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arXiv:quant-ph/0505032

Unusual scalar products in Hilbert space of Quantum Mechanics: non-Hermitian square-well model with two coupled channels

Miloslav Znojil

Published May 6, 2005• 0 citations

Abstract

A pseudo-Hermitian coupled-channel square-well model is proposed, solved and discussed. The domain of parameters is determined where all the bound-state energies (twice degenerate with respect to the second observable which we call "spin") remain real. The standard probabilistic interpretation of the wave functions is achieved by the transition from the original elementary non-physical indefinite pseudo-metric $θ$ to another, more involved but correct positive-definite physical metric $Θ$. In our model the latter step remains comparatively easily feasible. Another fortunate circumstance emerges during the perturbative construction of the metric $Θ$ and, hence, of the correct scalar product and of the physical norm. One finds that at the higher energy levels the influence of all the non-Hermiticities of our model becomes very strongly and progressively suppressed.

DOI: 10.48550/arxiv.quant-ph/0505032

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A pseudo-Hermitian coupled-channel square-well model is proposed, solved and discussed. The domain of parameters is determined where all the bound-state energies (twice degenerate with respect to the second observable which we call "spin") remain real. The standard probabilistic interpretation of the wave functions is achieved by the transition from the original elementary non-physical indefinite pseudo-metric

θ

to another, more involved but correct positive-definite physical metric

Θ

. In our model the latter step remains comparatively easily feasible. Another fortunate circumstance emerges during the perturbative construction of the metric

Θ

and, hence, of the correct scalar product and of the physical norm. One finds that at the higher energy levels the influence of all the non-Hermiticities of our model becomes very strongly and progressively suppressed.

A new Hermitizable quantum model is proposed in which the bound-state energies are real and given as roots of an elementary trigonometric expression while the wave function components are expressed as superpositions of two Chebyshev polynomials. As an -site lattice version of square well with complex Robin-type two-parametric boundary conditions the model is unitary with respect to the Hilbert space metric Θ which becomes equal to the most common Diracʼs metric in the conventional textbook Hermitian–Hamiltonian limit. This metric is constructed in closed form at all .

A discrete N-point Runge–Kutta version H(N)(λ) of one of the simplest non-Hermitian square-well Hamiltonians with real spectrum is studied. Its possible Hermitizations mediated by nontrivial (often called “non-Dirac”) metrics Θ≠I are considered as a source of nonequivalent standard probabilistic interpretations of this quantum model. A complete set of these alternative, multiparametric metrics Θ=Θ(a,b,…)(N)(λ) defining all the eligible Hamiltonian-dependent representations of the physical Hilbert space of states is constructed, in closed form, for any coupling λ∊(−1,1) and for any matrix dimension N.

A coupled pair of PT-symmetric square wells is studied as a prototype of a quantum system characterized by two manifestly non-Hermitian commuting observables. Via the diagonalization of our Hamiltonian H ≠ H † and spin-like observable Ω ≠ Ω†, we demonstrate that there exists a domain of couplings where both the respective sets of eigenvalues E n , n = 0, 1, ... (energies), and σ = ±1 ('spin projections') remain real . In such a 'measurable' regime, the model acquires a consistent probabilistic interpretation mediated by our selection of one of many available interaction-dependent scalar products.

A discrete N-point Runge–Kutta version H(N)(λ) of one of the simplest non-Hermitian square-well Hamiltonians with real spectrum is studied. Its possible Hermitizations mediated by nontrivial (often called “non-Dirac”) metrics Θ≠I are considered as a source of nonequivalent standard probabilistic interpretations of this quantum model. A complete set of these alternative, multiparametric metrics Θ=Θ(a,b,…)(N)(λ) defining all the eligible Hamiltonian-dependent representations of the physical Hilbert space of states is constructed, in closed form, for any coupling λ∊(−1,1) and for any matrix dimension N.

A class of pseudo-Hermitian quantum system with an explicit form of the positive-definite metric in the Hilbert space is presented. The general method involves a realization of the basic canonical commutation relations defining the quantum system in terms of operators that are Hermitian with respect to a pre-determined positive-definite metric in the Hilbert space. Appropriate combinations of these operators result in a large number of pseudo-Hermitian quantum systems admitting entirely real spectra and unitary time evolution. The examples considered include simple harmonic oscillators with complex angular frequencies, Stark (Zeeman) effect with non-Hermitian interaction, non-Hermitian general quadratic form of N boson (fermion) operators, symmetric and asymmetric XXZ spin chain in the complex magnetic field, non-Hermitian Haldane–Shastry spin chain and Lipkin–Meshkov–Glick model.