Normalization, Orthogonality, and Completeness for the Finite Step Potential

Abstract

Modern heterostructure devices as esgc heterojunctions, sing.le quantum wells, and s+erlattices are often isotropic in one plane s o that effectively a onedimensional problem remains only. Therefore exact solutions of the one-dimensional SchrBdinger equation and their properties are of fundamental interest. In recent times the continuum states play an essential role, especially in resonant tunneling structures /1 t o 4/. In these calculations the infinite space is usually preferred rather than the large box quantization because the handling of the former one is easier. But, on the other hand, the powerful results concerning (regular ) Sturm-Liouville problems /5/ a r e not applicable, and one has t o resort to the more elaborate theories of Weyl, Titchmarsh, and Kodaira /6 t o 8/. But the additional suppositions on the potential (continuity o r absolute integrability) often made in these theories are sometimes not satisfied by the model systems which are studied in the applications mentioned above.Because there were controversial discussions regarding the normalization, orthogonality, and completeness of eigenfunctions in the infinite space /2, 4, 9/ we will treat these problems for a finite step potential explicitly. The restriction t o this potential is caused by brevity; a generalization t o more complicated quantum well o r quantum b a r r i e r structures and/or the inclusion of Ben Daniel-Duke Hamiltonians /lo/ with a stepwise constant potential is straightforward. (For instance, one can prove completeness for the potential (6) of /4/ with only slight modifications of the procedure described in the following.) F o r potentials with bound states one has t o take into account these onea in the contour integration (see below) in such a way as in /ll/-Our model system is Hy&)= %*(d 1) PSF 327, DDR-6300 Ilmenau, GDR.

Completeness of scattering states of the Dirac Hamiltonian with a step potential

“…As described in Introduction, though it is generally taken for granted that eigenfunctions of a Hamiltonian constitute a complete orthonormal set, which can be shown explicitly when it has only a finite number of discrete eigenstates, to prove it for a Hamiltonian endowed with a continuous spectrum can be another, nontrivial task. In the present case, we need to show that the left-hand side of (40) is diagonal in both the coordinate and spinor spaces, which would make the proof more involved than the non-relativistic cases [4].…”

“…The presentation here, though rather involved and not elegant, would be just what such people is looking for. Actually, just as in [4] for the non-relativistic cases, the integrations over momentum have been carried out straightforwardly and explicitly, resulting in the delta function that represents the completeness relation (40). It would be interesting and instructive to see how such an intuitive approach does work even for this relativistic case.…”

“…Apparently these potentials do not satisfy some of the premises assumed for the general proofs and one has to work out such cases separately even though the potentials are very simple and considered rather elementary. In this paper, we shall demonstrate that the scattering wave functions of a Dirac Hamiltonian for a system with a step potential form a complete orthonormal set, along the same line of thought as [4], by summing up all of them. Though the completeness of them in the same system is already shown in [6] in a mathematically rigorous way, it is still of interest and instructive to see explicitly that the eigenfunctions of an interacting Dirac Hamiltonian can replace the plane waves in (1) to play the same role.…”

“…Surprisingly enough, the situation is not at all better for cases with simple potentials, say, the step or well potentials in one dimension. In fact, the proof of the completeness of one-dimensional scattering wave functions off a step potential appeared only in the late 80s [4] and a mathematically more rigorous treatment for step and well potentials has to be waited until 2010 [5].…”

“…Here and in the followings, the dependence on the spin s is and will be suppressed in the wave functions. The continuity of the current defined as j z z Scattering states are categorized according to whether a particle is incident from the left A: y ( ) or right B: f ( ) of the step potential and to the ranges of their energies(1)(2)(3)(4). No left(right)-incident scattering solution exists for the energy range…”