Tunneling splitting in a double well

Coleman s method can be applied to finding the ground state tunneling splitting in a symmetric double well [Vainshtein et al. 1982], for some... 

The discussion of Section 2.3, which is concerned with the destruction of coherence by interaction with the environment, applies to this case. As in the case of tunneling in a double well, A(f) 2 can be represented as a product of the attempt frequency of hitting the turning point and the barrier transparency (see Section 3.4). The tunneling splitting is determined by the same parameters and contains only an additional prefactor 3/2 because of the symmetry. 

In the first part of this section, the instanton theory [2] is explained by taking the motion of a particle of mass m in one-dimensional potential V x). Tunneling splitting in a symmetric double well potential and decay of metastable state by tunneling through a potential barrier are employed as examples. In the second subsection, it is shown that the results can be reproduced by the WKB method with slight modification. 

In this subsection, we show that the same results as those obtained by the instanton theory in the previous subsection can be derived by the WKB theory with a slight modification [43,46]. We consider the tunneling splitting in a symmetric double well potential and as usual we use the asymptotic WKB wave function localized in one of the wells—say, left-side well ... 

Aside from merely calculational difficulties, the existence of a low-temperature rate-constant limit poses a conceptual problem. In fact, one may question the actual meaning of the rate constant at r = 0, when the TST conditions listed above are not fulfilled. If the potential has a double-well shape, then quantum mechanics predicts coherent oscillations of probability between the wells, rather than the exponential decay towards equilibrium. These oscillations are associated with tunneling splitting measured spectroscopically, not with a chemical conversion. Therefore, a simple one-dimensional system has no rate constant at T = 0, unless it is a metastable potential without a bound final state. In practice, however, there are exchange chemical reactions, characterized by symmetric, or nearly symmetric double-well potentials, in which the rate constant is measured. To account for this, one has to admit the existence of some external mechanism whose role is to destroy the phase coherence. It is here that the need to introduce a heat bath arises. 

It is worthwhile to note that the charged Bose gas trapped in the double well potential Ua of Eq. 50 behaves as an inverted Josephson junction (N-S-S-N). The super-current, which accompanies the matter wave coherence, is induced between the degenerate resonance states of the adjacent wells at the frequency of the tunnel splitting A response time, as is typical of tunnel junctions (whose frequency cutoff is much smaller than the vibrational frequency even for nano junctions). The coherent oscillations of the Josephson current can be observed by virtue of their slow frequency A V which is robustly controlled by the bias voltage. 

Throughout the domain Rg < R < oo, tunneling through the barrier between the two minima occurs. However, as D and hence the effective mass 2 increases, tunneling diminishes markedly. Our chief aim is to evaluate the splitting AEd(R) between the lowest two eigenvalues of Eq.(l), produced by timneling in the double well domain. 

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