A tunnelling formalism

In electron tunnelling from channel to gate in a field effect transistor, and in alpha-particle emission from atomic nuclei, the tunnelling transition is from a quasi-level to a continuum of states, to which conventional tunnelling theory does not apply. Gamow, in the alpha-emission case, proposed to consider particle states with an imaginary component of energy, -1/2ℏλ, where λ, is the tunnelling decay rate. It is shown in this paper, in terms of an appropriate real wavefunction ψ(x), that λ is then given by u/λ, = ∫0∞ r2dx, where r(x) = ψ(x)/ψ(0) and where u = ℏk/m is the velocity of the tunnelling particle. The application of this formula is discussed. The connection with a quantum resonance of the electron system is shown.