Alternative analysis to perturbation theory in quantum mechanics

Time-dependent perturbation theory in quantum mechanics is divergent at long times when the perturbation induces a resonance between two eigenstates of the unperturbed Hamiltonian. Divergences in perturbation theory are also common in quantum field theory and in critical phenomena. The renormalization group (RG) was designed to deal with these divergences. In the last two decades, this procedure has been extended to dynamical systems where the perturbation theory diverges in the long-time limit. In this article, we first review the connection between RG in the context of field theory and RG in the context of dynamical systems. We then show that the long-time divergence in the resonant situation in the time-dependent perturbation theory in quantum mechanics can be removed by using a RG-aided calculational scheme.

We derive the formulas for the energy and wavefunction of the time-independent Schrödinger equation with perturbation in a compact form. Unlike the conventional approaches based on Rayleigh–Schrödinger or Brillouin–Wigner perturbation theories, we employ a recently developed approach of matrix-valued Lagrange multipliers that regularizes an eigenproblem. The Lagrange-multiplier regularization makes the characteristic matrix for an eigenproblem invertible. After applying the constraint equation to recover the original equation, we find the solutions of the energy and wavefunction consistent with the conventional approaches. This formalism does not rely on an iterative way and the order-by-order corrections are easily obtained by taking the Taylor expansion. The Lagrange-multiplier regularization formalism for perturbation theory presented in this paper is completely new and can be extended to the degenerate perturbation theory in a straightforward manner. We expect that this new formalism is also pedagogically useful to give insights on the perturbation theory in quantum mechanics.

Perturbation series at large orders in quantum mechanics and field theories: Application to the problem of resummation

A general comparison is presented between random electrodynamics and quantum electrodynamics for the two systems which can be solved exactly in both theories, free electromagnetic fields and point dipole oscillators. The $N$-point correlation functions of the fields are computed in both theories and are found to differ in general because of the dependence upon the order of the quantum operators within products of operators. However, if all products of quantum operators are symmetrized by taking all permutations of the operator order, then the two theories give identical results for the correlation functions. Analogous results hold to all orders in the fine-structure constant for dipole oscllators in quantum and random electrodynamics. The theories agree only if the quantum operator products are symmetrized. In the limit that the oscillator couplings to the radiation fields vanish, th oscllators can be regarded as mechanical oscillators in quantum mechanics and in random mechanics. The theory of random mechanics is defined in terms of this limit which uncouples a mechanical oscillator from the radiation field. The average values of oscillator variables in random mechanics agree with those of symmetrized products in quantum mechanics. The question is then raised as to the physical significance of the many quantum operators which differ only in the order of their factors. It is pointed out that some operator products which are regarded as physically important, such as the square of the angular momentum, indeed involve unsymmetrized products of operators. On this account the average values of the angular momentum squared in the ground state of an isotropic three-dimensional harmonic oscillator differ between the random-mechanical and quantum-mechanical descriptions. However, there seems to be no case in which experiments have shown that the (unsymmetrized) quantum operator value is to be preferred to that provided by random mechanics. The presence of thermal radiation is next treated for free electromagnetic fields and for dipole-oscillator systems. Despite extraordinary differences in the points of view toward thermal radiation taken by the two theories, the conclusion is the same as that found for zero temperature; the two theories agree in their average values if all products of quantum operators are symmetrized. Finally, as a further example of the power of random electrodynamics to give an account of phenomena where Lorentz's classical electron theory failed, we investigate the diamagnetism of a charged three-dimensional isotropic oscillator. The mathematical descriptions at finite temperature are developed in full random electrodynamics and quantum electrodynamics and in second-order perturbation theory in quantum mechanics.

Quantum mechanical and classical methods for theoretical analysis of the emission spectrum due to radiative association are presented. Quantum mechanical perturbation theory is employed to obtain the spectra when the diatomic molecule HF forms by transitions within the electronic ground state and when it forms by transitions between two electronic states. We contrast these spectra with each other. The former peaks in the infrared, while the latter peaks in the ultraviolet. The classical spectrum, which concerns transitions within the electronic ground state, is also calculated and found to favorably compare with that from quantum mechanical perturbation theory. The emission stemming from resonance mediated radiative association is also discussed.

The obvious fact that the eigenfunctions \(\ensuremath \psi(\lambda)\) of the Hamiltonian \(\ensuremath H(\lambda)=H_0+\lambda V\) are determined up to a phase factor \(\ensuremath \exp(i\alpha(\lambda))\) can be used to exhibit a simple method to find the correction \( \epsilon_{n}^{}\) of order n to a non-degenerate energy level. Rules are given to write down all the terms which build up \(\ensuremath \epsilon_n\) in terms of \(\ensuremath \epsilon_k\) , \(\ensuremath k \leq n-2\) , and series of matrix elements of the perturbation V .

Neste trabalho, apresentamos uma revisão da teoria de perturbação na Mecânica Quântica e mostramos que a série perturbativa de Rayleigh-Schrödinger não converge para o resultado exato obtido por integração numérica para o caso do oscilador anarmônico do tipo x4 e do efeito Stark no átomo de hidrogênio. Abordamos um método de renormalização da energia perturbativa, denominado aproximante de Padé, para contornar o problema da convergência das séries de potências. Como ilustração, tratamos o caso da teoria de perturbação de quarta ordem no oscilador anarmônico lambdax4 e verificamos a eficácia do aproximante de Padé em comparação à solução numérica.

Translation invariant calculation of nonperturbative quarkonium mass shifts

In order to calculate the potential energy of a collection of a large number of atoms it is necessary to use the quantum mechanical perturbation theory. The choice of the initial wave functions with which the perturbation calculation is to be carried out, is equivalent to deciding what model of the system shall be taken as the starting point. In the case of crystals the model which involves the simplest assumptions is that in which the crystal is regarded initially as a large number of atoms in their lowest energy state, arranged in a lattice; the lattice constant being great. The perturbation theory is then applied to find how the energy of the system changes as the atoms are brought slowly together; the lattice retaining its original form. It is not necessarily true, however, that when the separation has been reduced to that actually occurring in a given crystal that the system of normal atoms, adiabatically brought together, will be identical with the crystal itself. Thus, for example, Hertzberg has shown that the deepest state of N 2 + does not arise from the adiabatic approach of a normal N atom, and a normal N + ion. When the atoms of the lattice are still well separated it is possible to calculate the mean first order energy using a method given by Heitler. The determinant of the secular equation can be reduced, as Wigner has shown, to a number of irreducible sub-determinants to each of which corresponds a particular term system, and of these, those which satisfy the exclusion principle determine a given total spin moment. The sum of all the energies belonging to one-term system is then given by summing along the diagonal of the corresponding sub-determinant. This procedure, however, assumes that the initial waver functions form an orthogonal set. In the case of a crystal the initial wave functions are to be taken as the product of the wave functions of the separate atoms, and these will only be even approximately orthogonal when the distance between nearest atoms is great.

RF accelerating structures which consist of several more or less identical cavity resonators (cells) placed end to end and strongly coupled electrically to their neighbors are in use or planned for several electron synchrotrons and storage rings, e.g., DESY, SPEAR II, DORIS, PEP, PETRA and CESR. They are commonly designed to be operated in the standing-wave pi-mode with the fields in adjacent cells in antiphase and a physical separation of one half of a free-space wavelength between gap centers. These structures exhibit a band of N closely spaced resonances, where N is the number of cells. These resonances, of which the pi-mode is one, may be identified with the electromagnetic normal modes of the structure and the behavior of the system can be analyzed in terms of these modes. There also exist other bands of modes at higher frequencies which are of importance to the beam dynamics of the machine, but they do not materially affect the response of the accelerating structure at the driving frequency and will be ignored here. The purpose of this note is to characterize the multi-cell structure by means of an equivalent circuit which is valid in the neighborhood of the band, to cast the normal mode equations as a matrix eigenvalue problem, to find the solutions for the ideal (unperturbed) circuit, and to calculate properties of the imperfect (perturbed) circuit using these solutions as base vectors in first order perturbation theory. This approach has been used before. Our aim here is to present it in a somewhat different manner which invites comparison with quantum mechanical perturbation theory and to apply it somewhat differently.

In this work we launch a systematic theory of superconformal blocks for four­point functions of arbitrary supermultiplets. Our results apply to a large class of superconformal field theories including 4-dimensional models with any number N of supersymmetries. The central new ingredient is a universal construction of the relevant Casimir differential equations. In order to find these equations, we model superconformal blocks as functions on the supergroup and pick a distinguished set of coordinates. The latter are chosen so that the superconformal Casimir operator can be written as a perturbation of the Casimir operator for spinning bosonic blocks by a fermionic (nilpotent) term. Solu­ tions to the associated eigenvalue problem can be obtained through a quantum mechanical perturbation theory that truncates at some finite order so that all results are exact. We illustrate the general theory at the example of d = 1 dimensional theories with N = 2 supersymmetry for which we recover known superblocks. The paper concludes with an outlook to 4-dimensional blocks with N = 1 supersymmetry.

Cross sections and rate coefficients for the formation of BeH+ and BeD+ molecules in Be+ + H/D collisions through radiative association are calculated using quantum mechanical perturbation theory and Breit-Wigner theory. The local thermodynamic equilibrium limit of the molecule formation is also studied, since the process is also relevant in environments with high-density and/or strong radiation fields. The obtained rate coefficients may facilitate the kinetic modelling of BeH+/BeD+ production in astrochemical environments as well as the corrosion chemistry of thermonuclear fusion reactors.

The hydrogen fluoride vibrational frequency shift in the NeHF and ArHF complexes in their linear equilibrium configurations has been calculated using a theoretical model derived from quantum mechanical perturbation theory and involving first and second derivatives of the inert gas interaction energy with respect to displacements of the HF internuclear distance from equilibrium in the free molecule. These derivatives were obtained from ab initio calculations at the self-consistent field (SCF) and Moller-Plesset (MP)2 levels of theory. The calculated shifts are compared with harmonic frequency shifts computed from ab initio analytic second derivatives at the SCF and MP2 levels of theory. The frequency shifts for these complexes obtained by these two theoretical approaches are in reasonable agreement for ArHF but differ for NeHF whose shift is nearly zero. The basis set dependence of the shift was investigated at the SCF level for NeHF; large basis sets are required to reproduce the correct sign of the shift...

A formula for the cross section of radiative association where no electronic transitions take place is derived and tested for diatomic molecules. The approach is based on classical mechanics and therefore it is valid for direct, i.e., non-resonant, radiative association. For the formation of carbon monoxide (CO) and the cyano radical (CN), in the X1Σ+ and A1Π states, respectively, the treatment reproduces the baselines of the cross sections obtained using quantum mechanical perturbation theory. The method overestimates the formation cross section of potassium sodide (NaK) by about 8%. For the lower mass diatoms hydrogen fluoride (HF) and deuterium hydride (HD), the formula overestimates the cross sections by 12% and 60%, respectively. The formula can be used alone for estimates of radiative association rate constants, or in combination with Breit-Wigner theory to include resonance contributions.

Classical perturbation theory provides a particularly promising route to EBK quantization of nonseparable systems. However, the number of terms generated when implementing perturbation theory for systems with more than two degrees of freedom can prove too large for general purpose symbolic manipulators to handle. We describe PERTURB, a specialized algebraic manipulation program written in C for quantization of multidimensional systems. A review of operator based classical perturbation theory is given, and the relationship between this type of perturbation theory and quantum mechanical Van Vleck perturbation theory discussed. The relative performance of the Dragt–Finn and Lie transform algorithms is assessed.