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arXiv:1205.2058 · Physical Review E

Specific heats of quantum double-well systems

Hideo Hasegawa

Published Dec 5, 2012• 14 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreve.86.061104

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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.
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.
The report investigates how the quantum mechanical model of a particle in one dimension underthe influence of the double well potential V (x) = λ(x^2 − x_0^2)^2 can be approximated using aneffective model, where the system is approximated with a system of truncated dimension. Theinfinite dimensional space of energy eigenfunctions to the double well potential was estimatedby two finite dimensional spaces, ones which were spanned by the ground- and first excitedeigenstates of the harmonic oscillators corresponding to the two minima of the potential. Theenergy eigenvalues given by these models were then studied. Additionally, the effects on thepartition function, the internal energy and the heat capacity of the system were checked tofurther study the accuracy of the models. To determine the accuracy of the approximation ahigh order, highly resolved finite element method was used for comparison as stand in for anexact, analytical solution. All investigated quantities were found to be better approximated forlarger λ and lower temperatures, with the effective model of larger dimension achieving a betterapproximation. The result with respect to λ likely results from that, for low energy, the doublewell potential resembles two harmonic oscillator potentials more the higher the energy barrier,i.e. the higher the λ. The effective models being better approximations for lower temperaturescan be explained by the fact that at low temperature, the lowest energies have the strongestimpact on the behaviour of the system.
From the poles of a generalized semiclassical Green's function we derive expressions for the eigenvalues of 1D multiple well potentials. In the case of asymmetric and symmetric double wells, we also obtain analytical formulae for, respectively, the shift and splitting of energies. Our results are better than some approximations in the literature because they take more properly into account the tunnelling through the barriers forming the multiple well and depend on energy-dependent Maslov indices. We illustrate the good numerical precision of the method by discussing some case tests on double wells.
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 $〈m\ifmmode\pm\else\textpm\fi{}|{x}^{2l+1}|n\ensuremath{\mp}〉$.
Dedicated to Professor Józef Spa lek on the occasion of his 60th birthdayThe two-level version of the Richardson model presents a unique possibility to calculate numerically exactly thermodynamic properties of the system it describes.The point is that all energies and the degeneracies of the many-body system can be easily calculated.The energies are given by the eigenvalues of the small (of order of N × N , where 2N is a number of electrons in the system) tridiagonal matrices.Here we numerically obtain a complete spectrum of the interacting two-level model and calculate the specific heat and the pairing energy of the small system at finite temperatures.