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arXiv:0805.1656 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

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

Hugh F Jones

Published Sep 19, 2008• 66 citations
arXivHTMLPDF

Abstract

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 $L$ 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 $\ensuremath{\eta}$, 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.

DOI
10.1103/physrevd.78.065032

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We consider the interaction between the Hermitian world, represented by a real delta-function potential \ensuremath−\ensuremathα\ensuremathδ(x)\ensuremath{-}\ensuremath{\alpha}\ensuremath{\delta}(x)\ensuremath−\ensuremathα\ensuremathδ(x), and the non-Hermitian world, represented by a PT-symmetric pair of delta functions with imaginary coefficients i\ensuremathβ(\ensuremathδ(x\ensuremath−L)\ensuremath−\ensuremathδ(x+L))i\ensuremath{\beta}(\ensuremath{\delta}(x\ensuremath{-}L)\ensuremath{-}\ensuremath{\delta}(x+L))i\ensuremathβ(\ensuremathδ(x\ensuremath−L)\ensuremath−\ensuremathδ(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 \ensuremathη\ensuremath{\eta}\ensuremathη, and consequently a new wave function \ensuremathΨ(x)\ensuremath{\Psi}(x)\ensuremathΨ(x), defined in terms of the original wave function \ensuremathψ(x)\ensuremath{\psi}(x)\ensuremathψ(x) by means of \ensuremathη\ensuremath{\eta}\ensuremathη. In this picture, working perturbatively in \ensuremathβ\ensuremath{\beta}\ensuremathβ, the bound-state wave function is actually unchanged from its unperturbed form for ∣x∣\ensuremath≪L|x|\ensuremath{\ll}L∣x∣\ensuremath≪L. However, the scattering wave function, for ∣x∣\ensuremath≫L|x|\ensuremath{\gg}L∣x∣\ensuremath≫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.
In conventional Schrodinger representation the unitarity of the evolution of bound states is guaranteed by the Hermiticity of the Hamiltonian. A non-unitary isospectral simplification of the Hamiltonian, h→H=Ω h Ω≠H†\mathfrak{h} \to H=\Omega\,\mathfrak{h}\,\Omega \neq H^\daggerh→H=ΩhΩ=H† induces the change L→K{\cal L} \to {\cal K}L→K of the Hilbert space of states, reflected by the loss of the Hermiticity of H≠H†H\neq H^\daggerH=H†. In such a reformulation of the theory the introduction of an {\it ad hoc} inner-product metric reconverts K{\cal K}K into the third, correct physical Hilbert space H{\cal H}H, unitarily equivalent to L{\cal L}L. The situation encountered, typically, in PT−{\cal PT}-PT−symmetric or relativistic quantum mechanics is shown more complicated after an inclusion of perturbations. The formulation and solution of the problem are presented. Some of the consequences relevant, e.g., in the analysis of stability are discussed.
Attempts to understand the quantum mechanics of non-Hermitian Hamiltonian systems can be traced back to the early days, one example being Heisenberg's endeavour to formulate a consistent model involving an indefinite metric. Over the years non-Hermitian Hamiltonians whose spectra were believed to be real have appeared from time to time in the literature, for instance in the study of strong interactions at high energies via Regge models, in condensed matter physics in the context of the XXZ-spin chain, in interacting boson models in nuclear physics, in integrable quantum field theories as Toda field theories with complex coupling constants, and also very recently in a field theoretical scenario in the quantization procedure of strings on an AdS 5 x S 5 background. Concrete experimental realizations of these types of systems in the form of optical lattices have been proposed in 2007. In the area of mathematical physics similar non-systematic results appeared sporadically over the years. However, intensive and more systematic investigation of these types of non- Hermitian Hamiltonians with real eigenvalue spectra only began about ten years ago, when the surprising discovery was made that a large class of one-particle systems perturbed by a simple non-Hermitian potential term possesses a real energy spectrum. Since then regular international workshops devoted to this theme have taken place. This special issue is centred around the 6th International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics held in July 2007 at City University London. All the contributions contain significant new results or alternatively provide a survey of the state of the art of the subject or a critical assessment of the present understanding of the topic and a discussion of open problems. Original contributions from non-participants were also invited. Meanwhile many interesting results have been obtained and consensus has been reached on various central conceptual issues in the growing community of this subject. It is, for instance, well understood that the reality of the spectrum can be attributed either to the unbroken PT-symmetry of the entire system, that is, invariance of the Hamiltonian and the corresponding wavefunctions under a simultaneous parity transformation and time reversal, or more generally to its pseudo-Hermiticity . When the spectrum is real and discrete the Hamiltonian is actually quasi-Hermitian, with a positive-definite metric operator, and can in principle be related by a similarity transformation to an isospectral Hermitian counterpart. For all approaches well-defined procedures have been developed, which allow one to construct metric operators and therefore a consistent description of the underlying quantum mechanical observables. Even though the general principles have been laid out, it remains a challenge in most concrete cases to implement the entire procedure. Solvable models in this sense, some of which may be found in this issue, remain a rare exception. Nonetheless, despite this progress some important questions are still unanswered. For instance, according to the current understanding the non-Hermitian Hamiltonian does not uniquely define the physics of the system since a meaningful metric can no longer be associated with the system in a non-trivial and unambiguous manner. A fully consistent scattering theory has also not yet been formulated. Other issues remain controversial, such as the quantum brachistochrone problem, the problem of forming a mixture between a Hermitian and non-Hermitian system, the new phenomenological possibilities of forming a kind of worm-hole effect, etc. We would like to acknowledge the financial support of the London Mathematical Society, the Institute of Physics, the Doppler Institute in Prague and the School of Engineering and Mathematical Science of City University London. We hope this special issue will be useful to the newcomer as well as to the expert in the subject. Participants of the 6th International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics.
Excited bound states are often understood within scattering-based theories as resulting from the collision of a particle on a target via a short-range potential. We show that the resulting formalism is non-Hermitian and describe the Hilbert spaces and metric operator relevant to a correct formulation of such theories. The structure and tools employed are the same that have been introduced in current works dealing with PT-symmetric and quasi-Hermitian problems. The relevance of the non-Hermitian formulation to practical computations is assessed by introducing a non-Hermiticity index. We give a numerical example involving scattering by a short-range potential in a Coulomb field for which it is seen that even for a small but non-negligible non-Hermiticity index the non-Hermitian character of the problem must be taken into account. The computation of physical quantities in the relevant Hilbert spaces is also discussed.
To the existing list of alternative formulations of quantum mechanics, a new version of the non-Hermitian interaction picture is added. What is new is that, in contrast to the more conventional non-Hermitian model-building recipes, the primary information about the observable phenomena is provided not only by the Hamiltonian but also by an additional operator with a real spectrum (say, R(t)) representing another observable. In the language of physics, the information carried by R(t)≠R†(t) opens the possibility of reaching the exceptional-point degeneracy of the real eigenvalues, i.e., a specific quantum phase transition. In parallel, the unitarity of the system remains guaranteed, as usual, via a time-dependent inner-product metric Θ(t). From the point of view of mathematics, the control of evolution is provided by a pair of conjugate Schrödiner equations. This opens the possibility od an innovative dyadic representation of pure states, by which the direct use of Θ(t) is made redundant. The implementation of the formalism is illustrated via a schematic cosmological toy model in which the canonical quantization leads to the necessity of working with two conjugate Wheeler-DeWitt equations. From the point of view of physics, the “kinematical input” operator R(t) may represent either the radius of a homogeneous and isotropic expanding empty Universe or, if you wish, its Hubble radius, or the scale factor a(t) emerging in the popular Lemaitre-Friedmann-Robertson-Walker classical solutions, with the exceptional-point singularity of the spectrum of R(t) mimicking the birth of the Universe (“Big Bang”) at t=0.
The coupling of non-Hermitian -symmetric Hamiltonians to standard Hermitian Hamiltonians, each of which individually has a real energy spectrum, is explored by means of a number of soluble models. It is found that in all cases the energy remains real for small values of the coupling constant, but becomes complex if the coupling becomes stronger than some critical value. For a quadratic non-Hermitian -symmetric Hamiltonian coupled to an arbitrary real Hermitian -symmetric Hamiltonian, the reality of the ground-state energy for small enough coupling constant is established up to second order in perturbation theory.