At present, the general problem of anharmonic vibrations commands a significant place among scientific investigations in various branches of experimental and theoretical physics. The phenomena of anharmonicity are displayed in the vibrations of molecules and crystals, in the mechanics of molecular rotations and librations, in the resonance interaction of vibrational levels, in electro-optical effects, in non-linear spin interaction, and so on.

Behind the scenes of various phenomena, a unique essence is hidden. Through the absence of a perfect harmony, the various physical effects become possible. A harmony fails to be a tendency to a simple ideal, but it is a capacity of the nature to order an anharmonicity. The understanding of the natural beauty leads us to the anharmonic world. Under the constraint of physical laws, the anharmonicity bears the harmony…

The simplest choice to describe the vibrations of an arbitrary system in quantum mechanics is a model of harmonic oscillators. A harmonic case is certainly only an idealization of vibrations of a real system. The potential energy of vibrations, as in classical physics, is generally written in a form of an expansion in terms of normal coordinates; in this expansion, the linear terms disappear through the fact that the first derivative of the potential equals zero at the equilibrium condition. A set of harmonic oscillators corresponds to a first approximation. This model is only qualitatively correct: vibrations, failing to conform to a harmonic law, are anharmonic. To describe correctly the vibrations, apart from the quadratic part of the potential energy, one must therefore take into account the normal coordinates to greater than quadratic powers in an expansion of the potential. These terms additional to the harmonic Hamiltonian are defined by anharmonicity coefficients and characterize the interactions among various vibrational modes. The calculation of the corresponding corrections is generally performed with a perturbation theory for stationary states of a perturbed Hamiltonian, in which the perturbation function represents an expansion in powers of a small parameter. A formalism in terms of the polynomials of quantum numbers might serve as one example of the perturbation methods. In the book ‘Uncommon Paths in Quantum Physics’, we consider the polynomial formalism in detail.

First, the polynomials form, with the required accuracy, all necessary physical observables of the anharmonicity problem. The desired quantities are obtained immediately on solving or opening the recurrence equations or relations avoiding conventional intermediate manipulations. We compare two schemes to construct the stationary perturbation theory:

1) Schrödinger equation → eigenfunctions and eigenvalues →

→ matrix elements;

2) recurrence equations → eigenvalues and matrix elements.

The first scheme is conventional, whereas we proposed the second scheme. The main disadvantage of the conventional scheme is that, at each stage, one must return virtually to the beginning — to the Schrödinger equation — to improve the eigenfunctions by increasing the order of the perturbation calculation. Only after these calculations is one in a position to evaluate the matrix elements. In our method, intermediate calculations are performed on an equal footing; i.e., the procedures to calculate the eigenvalues and arbitrary matrix elements are performed simultaneously.

Second, the proposed theory automatically keeps track of non-zero contributions of the total perturbation to the result sought, and takes into account the history of the calculations, i.e., the intermediate calculations. This advantage is achieved on expanding, in a small parameter, the derivatives of the energies and their wave functions, rather than by expanding the eigenfunctions and eigenvalues as is done traditionally. In this sense, the expansion in exact eigenvectors plays a principal role [K. V. Kazakov, Electro-optics of molecules, *Opt. Spectrosc.*, **97**, 725—734, 2004],

because it ensures a full use of the history of the calculations and, consequently, significantly simplifies the general solution algorithm. If the expansion is performed in terms of the exact eigenvectors, rather than in terms of zero-order basis functions, it is assumed that the former functions exist and are expressible algebraically, for example, with recurrence relations. In addition, one might avoid the renormalization of the function; this problem presents considerable difficulties in the traditional approach in which the function should be renormalized upon passing from one perturbation order to the next.

Other advantages of this method appear in various applications of this perturbation theory [K. V. Kazakov, *Quantum Theory of Anharmonic Effects in Molecules*, 2012, Elsevier]. For example, in a framework of the polynomial formalism, one might consider the problem of electro-optical anharmonicity; this problem involves an electric dipolar-moment function in a non-linear form, and its solution requires evaluation of matrix elements. The absolute values of dipolar-moment derivatives might be unknown beforehand, which complicates the problem. In the traditional formalism, the consideration proceeds, as a rule, from the wave function of a definite order, which leads to the loss of significant contributions. In the polynomial formalism, we consider separately each term in an expansion of the dipolar-moment function, and, consequently, calculate the entire matrix element in a given order in a small parameter.

** Beyond the Predictable Trend**

A prospectively fruitful direction for further investigation is to proceed beyond solutions with perturbation theory. We assume that the effective internuclear potential is a real function that is represented as an expansion in a power series in terms of the normal coordinates. In this case, the procedure of quantization, i.e., the calculation of matrix elements of an arbitrary coordinate function, taking into account the influence of anharmonicity, is reduced to the sum of polynomials multiplied by factor √g [K. V. Kazakov, *Uncommon Paths in Quantum Physics*, 2014, Elsevier]:

Expanding here the polynomials in terms of quantum numbers, we obtain this intriguing formula,

for the one-dimensional case, and this one,

for the many-dimensional case.

The derived expansions in terms of quantum numbers hold for the matrix elements of an arbitrary physical function that is represented as an expansion in a power series in terms of creation and destruction operators. This consequence of perturbation-theory calculations is trivial. The values of energy *E _{n}* are expressible also from the formula for (

in which Ω-coefficients_{ }are the mechanical anharmonicity parameters. To generalize our theory, we assume that quantity *E _{n}* is a function of quantum numbers

We can determine heuristically a function Φ for the matrix element of a particular physical quantity *ƒ*(*ξ*), for instance, the dipolar moment, as a dependence on quantum number *n+k*/2+1/2 :

Functions Φ* _{k}* are here arbitrarily expressible, for example,

with parameters θ_{k }_{ }and ϕ* _{k }*determined from experiment. In the present formalism, one might also construct phenomenologically function Φ for a system with

Konstantin’s books *Uncommon Paths in Quantum Physics* and *Quantum Theory of Anharmonic Effects in Molecules* are both available on the Elsevier Store. Use discount code “STC215” and **save up to 30%** on your very own copies!

**About the Author**

Konstantin V. Kazakov obtained a Dr. Sc. in Physics and Mathematics at the St. Petersburg State University. He has published papers in internationally scientific journals, communications at scientific symposia and congresses, as well as 3 books.

]]>**On The Theory of Anharmonic Effects in Quantum Physics**

Vibrational phenomena have always fascinated scientists and engineers. A molecule constitutes a vibrational system of an important class that is mainly the subject of our present concern. High-resolution infrared absorption spectra provide information about the distribution of vibration-rotational energy levels and the transition probabilities of real molecules. Spectral lines command physical interest through their interpretation with the aid of physical models, i.e., the relation of frequencies and intensities of spectral lines to molecular motions of various types. As the precision of measurements made with various experimental techniques increases relentlessly, the interpretation of observed spectra becomes correspondingly challenging. This condition stimulates the search for, and development of, innovative methods to investigate vibrational systems for which a conventional description fails.

Intuitively, the most natural model of intramolecular motions involves interacting anharmonic oscillations of atomic centers, but this simple physical model lacks a mathematically exact solution. The use of perturbation theory, however, solves the problem. This classical method is simple and clear, but its application is generally limited to the first few orders of theory that any textbook on quantum mechanics describes. The determination of corrections of higher orders becomes complicated through the sheer bulk of the calculations. The calculation of frequencies and intensities of spectral lines with an accuracy defined by experiment hence becomes difficult. A real spectrum of a sample containing even diatomic molecules of a particular chemical compound can comprise lines numbering a few thousands. Despite these difficulties, some success in developing an adequate method of calculation has been achieved, embracing perturbation theory. In what follows in the book ‘Quantum Theory of Anharmonic Effects in Molecules’, we consider the development of techniques of perturbation theory applied to problems of molecular spectroscopy to calculate the frequencies and intensities of vibration-rotational transitions.

Historically, a quantum-mechanical consideration of the anharmonicity of diatomic molecules began with Dunham’s work; deriving matrix elements for vibrational transitions up to the third derivative of the dipolar moment in terms of perturbation theory, he determined a numerical value for the second derivative of the dipolar moment function of a HCl molecule from the experimental distribution of intensities in the infrared spectrum of a gaseous sample. Using various computational methods and varied initial assumptions about functions for potential energy and dipolar moment, other authors have subsequently tried to improve the techniques of calculations. In this regard, we mention specifically the hypervirial theorem, the method of Feynman diagrams and the canonical or contact transformations. The objective of the respective authors was typically the eventual results; the procedure of the calculations was thus afforded little attention.

Although for diatomic molecules an application of the hypervirial theorem was fruitful in calculations of matrix elements of a one-dimensional anharmonic oscillator through recurrence relations, this method is inefficient for polyatomic molecules.

The method of Feynman graphs enables one to eliminate the recurrence scheme of perturbation theory. Circumventing calculations of preceding orders, one might work directly with expressions for wave functions and energy of arbitrary order; this capability is a great advantage of this method. A characteristic of problems in molecular spectroscopy is, however, that one must initially calculate corrections of low order and only then proceed to approximations of higher order. The stated advantage for calculations of low order is rapidly lost in corrections of higher order. For example, the conversion of a diagram of twentieth order into an algebraic expression becomes a complicated procedure, for which one must be concerned about the risk of error.

A systematic investigation of vibration-rotational spectra of polyatomic molecules has been conducted mainly with the method of contact transformations, which allowed the retention of the *q*-number approach and eliminated a problem of superfluous summation over the matrix elements. Although corrections in canonical transformation theory are considered to be equivalent to approximations of the common perturbation theory, this point of view is inaccurate; rather, this method can be used to choose an effective hamiltonian. For instance, Watson proposed an hypothesis that there exist many rotational hamiltonians, which all describe experimental data equally validly. Choosing an initial hamiltonian, by means of a convenient canonical transformation we obtain another hamiltonian that yields the same eigenvalues and has a simple parameterization for the interpretation of experimental data. The principal deficiency of this method is that it lacks a clear form of all expressions; as a result, formulae become much too bulky, which impede a clear understanding by experimenters.

Dunham’s practice of standard perturbation theory can be extrapolated to polyatomic molecules, but alternative algorithms of perturbation theory for the pure vibrational problem have been developed. According to the book ‘Uncommon Paths in Quantum Physics’, a novel method within a formulation of quantum theory is based on differentiation with respect to coupling parameters; it produces simple and clear equations for matrix elements. Essentially a recurrence scheme, it represents a form of solution involving polynomials of quantum numbers. This formalism allows one to generate rules to calculate observable matrix elements, which determine the frequencies and intensities of vibrational transitions. This approach is reminiscent of Feynman diagrams: we calculate all desired polynomials, make convenient tables and then express physical quantities in terms of the polynomial quantities.

The principal objective of this formalism is to simplify the traditional perturbation theory. According to this polynomial method, we accrue all advantages and avoid all shortcomings of the preceding techniques. Efficient for both diatomic and polyatomic molecules, this method is free from the problem of superfluous summation. A convenient recurrence scheme implemented with contemporary computers allows one to optimize all calculations and to decrease greatly the duration of calculations of vibrational frequencies and intensities. When we allude here to approximations of higher order, we have in mind perturbation theory in the tenth or twentieth orders.

Introducing this formalism certainly does not solve all problems: many specific questions, such as those concerned with the effects of vibration-rotational interaction, remain. For instance, a theorem of extraneous quantum numbers has been formulated; with its help an exact solution for coefficients of the Herman—Wallis factor has been obtained — this method is highly original. As a result, we greatly simplify the calculation of intensities for diatomic molecules. For arbitrary linear polyatomic molecules, a comparable success is foreseen, but the possibility of extending this theorem to describe the vibration-rotational spectra of nonlinear molecules has yet to be investigated.

**The Creation and Destruction of States and Quanta **

To solve most equations of quantum mechanics, one generally applies the powerful apparatus of mathematical physics, which is based on traditional methods of the theory of integro-differential equations. Many problems might otherwise be solved in a purely algebraic manner. For instance, to describe a vibrational system in quantum mechanics, one uses the model of an anharmonic oscillator. This simple model might be regarded as founded on an exact solution of the problem for a harmonic case that substantially represents the description of some physical system in an approximation of zero order.

As the necessity to take into account the influence of anharmonicity increases, this solution becomes improved through the pertinent methods of perturbation theory. The non-zero matrix elements between the corresponding states of a perturbed system determine the observable quantities. In the case of the first few orders of the theory, the matrix elements are readily calculated in an algebraic manner, for instance, in the framework of a formalism of creation and destruction operators that follows from the classical work by Fock and Dirac. The calculations of higher orders are performed with the aid of special methods; the recurrence formalism of the perturbation theory in terms of the polynomials of quantum numbers might serve as an example of one among these special devices.

In some applications it is convenient to use, instead of the harmonic oscillator, the Morse oscillator as a zero-order approximation. Applying in this case the recurrence formalism of the perturbation theory, one might, in a manner analogous to the solution in the form of the polynomials of quantum numbers, evaluate the influence on the energy levels and the matrix elements of a term additional to the anharmonic field of the Morse potential that plays the role of a perturbation. Similar conclusions are applicable to another important case in which a non-perturbed system is described through the states in the form of the solutions of Schrödinger’s equation for the Pöschl—Teller potential. Moreover, in a search for a solution of each such problem, the methods of non-commutative algebra according to the language of so-called ladder operators, which are substantially the same of those as the creation and destruction operators, become applicable.

For a concrete physical problem, the terminology of ladder operators might be introduced in not just one way. For instance, a traditional analysis involves appropriate recurrence relations for special functions corresponding to the exact solutions of the Schrödinger’s equation, in which in the role of the potential appear the well known functions of type Morse and Pöschl—Teller, and also a series of other simple potential functions. The purely algebraic methods of factorization are applied less commonly. Among the latter algebraic enunciations is a technique of factorization described by Green that is simple and elegant: in the books ‘Quantum Theory of Anharmonic Effects in Molecules’ and ‘Uncommon Paths in Quantum Physics’, we consider it in detail.

**‘Dirac’s Theory Resolves the Paradoxes of Quantum Electrodynamics**

The anomalous magnetic moment of an electron and the Lamb shift of atomic levels of hydrogen have played a key role in the development of the contemporary quantum electrodynamics. The theoretical investigations were, however, far from immediately able to confirm the experimental facts unique at that moment of time. For instance, the calculation of the self-energy of an electron, as in classical electrodynamics, yielded an infinite value. The situation was analogous to the calculation of the electromagnetic shift of atomic levels. As Bethe said, “one might overlook the Lamb shift, because the latter was infinite in all the then existing theories”. This scenario was generally applicable to all infinities, which were simply discarded. Do the infinities lack a physical meaning?

It subsequently became clear that the reply to the appearance of infinities was hidden directly within them. On quantizing an electromagnetic field, we discard an infinite -number, associating it with the zero-point vibrations of the field or, as one says, with an electromagnetic vacuum. One might ignore the vacuum, but only when we consider the transitions of an electron between excited states. Moreover, there exists an electron-positron vacuum that is otherwise called the Dirac sea. To calculate the radiative corrections, we must hence take into account the interaction of electrons with the virtual electron-positron pairs of the Dirac sea, not only with the zero-point vibrations of an electromagnetic field. As a result of the interaction with the «vacuum», there arise the remarkable effects of quantum electrodynamics — the anomalous magnetic moment and the Lamb shift of atomic levels.

Discussing the fields, we imply real physical objects, for instance, the electrons and photons. At present, a field formalism extends far beyond the relativistic quantum mechanics: it is applicable in the physics of solids, in the theory of an atomic nucleus and in the theory of a plasma, and also in many other branches of physics, to describe the principal characteristics of quantum systems. It is well known that, to the vibrations of a crystalline lattice, one might ascribe a field of phonons, and, to describe ferromagnetic phenomena, one should apply a concept of spin waves that represent a field of magnons. One might continue this series, but it is already clear that the method of quantum fields is firmly entrenched in a prime place in all quantum physics. Even for some problems of the simplest atoms such as hydrogen and helium, it becomes simpler to apply a many-particle description than a clear traditional representation of a system in a form of several interacting particles. Through the fields we comprehend the particles.

In the book ‘Uncommon Paths in Quantum Physics’, our aim was to generate a first acquaintance with the field point of view that allows one to enter the general range of questions of quantum electrodynamics. In the Heisenberg picture, we have obtained the information about an electron when considering the temporal variation of its creation operator. One might perform something similar also for a creation operator of a photon, and further, for example, for an operator of creation of a muon, under a condition, of course, that it becomes possible to introduce a muon field into the theory. We fail, however, to derive thereby special dividends. For instance, the self-energy of a photon becomes infinite. Yes, we might apply the procedure of a regularization to ensure that this energy becomes large but finite; we cannot, however, eliminate it entirely. If a Hamiltonian comprises a mass parameter, we can manipulate it, suggesting that there exist an electromagnetic part of a mass, a meson part and other. For a photon, this scenario is questionable.

Beginning with the Bethe formula, there appeared many works aiming at seeking practical rules of a game with expressions that generally diverge, but not at a general comprehensive theory. We concentrated our attention on the theory that was developed by Dirac, which is, in our opinion, suitable for a first acquaintance with the principles of quantum electrodynamics. Despite the fact that there was a common preference to work in Schrödinger’s formalism, Dirac believed that it was much more logical to use Heisenberg’s equations. In this case, the theory retains the natural harmony and a reasonable sequence of conclusions.

This theory certainly does not lack shortcomings. Firstly, we failed to cope with ultraviolet divergences; as a result of the procedure of regularization of divergent integrals, the theory lost relativistic invariance. Secondly, in the calculations of observables, we neglected some quantities that were supposed to be small; this condition is not, however, entirely obvious, and many approximations are questionable in a strict sense. Thirdly, a gauge-invariant Hamiltonian was formally separated into two parts that are related to the free fields and their interaction respectively; the second part was considered a perturbation. We have thereby put both parts on unequal footings and lost partly the sense of gauge invariance.

In other respects, the theory is satisfactory. Solving Heisenberg’s equations, Dirac obtained general quantities that describe the interaction of fields according to perturbation theory. A regularization organically enters a computation, simply excluding the region of high energies from consideration. The approximations of various kinds are necessary to achieve an agreement with experiment. How should we understand such an approach? Likely, the theory cannot be perfect, such that subsidiary rules of a game become necessary. We arrive ultimately at the results in a form of expansions with respect to a coupling constant. This form is convenient for a comparison with experiment, but at the same time it forces us to work in a framework of the method of perturbation theory. One might think that, in the course of the calculations, we neglect instinctively some quantities only to convert our solution into a form resembling a reasonable expansion with respect to a coupling constant. Our actions are justifiable because this constant is small. If, further, it becomes possible to represent a solution in a form of some function of coupling constant without applying perturbation theory, a question regarding a gauge invariance of separate parts of a Hamiltonian becomes removed. One might thus conclude that quantum electrodynamics is based upon not only the equations and the methods to solve them but also the definite rules of the game to operate with the field variables.

Konstantin’s books *Uncommon Paths in Quantum Physics* and *Quantum Theory of Anharmonic Effects in Molecules* are both available on the Elsevier Store. Use discount code “STC3014” and **save up to 30%** on your very own copies!

**About the Author**

Konstantin V. Kazakov obtained a Dr. Sc. in Physics and Mathematics at the St. Petersburg State University.He published papers in internationally scientific journals, communications at scientific symposia and congresses, and 3 books.

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