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Acta Polytechnica

Generalized Continuity Equation for Quasi-Hermitian Hamiltonians

Amine B. Hammou

Published Jan 3, 2013• 0 citations
Publisher

Abstract

The continuity relation is generalized to quasi-Hermitian one-dimensional Hamiltonians. As an application we show that the reflection and transmission coefficients computed with the generalized current obey the conventional unitarity relation for the continuous double delta function potential.
DOI
10.14311/1803

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We consider the interaction between the Hermitian world, represented by a real delta-function potential $\ensuremath{-}\ensuremath{\alpha}\ensuremath{\delta}(x)$, and the non-Hermitian world, represented by a PT-symmetric pair of delta functions with imaginary coefficients $i\ensuremath{\beta}(\ensuremath{\delta}(x\ensuremath{-}L)\ensuremath{-}\ensuremath{\delta}(x+L))$. In the context of standard quantum mechanics, the effect of the introduction of the imaginary delta functions on the bound-state energy of the real delta function and its associated wave function is small for LLL large. However, scattering from the combined potentials does not conserve probability as conventionally defined. Both these problems can be studied instead in the context of quasi-Hermiticity, whereby quantum mechanics is endowed with a new metric $\ensuremath{\eta}$, and consequently a new wave function $\ensuremath{\Psi}(x)$, defined in terms of the original wave function $\ensuremath{\psi}(x)$ by means of $\ensuremath{\eta}$. In this picture, working perturbatively in $\ensuremath{\beta}$, the bound-state wave function is actually unchanged from its unperturbed form for $|x|\ensuremath{\ll}L$. However, the scattering wave function, for $|x|\ensuremath{\gg}L$, is changed in a significant manner. In particular, there are incoming and outgoing waves on both sides of the potential. One can then no longer talk in terms of reflection and transmission coefficients, but the total right-moving flux is now conserved.
We highlight the conceptual issues that arise when one applies the quasi-Hermitian framework to analyze scattering from localized non-Hermitian potentials, in particular complex square wells or delta functions. When treated in the framework of conventional quantum mechanics, these potentials are generally considered as effective theories, in which probability is not conserved because of processes that have been ignored. However, if they are treated as fundamental theories, the Hilbert-space metric must be changed. In order for the newly defined probability to be conserved, it must differ from the standard one, even at asymptotically large distances from the scattering center, and the mechanism for this is the nonlocality of the new metric, as we show in detail in the model of a single complex delta function. However, properties of distant bound-state systems, which do not interact physically with the non-Hermitian scattering potential, should not be affected. We analyze a model Hamiltonian that supports this contention.
The Schrödinger equation is reformulated as a universal continuity equation, which connects between changes in the particles probability density distribution to probability current densities (fluxes). The formulation of particle conservation in terms of stationary fluxes enables one to associate stationary wave functions also to open quantum systems characterized by stationary particle currents. These functions are (improper) solutions of the stationary Schrödinger equation, obtained under scattering boundary conditions. These boundary conditions can be fulfilled for any positive asymptotic kinetic energy, hence, the energy spectrum of the scattering states is continuous. We demonstrate flux calculations in scattering through a one-dimensional potential energy well/barrier, focusing on transmission and reflection probabilities. Nonclassical phenomena such as transmission at energies below a potential energy barrier (quantum tunneling), or reflections at energies above a potential energy well are analyzed. The phenomenon of full transmission through a double barrier structure (resonant tunneling) is introduced in the context of nanoscale transport.
Rarely noted paradoxes in applications of fundamental quantum relations are pointed out, with their resolution leading to emergent non-Hermitian behaviors due to boundary terms – even for closed systems and with real potentials. The role played by these non-Hermiticities on the consistency of quantum mechanical uncertainty relations is discussed, especially in multiply-connected spaces (more generally, for any system that satisfies the Bloch theorem of Solid State Physics). These subtleties – reflections of topological quantum anomalies – follow their own patterns (for any dimensionality, for both Schrödinger and Dirac/Weyl Hamiltonians and for either continuous or lattice (tight-binding) models): they can always be written as global fluxes of certain generalized current densities Jg. In continuous nonrelativistic models, these have the forms that had earlier been used by Chemists to describe atomic fragments of polyatomic molecules, while for Dirac/Weyl or other lattice models they have more interesting relativistic forms only recently worked out in graphene models. In spite of the deep mathematical origin as quantum anomalies examples are provided here, where such non-Hermiticities have a direct physical significance (for both conventional and topological materials). In all stationary state examples considered, these non-Hermitian boundary terms turn out to be quantized, this quantization being either of conventional or of a topological (Quantum Hall Effect (QHE)-type) origin. The latter claim is substantiated through direct application to a simple QHE arrangement (2D Landau system in an external in-plane electric field), where some particular Jg seems to be related to the well-known dissipationless edge currents. More generally, the non-Hermitian terms play a subtle role on Berry curvatures in solids and seem to be crucial for the consistent application of the so called Modern Theories of Polarization and Orbital Magnetization. It is emphasized that the above systems can be _closed_ (in multiply-connected space, so that the boundaries disappear, but the non-Hermiticity remains), a case in non-Hermitian physics that is not usually discussed in the literature; it is also stressed that a mapping between the above non-Hermiticity (for continuous systems) and the many recent available results in tight-binding solid state models (leading to the so-called exceptional points) is expected to promote enhanced understanding of quantum behavior at the most fundamental level.