The junction rule for probability flux

So far, we did not apply the WKB approximation yet. The assumptions on which the above results are based, the strong vibronic coupling, k2 A, and large energy gap in the electron spectrum, A 1, are typical for all tunneling problems in JT systems. If, as in the early papers of Bersuker [1,2], we substitute the oscillator ground-state wave functions in equation (17), then we come to the approximate results that can be obtained directly from the matrix element (12) with the oscillator functions. 

The WKB approximation was applied to the symmetric double-well potential so many times that it makes it difficult to provide a comprehensive list of references. In the context of the present paper, the most important references are the famous text of Landau and Lifshits [27], where tunneling in a symmetric double-well is given as a sample problem, and the paper of Auerbach and Kivelson [11] where the symmetric double-well potential is considered as one of the model cases. 

Although the shortest way to the tunneling gap 8 is the solution of Landau and Lifshits [27], here we consider the problem from a different perspective. Like in the theory of electric circuits, instead of a detailed consideration of each particle, one can apply some simple rules that provide enough equations to solve the problem. One is the junction rule. It is based upon the probability conservation law for a stationary state, PiQ, t). At any point Q in the domain of 77(2, t), the probability density, I PiQ, t) 2 remains constant, dl P(Q. f)P/df = 0. Consider the part of a vibronic state that is located in a potential well. In this region, the probability density, P(Q, t) 2, looks like an octopus with its tentacles extended into the restricted areas under the barriers.2 If we construct a closed surface S around the body of the octopus , then, due to conservation of probability density, the total flux of probability through the surface S must be equal to zero, 

Here we conventionally assume that the flux entering the junction is positive, whereas the terms that describe the leaving part are taken negative. The advantage of the junction rule is that it allows to avoid the detailed consideration of system s dynamics inside the junction (inside the body of the octopus ) and provides the results of tunneling through the tentacles. 

To show how the junction rule works, consider the above example of tunneling in the double-well potential. In this case we have two nodes connected by just one tunneling path. Let the starting position of the system be in the left well with the ground-state wave function P1 = C il (r) T (Q). Here both (/ i(r) and (I (Q) are assumed to be normalized, and C is the amplitude in the left well, so that I C I2 is the probability to find the system in this well. The corresponding tail of the WKB ground-state wave function under the barrier should decrease with Q exponentially, 

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