Born-Oppenheimer excited states correspond to what are normally known as vibrationally excited levels of the electronic ground state. The excited-state polarizability series are found to be more divergent in each corresponding order than the ground-state one, becoming increasingly so for the higher excited states. Methods for obtaining the excited-state polarizabilities to still higher order within the framework of PV RR are discussed.

High-order stark effect perturbation series for hydrogenic ions via the perturbational-variational Rayleigh-Ritz formalism

Perturbation theory combined with the variational principle in the form of the perturbational-variational Rayleigh-Ritz (PV RR) matrix formalism is a powerful method for applying large-order perturbation theory to the Stark effect for hydrogenic ions. A paradigm is derived and implemented which permits the precise construction of a variational ansatz from which the Stark Rayleigh-Schrodinger (RS) eigenvalue (polarizability) and eigenfunction perturbation series for the ground state are generated via PV RR to any prespecified order. We have also computed high-order RS series of the kinetic- and potential-energy components of the Stark eigenvalue series, and determined the Bender-Wu-type asymptotic formulae which govern the limiting behaviour of their large-order coefficients. PV RR is not restricted to one-electron systems.

Twentieth-order perturbation study of the nonadiabatic electric polarizabilities for H2 + via the perturbational-variational Rayleigh-Ritz formalism.

Large-order perturbation theory has been applied, for the first time, to the Stark effect for ${\mathrm{H}}_{2}^{+}$, yielding the Rayleigh-Schr\"odinger ground-state eigenvalue (polarizability) series through twentieth order; previous expansions were limited to fourth order. The calculations were performed nonadiabatically (i.e., without invoking the Born-Oppenheimer approximation) by means of the perturbational-variational Rayleigh-Ritz formalism. The leading terms of the Rayleigh-Schr\"odinger polarizability series so obtained provide the most accurate values thus far determined for ${\ensuremath{\alpha}}_{\mathrm{zz}}$ and ${\ensuremath{\gamma}}_{\mathrm{zzzz}}$.

A 30th-order perturbational study of the born-oppenheimer electronic polarizabilities for H2+ via the perturbational-variational Rayleigh-Ritz formalism

Large-order perturbation theory in the form of the perturbational-variational Rayleigh-Ritz formalism has been applied to the calculation of the Born-Oppenheimer electronic polarizabilities for H2+, yielding the Rayleigh-Schrödinger ground-state eigen-value (polarizability) series, at an internuclear separation of 2.0 au, through 30th order with unprecedented accuracy; previous Born-Oppenheimer polarizability expansions were limited to 4th order. Comparison of these purely electronic Born-Oppenheimer results with recent non-Born-Oppenheimer calculations, which include vibration, reveals the enormous effect of the nuclear motion on the high-order polarizabilities.

Large-order perturbation theory for the O(2) anharmonic oscillator with negative anharmonicity and for the double-well potential

For the O(2) anharmonic oscillator with negative anharmonicity and one-dimensional double-well potential U(x)=(x2-R2)2/8R2, R is a large parameter for both problems, conventional Rayleigh-Schrodinger expansions in power series of 1/R2 for the energy eigenvalues agree. The authors studied asymptotic expansions for the eigenvalues which contain, in addition to the perturbation series, exponentially small terms in R2 due to the tunnelling through corresponding potential barriers. Numerical calculations are also performed and comparisons with the asymptotic formulae are given.

Continued fractions and quantum-mechanical large-order perturbation theory: The anharmonic oscillator revisited

Continued fractions and quantum-mechanical large-order perturbation theory: The anharmonic oscillator revisited

On a field-theoretic non-polynomial interaction

We consider the field-theoretic potential ω 2 x 2 + gx 4/(1 + αgx 2), α > 0, g > 0, and show that there exists a class of exact eigenvalues and eigenfunctions when certain algebraic relations between g and α hold. Comparison with large-order perturbation theory is made and, contrary to recent results, an important difference between the above potential and the field-theoretic ω 2 x 2 + gx 4 interaction is discovered.

Extension of the perturbational-variational Rayleigh-Ritz formalism to large order

The previously derived perturbational-variational Rayleigh-Ritz (PVRR) matrix formalism for calculating the Rayleigh-Schrodinger (RS) perturbation series of the normal modes of astrophysical and other classical systems is greatly strengthened so as to make it a powerful tool for applying large-order perturbation theory to the study of both classical and quantum mechanical eigenvalue problems; the underlying generalised Sturm-Liouville type perturbed eigenvalue equation is encountered in many branches of physics. The original formalism is first modified by introducing a number of auxiliary matrixes which render it amenable to the recursive calculation of the RS series of eigenvalues, eigenvectors, and related expectation values. It is then extended to large perturbational order by deriving and incorporating the generalised PVRR remainder and Hellmann-Feynman theorems: these PVRR matrix theorems include the corresponding RS quantum mechanical theorems as special cases. The resultant modified and extended formalism is applicable to an arbitrary Rayleigh-Ritz (linear variational) Ansatz and enables, in a single computer run, the accurate recursive calculation to extremely high perturbational order of the PVRR series of all eigenstates.

Leading large order asymptotics for (?4)2 perturbation theory

We develop a rigorous semiclassical expansion to compute the radius of convergence of the Borel transform for the pressure in (φ4)2 field theory. This result gives a partial justification for the Lipatov method of finding large order perturbation theory asymptotics in quantum field theory.

Perturbation theory in large order for some elementary systems

AbstractReasons for understanding the general problem of perturbation theory in large order are discussed. It is shown that the behavior of perturbation theory in larger order is generally very simple because it reflects just the semiclassical content of the theory. Many simple examples are given, including some graph‐counting problems. The behavior of mass‐renormalized perturbation series for some very simple field‐theory models is examined. It is shown that mass renormalization hardly affects the large‐order behavior of a weak‐coupling perturbation series. However, in a strong‐coupling perturbation series mass renormalization has a dramatic effect.

Large order perturbation theory in the context of atomic and molecular physics—interdisciplinary aspects

AbstractThe paper is introduced by a description of the role of divergent series in mathematics and their particular use in quantum mechanics. It is emphasized that there is a formal similarity between divergent series in nonrelativistic quantum mechanics and those series occurring in quantum field theory. It is indicated that studies of divergent series in nonrelativistic quantum mechanics can be helpful in the verification of suminability techniques in quantum field theory. The general problems of the application of perturbation theory to the study of the hydrogen atom in external fields are discussed. It is shown how the problem of the continuous spectrum can be bypassed. Further it is shown that algebraic techniques are very useful for a transparent formulation of perturbation theory. The theory is applied to the study of the hydrogen atom in a magnetic field and in an electric field and also to the study of the hydrogen molecule ion. The results for the hydrogen atom in a magnetic field and the hydrogen molecule ion are discussed in detail and the large order behavior of perturbation terms is compared with analytic formulas. Finally the questions of the summability for the series corresponding to the quartic anharmonic oscillator and the hydrogen atom in a magnetic field are discussed. It is seen that large order perturbation theory can be built into a continued fraction representation of a divergent perturbation series. Numerical results are presented for the two perturbation series mentioned above.

Perturbation theory in large order

For many quantum mechanical models, the behavior of perturbation theory in large order is strikingly simple. For example, in the quantum anharmonic oscillator, which is defined by −″ + ( x 2 4 + ϵx 4 4 −E)y=0, y(±∞)=0 , the perturbation coefficients A n in the expansion for the ground-state energy E( ground state) ∼ ∑ n=0 ∞ A nϵ n simplify dramatically as n → ∞: A n∼−( 6 π 3 ) 1 2 Гn+ 1 2 ) . We use the methods of applied mathematics to investigate the nature of perturbation theory in quantum mechanics and we show that its large-order behavior is determined by the semiclassical content of the theory. In quantum field theory the perturbation coefficients are computed by summing Feynman graphs. We present a statistical procedure in a simple λϕ 4 model for summing the set of all graphs as the number of vertices → ∞. Finally, we discuss the connection between the large-order behavior of perturbation theory in quantum electrodynamics and the value of α, the charge on the electron.

Multipseudoparticles and perturbation theory in large order of the anharmonic oscillator

Multipseudoparticles are included in a calculation of the ground-state energy of the anharmonic oscillator in large orders of perturbation theory. Their inclusion verifies a conjecture of Langer and results in a partition function that reproduced the Bender-Wu result.

Anharmonic Oscillator. II. A Study of Perturbation Theory in Large Order

This paper is concerned with the nature of perturbation theory in very high order. Specifically, we study the Rayleigh-Schrodinger expansion of the energy eigenvalues of the anharmonic oscillator. We have developed two independent mathematical techniques (WKB analysis and difference-equation methods) for determining the large- n behavior of A K n , the n th Rayleigh-Schrodinger coefficient for the K th energy level. We are not concerned here with placing bounds on the growth of A K n , as n , the order of perturbation theory, gets large. Rather, we consider the more delicate problem of determining the precise asymptotic behavior of A K n , as n → ∞ for both the Wick-ordered and non-Wick—ordered oscillators. Our results are in exact agreement with numerical fits obtained from computer studies of the anharmonic oscillator to order 150 in perturbation theory.