Effect of Anharmonicity on the WKB Energy Splitting in a Double Well Potential

We investigate the effect of anharmonicity on the WKB approximation in a double well potential. By incorporating the anharmonic perturbation into the WKB energy splitting formula we show that the WKB approximation can be greatly improved in the region over which the tunneling is appreciable. We also observe that the usual WKB results can be obtained from our formalism as a limiting case in which the two potential minima are far apart. Typeset using REVTEX 1 It is well known that quantum tunneling leads to a splitting of degenerate energy levels in a symmetrical two-well potential. There are three approaches to the calculation of this energy splitting: the WKB approximation, the instanton method, and numerical calculation. From the comparison of the results from the WKB and instanton methods with those of numerical calculations, it was shown that the instanton method is better than the WKB approximation [1], because the WKB method is generally believed to have inherent errors associated with the connection formula [2]. The modified-barrier [3] and modified-well [4] formalisms have been proposed for the improvement of the WKB approximation. Recently the authors in Ref. [5] have shown that a careful account of the phase changes in connection formula improves the accuracy of the WKB wave function. In this letter we propose another formalism whereby the energy splitting within the WKB approximation becomes consistent with the instanton result. Unlike many of the WKB formalisms, the present work incorporates the anharmonicity into theWKB formalism, which gives a more realistic model, and hence more improved energy splitting result. In other words, the incorporation of anharmonicity results in a level shift due to the perturbation in each well. Consider a particle of mass m in a one-dimensional symmetrical two-well potential V (x) = mω 8a2 (x− a)(x+ a), (1) where ω is the angular frequency in each well when the two wells are far apart, and ±a are the positions of the two potential minima. For a tunneling to occur the separation between the two minima should be large enough so that the height of the barrier mω a 8 is higher than the lowest energy level in each well. In the limit a → ∞, the potential is divided into two independent harmonic oscillator potentials in which the lowest energies are the same and given by E0 = 1 2 h̄ω. When these two harmonic oscillator potentials approach each other, they become an anharmonic potential, so that the lowest energies are no longer E0 because of the anharmonic perturbation. To evaluate the lowest perturbation energies we expand V (x) around each minima ±a. 2 Since the potential is symmetric, we consider one of either positions. For the minimum at x = a we have V (x) = mω 2 (x− a) [ 1 + x− a a + 3(x− a) a2 ] . (2) Following a standard perturbation theory it is straightforward to show that the perturbation energy to second order correction is E = E0 [1 + ǫ(η)] , (3) where ǫ(η) is defined as ǫ(η) = η 16 (25− 189η), and we have introduced a dimensionless parameter η = √ h̄ mωa2 which is small for large a. We see that the first term in Eq.(3) corresponds to the lowest energy of the unperturbed harmonic potential and ǫ(η) is the correction term which was ignored in the previous studies. In the following, we demonstrate that this correction term plays an important role in the improvement of the WKB approximation. Using Eq.(3) we write the WKB level splitting formula as [6] ∆EWKB = 2h̄ T e, (4) where S = 1 h̄ ∫ α −α √ 2m(V (x)− E)dx, T = ∫ γ α √ 2m