Interface between Hermitian and non-Hermitian Hamiltonians in a model calculation

Abstract

We consider the interaction between the Hermitian world, represented by a real delta-function potential $-\alpha \delta (x)$, and the non-Hermitian world, represented by a PT-symmetric pair of delta functions with imaginary coefficients $i\beta (\delta (x-L)-\delta (x+L))$. 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 L 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 $\eta$, and consequently a new wave-function $\Psi (x)$, defined in terms of the original wave-function $\psi (x)$ by means of $\eta$. In this picture, working perturbatively in $\beta$, the bound-state wave-function is actually unchanged from its unperturbed form for $|x|<<L$. However, the scattering wave-function, for $|x|>>L$, 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.

In this revised version we use a single metric operator for both bound and scattering states. The Discussion is amplified and extended

References in corpus (18)

• Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• Complex Extension of Quantum Mechanics
• Pseudo-Hermiticity versus PT Symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian
• Faster than Hermitian Quantum Mechanics
• Exact PT-Symmetry Is Equivalent to Hermiticity
• Physical Aspects of Pseudo-Hermitian and $PT$-Symmetric Quantum Mechanics
• Extension of PT-Symmetric Quantum Mechanics to Quantum Field Theory with Cubic Interaction
• On Pseudo-Hermitian Hamiltonians and Their Hermitian Counterparts
• An Equivalent Hermitian Hamiltonian for the non-Hermitian -x^4 Potential
• PT-Symmetric Cubic Anharmonic Oscillator as a Physical Model
• Quantum Brachistochrone Problem and the Geometry of the State Space in Pseudo-Hermitian Quantum Mechanics
• Projective Hilbert space structures at exceptional points
• Conceptual Problems in Scattering from Localized non-Hermitian Potentials
• Delta-Function Potential with a Complex Coupling
• Scattering theory with localized non-Hermiticities
• The quantum brachistochrone problem for non-Hermitian Hamiltonians
• Differential Realization of Pseudo-Hermiticity: A quantum mechanical analog of Einstein's field equation
• Comment on the Quantum Brachistochrone Problem

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