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arXiv:2308.15043

Quasi-Hermitian quantum mechanics and a new class of user-friendly matrix Hamiltonians

Olaf Lechtenfeld, Miloslav Znojil

Published Aug 29, 2023• 0 citations
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Abstract

In the conventional Schrödinger's formulation of quantum mechanics the unitary evolution of a state $ψ$ is controlled, in Hilbert space ${\cal L}$, by a Hamiltonian $\mathfrak{h}$ which must be self-adjoint. In the recent, ``quasi-Hermitian'' reformulation of the theory one replaces $\mathfrak{h}$ by its isospectral but non-Hermitian avatar $H = Ω^{-1}\mathfrak{h}Ω$ with $Ω^\daggerΩ= Θ\neq I$. Although acting in another, manifestly unphysical Hilbert space ${\cal H}$, the amended Hamiltonian $H \neq H^\dagger$ can be perceived as self-adjoint with respect to the amended inner-product metric $Θ$. In our paper motivated by a generic technical ``user-unfriendliness'' of the non-Hermiticity of $H$ we introduce and describe a specific new family of Hamiltonians $H$ for which the metrics $Θ$ become available in closed form.

DOI
10.48550/arxiv.2308.15043

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In the conventional Schrödinger's formulation of quantum mechanics the unitary evolution of a state ψψψ is controlled, in Hilbert space L{\cal L}L, by a Hamiltonian h\mathfrak{h}h which must be self-adjoint. In the recent, quasi-Hermitian'' reformulation of the theory one replaces $\mathfrak{h}$ by its isospectral but non-Hermitian avatar $H = Ω^{-1}\mathfrak{h}Ω$ with $Ω^\daggerΩ= Θ\neq I$. Although acting in another, manifestly unphysical Hilbert space ${\cal H}$, the amended Hamiltonian $H \neq H^\dagger$ can be perceived as self-adjoint with respect to the amended inner-product metric $Θ$. In our paper motivated by a generic technical user-unfriendliness'' of the non-Hermiticity of HHH we introduce and describe a specific new family of Hamiltonians HHH for which the metrics ΘΘΘ become available in closed form.
Abstract In the conventional Schrödinger’s formulation of quantum mechanics the unitary evolution of a state ψ is controlled, in Hilbert space <?CDATA L{\mathcal{L}}L?> L , by a Hamiltonian ɧ which must be self-adjoint. In the recent, “quasi-Hermitian” reformulation of the theory one replaces ɧ by its isospectral but non-Hermitian avatar H = Ω−1 𝖍Ω with Ω†Ω = Θ ≠ I. Although acting in another, manifestly unphysical Hilbert space <?CDATA H{\mathcal{H}}H?> H , the amended Hamiltonian H ≠ H † can be perceived as self-adjoint with respect to the amended inner-product metric Θ. In our paper motivated by a generic technical “user-unfriendliness” of the non-Hermiticity of H we introduce and describe a specific new family of Hamiltonians H for which the metrics Θ become available in closed form.
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.
Abstract In 1992, Scholtz et al. ( Ann. Phys., 213 (1992) 74) showed that a set of non-Hermitian operators can represent observables of a closed unitary quantum system, provided only that its elements are quasi-Hermitian ( i.e. , roughly speaking, Hermitian with respect to an ad hoc inner-product metric). We show that such a version of quantum mechanics admits a simultaneous closed-form representation of the metric and of the observables , in terms of auxiliary operators Z k with . At N = 2 the formalism degenerates to the well-known quantum mechanics using factorized metric , where is parity and where is charge.
In quite a few recent quantum models one is allowed to make a given Hamiltonian H self-adjoint only after an ad hoc generalization of Hermitian conjugation, H†→H‡:= Θ −1H†Θ wherethe suitable operator Θ is called Hilbert-space metric. In the generalized, hidden-Hermiticity scenario with nontrivial metric Θ≠ I the current concept of solvability (meaning, most often, the feasibility of a non-numerical diagonalization of H) requires a generalization (allowing for a non-numerical tractabilityof Θ). A few very elementary samples of "solvable" quantum models of this new type are presented.
In quite a few recent quantum models one is allowed to make a given Hamiltonian H self-adjoint only after an ad hoc generalization of Hermitian conjugation, H†→H‡:= Θ −1H†Θ wherethe suitable operator Θ is called Hilbert-space metric. In the generalized, hidden-Hermiticity scenario with nontrivial metric Θ≠ I the current concept of solvability (meaning, most often, the feasibility of a non-numerical diagonalization of H) requires a generalization (allowing for a non-numerical tractabilityof Θ). A few very elementary samples of "solvable" quantum models of this new type are presented.
A diagonalizable non-Hermitian Hamiltonian having a real spectrum may be used to define a unitary quantum system, if one modifies the inner product of the Hilbert space properly. We give a comprehensive and essentially self-contained review of the basic ideas and techniques responsible for the recent developments in this subject. We provide a critical assessment of the role of the geometry of the Hilbert space in conventional quantum mechanics to reveal the basic physical principle motivating our study. We then offer a survey of the necessary mathematical tools, present their utility in establishing a lucid and precise formulation of a unitary quantum theory based on a non-Hermitian Hamiltonian, and elaborate on a number of relevant issues of fundamental importance. In particular, we discuss the role of the antilinear symmetries such as [Formula: see text], the true meaning and significance of the so-called charge operators [Formula: see text] and the [Formula: see text]-inner products, the nature of the physical observables, the equivalent description of such models using ordinary Hermitian quantum mechanics, the pertaining duality between local-non-Hermitian versus nonlocal-Hermitian descriptions of their dynamics, the corresponding classical systems, the pseudo-Hermitian canonical quantization scheme, various methods of calculating the (pseudo-) metric operators, subtleties of dealing with time-dependent quasi-Hermitian Hamiltonians and the path-integral formulation of the theory, and the structure of the state space and its ramifications for the quantum Brachistochrone problem. We also explore some concrete physical applications and manifestations of the abstract concepts and tools that have been developed in the course of this investigation. These include applications in nuclear physics, condensed matter physics, relativistic quantum mechanics and quantum field theory, quantum cosmology, electromagnetic wave propagation, open quantum systems, magnetohydrodynamics, quantum chaos and biophysics.