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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Tunneling contributions to matrix elements of the anharmonic oscillator

John B. Bronzan

Published Feb 15, 1982• 12 citations
Publisher

Abstract

We study the double-well anharmonic oscillator when the barrier between the two wells is broad and high. In this limit, the energy gap between even- and odd-parity eigenstate pairs is exponentially small. We develop equations for similar exponentially small tunneling contributions to arbitrary matrix elements of arbitrary operators. As an example, we calculate the matrix elements $〈m\ifmmode\pm\else\textpm\fi{}|{x}^{2l}|n\ifmmode\pm\else\textpm\fi{}〉$ and $〈m\ifmmode\pm\else\textpm\fi{}|{x}^{2l+1}|n\ensuremath{\mp}〉$.

DOI
10.1103/physrevd.25.1065

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We study the double-well anharmonic oscillator when the barrier between the two wells is broad and high. In this limit, the energy gap between even- and odd-parity eigenstate pairs is exponentially small. We develop equations for similar exponentially small tunneling contributions to arbitrary matrix elements of arbitrary operators. As an example, we calculate the matrix elements 〈m\ifmmode±\else\textpm\fi∣x2l∣n\ifmmode±\else\textpm\fi〉〈m\ifmmode\pm\else\textpm\fi{}|{x}^{2l}|n\ifmmode\pm\else\textpm\fi{}〉〈m\ifmmode±\else\textpm\fi∣x2l∣n\ifmmode±\else\textpm\fi〉 and 〈m\ifmmode±\else\textpm\fi∣x2l+1∣n\ensuremath∓〉〈m\ifmmode\pm\else\textpm\fi{}|{x}^{2l+1}|n\ensuremath{\mp}〉〈m\ifmmode±\else\textpm\fi∣x2l+1∣n\ensuremath∓〉.
Specific heats of quantum systems with symmetric and asymmetric double-well potentials have been calculated. In numerical calculations of their specific heats, we have adopted the combined method which takes into account not only eigenvalues of ϵn\epsilon_nϵn​ for 0≤n≤Nm0 \leq n \leq N_m0≤n≤Nm​ obtained by the energy-matrix diagonalization but also their extrapolated ones for Nm+1≤n<∞N_m+1 \leq n < \inftyNm​+1≤n<∞ (Nm=20N_m=20Nm​=20 or 30). Calculated specific heats are shown to be rather different from counterparts of a harmonic oscillator. In particular, specific heats of symmetric double-well systems at very low temperatures have the Schottky-type anomaly, which is rooted to a small energy gap in low-lying two-level eigenstates induced by a tunneling through the potential barrier. The Schottky-type anomaly is removed when an asymmetry is introduced into the double-well potential. It has been pointed out that the specific-heat calculation of a double-well system reported by Feranchuk, Ulyanenkov and Kuz'min [Chem. Phys. 157, 61 (1991)] is misleading because the zeroth-order operator method they adopted neglects crucially important off-diagonal contributions.
Specific heats of quantum systems with symmetric and asymmetric double-well potentials have been calculated. In numerical calculations of their specific heats, we have adopted the combined method, which takes into account not only eigenvalues of ε(n) for 0 ≤ n ≤ N(m) obtained by the energy-matrix diagonalization but also their extrapolated ones for N(m) + 1 ≤ n <∞ (N(m) = 20 or 30). Calculated specific heats are shown to be rather different from counterparts of a harmonic oscillator. In particular, specific heats of symmetric double-well systems at very low temperatures have the Schottky-type anomaly, which is rooted to a small energy gap in low-lying two-level eigenstates induced by a tunneling through the potential barrier. The Schottky-type anomaly is removed when an asymmetry is introduced into the double-well potential. It has been pointed out that the specific-heat calculation of a double-well system reported by Feranchuk, Ulyanenkov, and Kuz'min [Chem. Phys. 157, 61 (1991)] is misleading because the zeroth-order operator method they adopted neglects crucially important off-diagonal contributions.
This is an investigation of an anharmonic oscillator characterized by the potential (ω02/ 2) q2 + λ′ q4. By using the exact relations obtained by evaluating ≪i|[H, O]∓|j > where O is an arbitrary operator, H is our total Hamiltonian and |i > and |j > are exact eigenstates of H, we derive an exact recurrence formula. This formula allows us to express τ-functions with a higher power of the variables through τ-functions with a lower power of the variables and energy eigenvalues. In this way we derive several exact relations, which are, in a sense, generalizations of the virial theorem and sum rules (Eqs. (4·5), (4·6) and (5·4)). These exact relations are the central equaions of this paper. On the basis of these exact relations we propose our “nearest neighbour level” (N. N. L) approximation, which seems to provide a good approximation scheme. We can also use our exact relations to test the validity of various approximation methods, and as an example, we discuss the “New-Tamm-Dancoff” (N. T. D)-type of approximation in detail.
In some molecules such as ammonia, double vibrations are observed at small energies. These small vibrations are thought to arise from double well potentials. To explain these small energy gaps, double potential wells are investigated, and complex calculations are made. Until recently, since the exact solution of the Schrödinger equation was not known, approximate solutions were always obtained. The exact solution of the Schrödinger equation is now known. In previous studies, a simple procedure for the general solution of the radial Schrödinger Equation has been found for spherical symmetric potentials without making any approximation. In this article, the Schrodinger equation was solved with this new solution method. It has been applied to the oscillations through a barrier tunneling potential for a particle in a central quadratic deformed potential well (double potential well). It is seen that there is no need to look for complex potentials to explain the double wells observed in small vibrations in ammonia and similar molecules, they were compared with experimentally measured energies. The results were found to be fully compatible. It has been seen that there can always be double well potentials at small energies for all molecules, atomic nuclei, and similar particles with angular momentum other than zero.
Abstract A formalism is presented to obtain approximate analytic expressions for the eigenstates and eigenvalues of a quantum double oscillator (QDO). The matrix elements of a large class of operators with respect to states of different double oscillators result as finite sums of explicit functions of the respective parameters. Matrix elements between states of a harmonic oscillator and a double oscillator are also determined. The analytic expressions were used to calculate Franck-Condon factors for electronic transitions including double oscillator anharmonicities.