The Highlights of the Polynomial Formalism

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 (n/ƒ/n+k) in which ƒ = H . Assuming k=0 we obtain:

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 n₁ + ½, n₂ + ½,…, n_r + ½ . Together with this dependence on the quantum numbers, energy E_n depends parametrically on some coefficients that govern the extent of anharmonicity. Choosing appropriately these coefficients and an explicit form of quantum function E_n , one might obtain a pertinent representation for anharmonicity.

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 r variables. From the solution of the Schrödinger equation according to perturbation theory follows, however, not the functions themselves but their expansions in terms of quantum numbers with coefficients  Φ_kⁱ  that characterize the exact influence of anharmonicity. These coefficients have no dependence on quantum numbers and have the dimension of initial function ƒ . The introduction of the functions of quantum numbers is essentially a conversion to an ‘anharmonicity representation’, which transcends the solution according to perturbation theory. The study of these functions of quantum numbers with pertinent laws represents a special interest in physics today.

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.