Quantum Mechanical Perturbation Theory Research Articles

General connection between random electrodynamics and quantum electrodynamics for free electromagnetic fields and for dipole oscillator systems

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.

Residue-squaring iterative diagonalization method for perturbation problems

We present a new method for the evaluation of the change in eigenvalues due to a perturbation of strength λ. It is a fast converging iterative method which, at thenth step, gives results accurate to order (2 n+1−1) in λ. Unlike the Rayleigh-Schrodinger perturbation theory in quantum mechanics, which becomes prohibitively cumbersome when carried to higher orders, the present method involves a routine which remains stralghtforward at all stages.

The interaction energy between hydrocarbon chains

A method is proposed for the determination of the interaction energy between isolated, parallel oriented, saturated, aliphatic hydrocarbon chains and the calculation is extended to include the monolayer case. The treatment is based upon the use of the second-order perturbation theory of quantum mechanics for the determination of the attractive (dispersion) component of the interaction potential. It is made applicable to long saturated hydrocarbon chains by utilizing the bond polarizability approximation method. The results compared favorably with those obtained by other investigators. An approximate form for the repulsive component of the interaction potential is provided. The total interaction potential obtained is approximately of the (5–11) form. The minimum energy position is calculated for the hydrocarbon chain crystal model allowing free rotation around the molecular axes by rotational averaging of the polarizability. The minimum energy position may occur at 5.0 Å interchain separation for the hexagonally packed hydrocarbon monolayer with 18 carbons per chain.

Multipolar expansions for magnetic shielding near anisotropic molecules

The McConnell-Buckingham-Stiles expansion Ʃ ∞ L = 2 Ʃ L M = 0 ( A LM cos M Φ + B LM sin M Φ) P M L (cos Ɵ)/ R L +1 describes magnetic shielding at the point P ( R , Ɵ, Φ) outside a molecule in terms of anisotropies A LM and B LM in the magnetic multipolar susceptibilities and the associated Legendre polynomials P M L (cos Ɵ). This expansion is restricted to regions further from an internal origin than the most distant part of the molecular electron-density. It is shown that shieldings in other regions adjoining the molecule are conveniently calculated using the expansion Ʃ ∞ L = 0 Ʃ L M = 0 ( F LM cos M Φ + G LM sin M Φ) R L P M L (cos Ɵ) about an external origin selected to lie closer to field-points of interest than to the molecule. Second-order quantum-mechanical perturbation theory has been used to derive formulae for the observables A LM , B LM , F LM and G LM , each of which is a sum of diamagnetic and para-magnetic components. The orbital component of the dipolar term F 00 in the latter expansion is the Lamb-Ramsey expression for magnetic shielding.

Quantum theory of concerted proton transfer reactions in polar media. Linear electronic terms

A quantum mechanical theory has been developed for concerted two-proton transfer reactions in polar media. The formulation is based on second order quantum mechanical perturbation theory; this assumes that the intermediate state, i.e., the state prevailing after the transfer of the first proton, is short-lived compared with the uncertainty time of the system. The energy of activation Ea of the overall process is determined by the value of Ea for either the first or the second proton transfer, and it is shown how, at least in principle, a distinction from a one-proton and a consecutive two-proton transfer reaction can be made. Important qualitative and semiquantitative correlations between energy of activation and overall energy of reaction are predicted, and the Brønsted coefficient and pre-exponential Arrhenius factor are discussed.

Theory of electrode processes

The transfer of an electron from a metal electrode to an electroactive species in the interface is central to electrochemical equilibrium and kinetic analyses. The purpose of this paper is to present a discussion of the theory of electron transfer based on a linear response analysis. We base our theoretical considerations on the Yamamoto reaction rate theory; Yamamoto's theory is an outgrowth of the Kubo linear response theory. The rate constant and ultimately the exchange current expressions we obtain are derived from a model representation of the electrode interface. This model is a generalization of one considered by Dogonadze and Chizmadzhev. In particular, we examine an outer sphere type transfer mechanism, but we allow for the following effects: (1) the effect of reactant center of mass motions and transport to a most favorable electron exchange configuration, (2) the effect of internal vibronic interactions within the electroactive species, and (3) the effect of strong electron exchange interactions. This last effect precludes the direct use of quantum mechanical perturbation theory to obtain the rate constant expressions. For our model system we obtain general rate expressions valid for adiabatic to nonadiabatic transfers. This follows from the generality of the Yamamoto theory. We find particularly that the effects of reactant transport and internal vibronic contributions can be cast into a virial form for the rate constant. Moreover, the activation energy, in the Arrhenius sense, contains new terms relating specifically to the new effects we consider, namely, transport and vibronic effects.

Approximate Functional Integral Methods in Statistical Mechanics. I. Moment Expansions

In this paper, four distinct ideas are combined, which under a wide range of circumstances can give very rapidly converging series expansions for functional integrals. (1) Expansion of the functional being integrated in functional Taylor series. In the familiar case arising in quantum statistical mechanics, that of the Wiener integral of exp[−∫V(x(t))dt],V(x) being the perturbing potential, this is equivalent to expanding the characteristic functional of the probability functional of V(x(t)) in central moments of V(x(t)). The lowest-order term of the series is the approximation obtained by Feynman and Hibbs through a variational method. (2) Transfer of the harmonic term of the potential, when the functional integral is the quantum-statistical density matrix (Green's function of the Bloch equation), to the weighting function. This transforms the functional integral from a Wiener to an Uhlenbeck-Ornstein integral. The formal expressions for the terms of the expansion are somewhat more complicated, but they can be worked out, and the result is a great improvement in the speed of convergence of the series with decreasing temperature and/or decreasing relative magnitude of the anharmonic part of the potential. (3) ``Reservation of variables'' in the integration. This amounts to breaking the averaging process down into an average over subsets of the distributed random function (conditional average), followed by an averaging of these averages. Any step of this kind (it may be repeated within the subsets, etc.) gives an improvement of accuracy. (4) When the quantity being evaluated through functional integration is the partition function, the device introduced by Feynman and Hibbs, of interchanging the functional integration with the integration of the Green's function over the equated initial and final configuration-space points, may be combined with the above techniques. This eliminates one integration in the terms of the expansion and seems to improve accuracy at the same time. The general series obtained is correlated with more conventional operator techniques of quantum-mechanical perturbation theory, in order to answer the perennial question, does the path-integral method bring with it anything that could not be derived by other methods? It is in some sense a Feynman-Dyson expansion of the Green's function, but one that is further modified mathematically in a way characteristic only of the path-integral point of view, and which, moreover, improves its accuracy. It thus appears unlikely that the result is merely one of standard type disguised as a functional integral result. Sample numerical calculations are given to assay the accuracy of the methods, which are shown to compare very favorably with the traditional approximation of finite subdivision of the time interval.

The Product-Integral Calculus Formulation in Quantum Mechanics

In this paper we investigate the advantages of using the product-integral calculus rather than the usual successive approximation perturbative expansion in three branches of quantum physics. The mathematics of the calculus is fully developed, and its superiority over the perturbative expansion is explored in specific examples from physics. All of time-dependent perturbation theory in quantum mechanics can be written in terms of the calculus, as well as the most general solution for the density operator in quantum statistical mechanics.

Determination de la densite electronique d'un plasma ionise a partir de la distance entre les pics de la raie ‘permise’ LiI [formula omitted] et de la raie ‘interdite’ LiI [formula omitted

The aim of this investigation is to verify the method of measuring electron concentrations in partially ionized plasmas on the basis of a comparison of an ‘allowed’ line [such as the LiI (2s 2 2p 2P−2s 2 4f 2D) line] and a ‘forbidden’ neighbouring line induced by the ionic field [such as the LiI (2s 2 2p 2P−2s 2 4f 2F) line]. First, the line profiles are calculated theoretically. They are then compared with the observed profiles. It is found that the separation Δ( N e ) between the intensity maxima of the two lines may be conveniently used for the evaluation of the electron concentration N e . This method is more convenient than one based on the measurement of the width of the ‘allowed’ line or on measurement of the maximum intensity ratios of the ‘allowed’ and ‘forbidden’ lines. The electromagnetic LiI and CsI field-free energies of excited levels, which have not been recently tabulated, have been calculated by using a hydrogenic model corrected for polarization potential. When an electromagnetic field ( F, H ) is present, application of quantum-mechanical perturbation theory yields the eigenvalues and eigen-vectors of the excited atomic wavefunctions and, therefore, the shifts, and intensities of components. When the magnetic field is absent, the profiles of the LiI 2 P−4 R ( R = P, D, F) and CsI 5D 3 2 −6L (L = F, G, H) lines broadened by the Stark effect are calculated for a range of electron densities from 1×10 15 cm -3 to 5×10 16 cm -3, and for electron temperatures from 4000 to 6000°K. The ionic effect is described by using the quasi-static approximation F = βF 0. The electron contribution is described by using a generalized impact theory corrected for the Lewis cut-off. We have taken into account the contribution of all sublevels l, m of the upper state, which is perturbed by the ionic field F. The electronic width and shift are calculated using closed form expressions.

Unitarity, Factorization Theorem and Removal of Level Degeneracy in Dual Resonance Model

It is shown that, by using a unitarity relation for a many-body scattering amplitude, a Regge residue appearing in this amplitude is factorized. This theorem is a generalization of that in the t~o-body case. According to this theorem the famous level degeneracy in dual resonance models is expected to be removed, if unitarity is taken into account. We propose an eguation for removing the degeneracy, which has a close resemblance to that in the quantum mechanical perturbation theory with degenerate levels. shown that the number of resonant states with mass Sn, sn being defined by n =a (sn), is given by exp (aVn), (a= 2n/ -/6), for a large n. In other words, the N-point Veneziano formula has the extremely large number of degenerate levels. On the other hand, we can see in § 2 that, by using a unitarity relation for a many-body scattering amplitude, a Regge residue appearing in this amplitude is factorized, that is, the number of the states having mass sn is only one. This theorem is a generalization of that in the two-body case. 2 > The Veneziano amplitude violates unitarity in a way similar to that in which the Born approximation does in the usual Feynman-Dyson theory. It may be expected, therefore, that the level degeneracy in the dual resonance model will be removed if unitarity is taken into account. If, for the purpose of imposing unitarity, we sum up all the higher order diagrams in the framework of the operator formalism, the poles of the amplitude appear as poles of the propagator. Though the self-energy part of this clothed propagator is strongly divergent, we expect that some kind of renormalization procedure will be· possible. 3 > (We do not consider this procedure in this paper.) ln order to obtain the observed masses, that is, the zeros of the inverse propagator, we propose in § 3 a method very similar to that in the quantum mechanical perturbation theory with degenerate levels.

Orbital symmetry rules and the mechanisms of inorganic reactions

In the last few years symmetry arguments have been used very effectively to predict the course of chemical reactions. The Woodward—Hoffmann rules are famous examples. A complete, but simple, theory of how symmetry enters into a chemical process can be given. Use is made of group theory and secondorder quantum mechanical perturbation theory. The resulting simple equations can be reduced even further to a consideration of the symmetry of the molecular orbitals of the reactants. The relevant orbitals are the highest filled (HOMO) and the lowest empty (LU MO) with the correct symmetries to match the symmetry of the reaction coordinate. The closer in energy these orbitals are, the lower the activation energy. An orbital symmetry forbidden reaction is one where no orbitals of the right symmetry exist within a reasonable energy range of each other. In the usual case it is unnecessary to know the molecular orbital scheme of the products. For bimolecular and trimolecular reactions, the reaction coordinate must be totally symmetrical, therefore the symmetry requirement for the HOMO and LUMO is that they have a net positive overlap. For unimolecular reactions, the reaction path need not be totally symmetrical. The direct product of the HOMO and LUMO symmetries determines the symmetry of the reaction coordinate. The HOMO and LUMO also must correspond to bonds that are to be broken and bonds that are to be made; if they are bonding MOs the reverse statement holds true for antibonding MOs. Examples are given for all of these rules. The development of so-called orbital symmetry rules for chemical reactions has had a great impact on organic chemistry'. Corresponding rules for inorganic reactions have not been extensively presented or used up to now. The attemptin the literature2 have dealt only with the d orbitals of transition metal complexes. The conclusions have been neither definitive nor consistent. While d orbitals are of great importance in coordination chemistry, it is unlikely that these are the only important orbitals in chemical reactions. Also much of inorganic chemistry deals with the non-transition elements. It is necessary to include molecular orbitals made up of s and p atomic orbitals to have a complete understanding. In this article we will show in the most general way how symmetry rules for all chemical reactions can be derived3. The procedure used is to consider the variation of potential energy with

Retarded Dispersion Energy between Macroscopic Bodies

We calculate the change in self-energy of the electromagnetic radiation field in the presence of two dielectric bodies $A$ and $B$. Starting from Maxwell's equations, the perturbed radiation field is expanded in terms of plane waves. The perturbed frequencies are obtained by applying quantum-mechanical perturbation theory. The sum over the perturbed minus the unperturbed frequencies, giving the retarded dispersion energy between the two bodies, is evaluated explicitly for the case of two spheres $A$ and $B$. It is shown to assume a finite value at zero separation $d$ of the spheres, so that no assumptions regarding an appropriate minimum separation are necessary. The total dispersion energy between bodies $A$ and $B$, which includes also the energy gain of the material modes, is found via a complex integral transform. The total dispersion energy at small and medium separations is in agreement with previous results, i.e., we obtain a ${d}^{\ensuremath{-}1}$ relationship at separations smaller than the radii of the spheres, and a ${d}^{\ensuremath{-}6}$ relationship at separations larger than the radii of the spheres. In the retarded case, i.e., at separations $d$ large compared with the characteristic wavelengths in the absorption spectra of the dielectrics, the dispersion energy is found to obey a ${d}^{\ensuremath{-}2}$ law at separations $d$ smaller than the radii of the spheres and the Casimir-Polder ${d}^{\ensuremath{-}7}$ law at separations $d$ larger than the radii.

An expansion of the master equation with applications to coupled atom-radiation systems

A straightforward extension of time-dependent quantum-mechanical perturbation theory is made to obtain the equivalent formulae for a statistical mixture of states. The 'master equation' resulting from this procedure is, in its lowest order, the analogue in quantum statistics of the 'Golden Rule' for transition rates and reduces exactly to this form in the limit of pure initial states. The authors give results to fourth order and illustrate their use by applying them to the problem of the resonant interaction of an electromagnetic field and a two-level atom.

Theory of Solvent Effects on Oscillator Strengths for Molecular Electronic Transitions

Abstract A theoretical expression has been presented for an oscillator strength of an electronic transition in a solute molecule immersed in an isotropic dielectric continuum of solvent. It is assumed that a charged particle in the solute is subjected to a force due to a cavity field. The approximate expression derived on the basis of a quantum-mechanical perturbation theory has been successfully tested for nitrobenzene and acetone.

Quantum statistical mechanics of message transport in nerves

In Mathematical Biosciences 5(1969), 495–509, I introduced a new theory of message transport in nerves based on an interpretation of the role played by the sodium ion diffusion in the transport process. In this paper I report the results of a quantum statistical mechanical theory of the fundamental message transport step. The theory is developed using the assumption that the message charge transfer step and the sodium ion diffusion from the Donnan nonequilibrium state are coupled through modes of the environmental dielectric continuum. The analysis is carried out by means of second-order, time-dependent, quantum mechanical perturbation theory. The analysis yields an expression for the message current which is of the form of a second-order kinetic process. The requirement of at least a second-order rate for the message transport mechanism was pointed out in the above mentioned paper.

Electron Eigenstates in Uniform Magnetic Fields

The purpose of this paper is to show in detail how the electron energy-eigenvalue spectrum changes from the form εB = (N+12)ħωc+Pz2/2M in a finite magnetic field to the free-particle form ε0 = P2/2M in zero magnetic field. We consider an electron confined to a “box” of finite volume and calculate the exact eigenstates for arbitrary magnetic field strengths. When the electron cyclotron radius rc = (2ħ/Mωc)1/2 is small compared to the dimensions of the box, we get the spectrum εB, and when it is large we get ε0. The key to this demonstration is a mathematical identity which relates the magnetic eigenfunctions to the free-particle eigenfunctions. Magnetic corrections to the spectrums ε0 are obtained without the use of quantum mechanical perturbation theory.

Upper and lower bounds in quantum mechanical perturbation theory using linear programming

Upper and lower bounds to quantum mechanical second order perturbation quantities are found using the method of Linear Programming. In this paper bounds on the polarizability of hydrogen, helium and neon are obtained using the sum rules popularized by Dalgarno.

Brillouin Light Scattering from Crystals in the Hydrodynamic Region

With the availability of modern lasers, light scattering can now be used as a probe of the energy, damping, and relative weight of the various hydrodynamic collective modes in anharmonic insulating crystals. We first give a general review of the distinction between hydrodynamic and high-frequency (or dynamic) vibrational modes. We then express the intensity and spectral distribution of scattered light in terms of the Fourier transform of the displacement-displacement correlation function ${\ensuremath{\chi}}^{\ensuremath{'}\ensuremath{'}}(\mathrm{K}\ensuremath{\Omega})$, which is the spectral weight of the phonon propagator. This follows some work of Loudon, who has discussed first-order Raman (or Brillouin) scattering using standard quantum-mechanical perturbation theory. Next we summarize what can be said about the spectral weight ${\ensuremath{\chi}}^{\ensuremath{'}\ensuremath{'}}(\mathrm{K}\ensuremath{\Omega})$ in the region of low frequencies (or small energy transfer). We use the model calculations of Kwok and Martin, as well as the standard theory of an elastic medium with a nonlocal form of Fourier's law of heat diffusion. In the case of pure isotropic anharmonic crystals ${\ensuremath{\chi}}^{\ensuremath{'}\ensuremath{'}}(\mathrm{K}, \ensuremath{\Omega})$ has resonances corresponding to first-sound (pressure) and second-sound (temperature) waves, in addition to that from transverse or shear elastic waves. Unless fairly restrictive conditions are met, the second-sound wave does not propagate and reduces to the ordinary thermal diffusion mode of Landau and Placzek. The special nature of second sound in He II is discussed in an Appendix.

Operator equations in perturbation theory

Quantum mechanical perturbation theory is discussed in terms of solutions W of commutator equations of the type [ h , W ] = A . The operators W lead to perturbation corrections for any eigenstate of the unperturbed system. The theory is illustrated by considering the linear harmonic oscillator perturbed by the N th power of the displacement.

Perturbation analysis of axially nonuniform electromagnetic structures using nonlinear phase progression.

A perturbation technique is developed to deal with axial, as well as transverse, perturbations of an electromagnetic propagating structure which is otherwise axially uniform. The method, an adaptation and extension of time-dependent perturbation theory of quantum mechanics, uses a nonlinear phase progression term in the axial propagating factor of the fields to accommodate and readily calculate, without secular terms, corrections to the progressive phase delay along the perturbed structure. The technique admits inhomogeneities and anisotropy and does not depend upon quasi-static or quasi-optic assumptions. It is particularly useful for axially dependent perturbations which give rise to a phase shift, or to radiation, or to any other effect of interest which is strongly dependent on the phase progression of the wave. The method is first illustrated by an analysis of simple axial loading of a waveguide, and then by a calculation of the radiation from a surface wave structure with a ridged dielectric binding medium. Finally, the casting of Maxwell's equations for general situations into a form appropriate for the application of this perturbation analysis is discussed.