Quantum Mechanical Reflection Resonances

Resonances in the reflection probability amplitude r(E) can occur in energy ranges in which the reflection probability R(E)=|r(E)|^2 is 1. They occur as the phase phi(E) defined by r(E) = t*(E)/t(E) = 1e^{i 2phi(E)} undergoes a rapid change of pi radians. During this transition the phase angle exhibits a Lorentzian profile in that d(phi(E))/dE ~= 1/[(E-E_0)^2+(hbar*gamma/2)^2]. The energy E_0 identifies the location of a quasi-bound state, gamma measures the lifetime of this state, and t(E) is a matrix element of the transfer operator. Methods for computing and measuring these resonances are proposed.