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arXiv:2608.16835 · APS Open Science

Classical-limit formula for matrix elements between bound states of distinct one-dimensional potentials

K. Beloy

quant-phphysics.atom-phSubmitted Aug 17, 2026• Announced Aug 18, 2026• 0 citations
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Abstract

A classical-limit formula is derived for matrix elements between bound states of distinct one-dimensional potentials. We leave open the physical interpretation of the potentials, but generally envision them to be Born-Oppenheimer-type potentials with the position variable $x$ identifying with a slowly evolving degree of freedom of the system. For instance, $x$ could represent the internuclear separation in a diatomic molecule, with the potentials being potential energy curves for different electronic states. In this scenario, the matrix elements could be, e.g., conventional Franck-Condon factors. To test the derived formula, we assume functional forms for the potentials and operator that afford analytical solutions for the matrix elements. As the classical limit is approached, the computed matrix elements exhibit a clear tendency towards the classical-limit formula, providing strong validation for the formula. In future work, we anticipate using the formula to model inhomogeneous excitation spectra of atoms in one-dimensional optical lattices, with an eye towards improved optical lattice clock performance.

Journal reference
APS Open Sci. 1, 000077 (2026)
DOI
10.1103/kyjq-2fc4
Comments
11 pages, 4 figures

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A classical-limit formula is derived for matrix elements between bound states of distinct one-dimensional potentials. We leave open the physical interpretation of the potentials, but generally envision them to be Born-Oppenheimer-type potentials with the position variable xxx identifying with a slowly evolving degree of freedom of the system. For instance, xxx could represent the internuclear separation in a diatomic molecule, with the potentials being potential energy curves for different electronic states. In this scenario, the matrix elements could be, e.g., conventional Franck-Condon factors. To test the derived formula, we assume functional forms for the potentials and operator that afford analytical solutions for the matrix elements. As the classical limit is approached, the computed matrix elements exhibit a clear tendency towards the classical-limit formula, providing strong validation for the formula. In future work, we anticipate using the formula to model inhomogeneous excitation spectra of atoms in one-dimensional optical lattices, with an eye towards improved optical lattice clock performance.
Abstract In this work we discuss Bohr’s correspondence principle in a particular physical system, namely the hydrogen atom. It is shown that while diagonal elements of the position operator are equal to zero, off-diagonal elements between two states of different but close energies have a classical limit. The classical limit is reached when both principal quantum numbers, and hence both energies, are large but their difference is small. In this case the matrix elements of the position operator tend to the Fourier component of a classical trajectory, whose parameters are found. The classical limit is proved by finding numerical values of the matrix elements and also by exact treatment of the analytical formulas. This discussion provides a deeper understanding of the correspondence between quantum and classical mechanics and contributes to a better understanding of hydrogen atom wave functions.
The semiclassical formula for matrix elements, sometimes called the Heisenberg correspondence principle, relates matrix elements (operators in energy representation) to phase space functions (Weyl representatives). The formula does not make sense for arbitrary operators; when it is valid it implicitly fixes the relative phases of the eigenfunctions of the Hamiltonian. The conventions, which have to be used for the wavefunctions in position or momentum representation, are given here in explicit form, and we present a class of operators related to coherent states of high energy for which the Heisenberg correspondence principle holds.
Heisenberg's form of the correspondence principle for non-relativistic matrix elements has been used to evaluate matrix elements for various potentials. These are compared with available quantum mechanical results to check the general validity of the use of classical mechanics and correspondence principles in the derivation of quantum mechanical expressions. Using a one dimensional harmonic oscillator potential matrix elements of the form (n mod qm mod n+s) and (n mod pm mod n+s) are worked out. A Morse potential is then considered and the matrix elements for position, momentum, and kinetic energy are evaluated. Using a Coulomb potential a similar procedure is used for functions of various position coordinates, from which dipole and quadrupole moments from transitions between n, l states are calculated. The agreement with quantum mechanics is found to be generally good, and in some cases identical results are obtained.
We introduce a computational method developed for study of long-range molecular Rydberg states of such systems that can be approximated by two electrons in a model potential of the atomic cores. Only diatomic molecules are considered. The method is based on a two-electron RRR-matrix approach inside a sphere centered on one of the atoms. The wave function is then connected to a Coulomb region outside the sphere via a multichannel version of the Coulomb Green's function. This approach is put into a test by its application to a study of Rydberg states of the hydrogen molecule for internuclear distances RRR from 20 to 400 bohrs and energies corresponding to nnn from 3 to 22. The results are compared with previous quantum chemical calculations (lower quantum numbers nnn) and computations based on contact-potential models (higher quantum numbers nnn).
Long-range states of diatomic molecules have average values of internuclear separations $\ensuremath{\langle}R\ensuremath{\rangle}$ at least one order of magnitude larger than the equilibrium value of RRR. For example, the helium dimer 4He2{{}^{4}\mathrm{He}}_{2}4He2​ has a single bound state with $\ensuremath{\langle}R\ensuremath{\rangle}$ of about 50 \AA{}. We show that the properties of these states, such as $\ensuremath{\langle}R\ensuremath{\rangle}$, the dissociation energy, or the sss-wave scattering length, can be related by simple, yet very accurate, formulas if a potential-energy curve is known. By examining a range of ab initio and empirical helium dimer potentials as well as scaling these potentials, we found that the formulas remain accurate, even if very approximate potentials were used. In addition to 4He2{{}^{4}\mathrm{He}}_{2}4He2​, we present results for 9Be2,20Ne2{{}^{9}\mathrm{Be}}_{2},\phantom{\rule{0.28em}{0ex}}{{}^{20}\mathrm{Ne}}_{2}9Be2​,20Ne2​, and KRb.