Tunneling Eigenstates

Neglecting the tunneling splitting we can assume the eigenstates Oct) to be localized in the wells of the deformation potential so that... 

For interacting electrons the calculation is a little bit more complicated. One should establish the relation between many-particle eigenstates of the system and single-particle tunneling. To do this, let us note, that the states /) and i) in the golden rule formula (83) are actually the states of the whole system, including the leads. We denote the initial and final states as... 

Here 6 is the Heaviside step function and the effective mass is M = m/Ak2L. In (6) we have divided U(q) between Ih,. whose eigenstates are strictly bound if their energy fix, < Um, and a potential V < 0 in the time-modulated perturbation e(t)V. This perturbation allows the particle to tunnel periodically from the bound state to the the unbound (continuum-energy) eigenstates of H whenever e(t) = 1 in Eq. (4). This form conforms to Eq. (1) and answers query (a) (Fig. 1-insets). 

Field, R.W. Jonas, D.M. Kinsey, J.L. Chen, Y.Q. Acetylene to vinylidene isomerization — tunneling resonances in eigenstate resolved spectra. In Abstracts of papers of the American Chemical Society 199, 156-PHYS Part 2, Apr 22, 1990. 

Fig. 9.5 A double barrier model of resonance tunneling. Starting from state 0) on the left, our problem is to compute the fluxes into the continua L and R, defined in terms of basis states that are restricted to the left and right sides ofthe barrier, respectively. Below the barrier energy such states can be taken as eigenstates of a particle moving in the potentials (a) and (b) respectively, which are shown on the right. The basis is supplemented by state 11), taken as the bound eigenstate of a particle in the potential (c)...

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