2 research outputs found
Compound transfer matrices: Constructive and destructive interference
Scattering from a compound barrier, one composed of a number of distinct
non-overlapping sub-barriers, has a number of interesting and subtle
mathematical features. If one is scattering classical particles, where the wave
aspects of the particle can be ignored, the transmission probability of the
compound barrier is simply given by the product of the transmission
probabilities of the individual sub-barriers. In contrast if one is scattering
waves (whether we are dealing with either purely classical waves or quantum
Schrodinger wavefunctions) each sub-barrier contributes phase information (as
well as a transmission probability), and these phases can lead to either
constructive or destructive interference, with the transmission probability
oscillating between nontrivial upper and lower bounds. In this article we shall
study these upper and lower bounds in some detail, and also derive bounds on
the closely related process of quantum excitation (particle production) via
parametric resonance.Comment: V1: 28 pages. V2: 21 pages. Presentation significantly streamlined
and shortened. This version accepted for publication in the Journal of
Mathematical Physic
Reformulating the Schrodinger equation as a Shabat-Zakharov system
We reformulate the second-order Schrodinger equation as a set of two coupled
first order differential equations, a so-called "Shabat-Zakharov system",
(sometimes called a "Zakharov-Shabat" system). There is considerable
flexibility in this approach, and we emphasise the utility of introducing an
"auxiliary condition" or "gauge condition" that is used to cut down the degrees
of freedom. Using this formalism, we derive the explicit (but formal) general
solution to the Schrodinger equation. The general solution depends on three
arbitrarily chosen functions, and a path-ordered exponential matrix. If one
considers path ordering to be an "elementary" process, then this represents
complete quadrature, albeit formal, of the second-order linear ODE.Comment: 18 pages, plain LaTe