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arXiv:2404.17430 · Journal of Physics A Mathematical and Theoretical

Classical echoes of quantum boundary conditions

Giuliano Angelone, Paolo Facchi A, Marilena Ligabò

Published Aug 27, 2024• 3 citations
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Abstract

Abstract We consider a non-relativistic particle in a one-dimensional box with all possible quantum boundary conditions that make the kinetic-energy operator self-adjoint. We determine the Wigner functions of the corresponding eigenfunctions and analyze in detail their classical limit, governed by their behavior in the high-energy regime. We show that the quantum boundary conditions split into two classes: all local and regular boundary conditions collapse to the same classical boundary condition, while a dependence on singular non-local boundary conditions persists in the classical limit.

DOI
10.1088/1751-8121/ad7428

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Abstract We consider a non-relativistic particle in a one-dimensional box with all possible quantum boundary conditions that make the kinetic-energy operator self-adjoint. We determine the Wigner functions of the corresponding eigenfunctions and analyze in detail their classical limit, governed by their behavior in the high-energy regime. We show that the quantum boundary conditions split into two classes: all local and regular boundary conditions collapse to the same classical boundary condition, while a dependence on singular non-local boundary conditions persists in the classical limit.
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We handle all the self-adjoint extensions of the minimal Schroedinger operator for the non-relativistic electron living in the one-dimensional configuration space with a junction. We are interested in every boundary condition corresponding to the individual self-adjoint extension. Thus, we clarify all the types of those boundary conditions of the wave functions of the non-relativistic electron. We find a tunnel-junction formula for the non-relativistic electron passing through the junction. Using this tunnel-junction formula, we propose a mathematical possibility of a tunnel-junction device for qubit.
The following conjecture is shown to be plausible: Consider a sequence of quantum theories with different values of ℏ. In each one, let be a non-degenerate self-adjoint operator, having a "classical limit" as ℏ → 0. Let z' be an eigenvalue of in the vicinity of a fixed real number z (independent of ℏ) and let W ( p, q ; z' ) be the corresponding Wigner function. Then, if we subject W ( p, q ; z' ) to a coarse graining over a scale ≫ ℏ, the result, in the limit ℏ → 0, is proportional to the classical distribution δ[ Z ( p, q ) - z ].
Non-self-adjoint quantum mechanical operators do not necessarily possess eigenvalues. Finite N ? N matrix representations of these operators, however, can be hermitian and therefore have a finite set of N real eigenvalues. Using the momentum operator, the kinetic energy operator, and the relativistic Hamiltonian of the Coulomb problem for the Klein?Gordon equation as examples, we examine analytically and also numerically the properties of the spectrum and eigenvectors in finite dimensional Hilbert spaces. We study the limit of N ? ? for which some eigenvalues cease to exist as the corresponding operators are not self-adjoint.
Abstract We explore the semi-classical structure of the Wigner functions Ψ(q,p) representing bound energy eigenstates | Ψ 〉 for systems with fdegrees of freedom. If the classical motion is integrable, the classical limit of Ψ is a delta function on the f-dimensional torus to which classical trajectories corresponding to |Ψ〉 are confined in the 2f-dimensional phase space. In the semi-classical limit of Ψ (ℏ small but not zero) the delta function softens to a peak of order Ψ−f and the torus develops fringes of a characteristic ‘Airy’ form. Away from the torus,Ψ can have semi-classical singularities that are not delta functions; these are discussed (in full detail when f = 1) using Thom's theory of catastrophes. Brief consideration is given to problems raised when is calculated in a representation based on operators derived from angle coordinates and their conjugate momenta. When Ψ the classical motion is non-integrable, the phase space is not filled with tori and existing semi-classical methods fail. We conjecture that (a) For a given value of non-integrability parameter ⋲,the system passes through three semi-classical régimes as ℏ diminishes. (b) For states |Ψ〉 associated with regions in phase space filled with irregular trajectories, Ψ will be a random function confined near that region of the ‘energy shell’ explored by these trajectories (this region has more thanks dimensions). (c) For ⋲ ≠ 0, ℏ blurs the infinitely fine classical path structure, in contrast to the integrable case ⋲ = 0, where ℏ imposes oscillatory quantum detail on a smooth classical path structure.