Connection between bound state and tunneling problems

The transfer matrix formalism is used to show that the problem of finding the bound states of a quantum well with an arbitrary one-dimensional potential energy profile Ec(x) can be reformulated as a tunneling problem. The following theorem is proved: for an electron confined to a quantum well of size W with an arbitrary conduction band energy profile Ec(x) and maximum depth V0, the bound state energies (E1, E2, E3,…) can be found by adding a barrier of width d and height V0 on either side of the quantum well and calculating the energies at which the transmission through the resulting resonant tunneling structure reaches unity. More precisely, the energies at which the transmission coefficient reaches unity converge towards the bound state energy levels when the thickness d tends to infinity. Numerically, the bound state energies can be determined with enough accuracy by using barrier thicknesses of a few nm.