Incoherent Tunneling and the Bell Model

As soon as bound states are considered there are only discrete energy levels. Nevertheless it was shown by Bell [77] that it is possible to employ approximately a continuum of energy levels for the calculations of the tunnel rates, which is adequate for the description of many experimental systems. In the simplest form (see Fig. 21.5) of the Bell model, the potential barrier is an inverted parabola. This allows the use of the known solution of the quantum mechanical harmonic oscillator for the calculation of the transition probability through the barrier. The corresponding Schrodinger equation is 

Here 2a is the width of the potential barrier at the ground state. Solving for Vq = cog/2 expresses the oscillation frequency via the ground state energy Eg and the width of the potential at Eg  

From this the probability for transition through the barrier 

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