Inner products of energy eigenstates for a 1-D quantum barrier

Abstract

The features of the standard inner products between all the types of real and complex-energy solutions of the Schrödinger equation for 1-dimensional cut-off quantum potentials are worked out using a Gaussian regularization. A general Master Solution is introduced which describes any of the above solutions as particular cases. From it, a Master Inner Product is obtained which yields all the particular products. We show that the Outgoing and the Incoming Boundary Conditions fully determine the location of the momenta respectively in the lower and upper half complex plane even for purely imaginary momenta (anti-bound and bound solutions). © 2013 Springer Science+Business Media New York.