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arXiv:1304.6277

Localization properties of squeezed quantum states in nanoscale space domains

Anton Sergeevich Trushechkin, I. V. Volovich

Published Apr 23, 2013• 5 citations
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Abstract

We construct families of squeezed quantum states on an interval (depending on boundary conditions, we interpret the interval as a circle or as the infinite square potential well) and obtain estimates of position and momentum dispersions for these states. A particular attention is paid to the possibility of proper localization of a particle in nanoscale space domains. One of the constructed family of squeezed states is based on the theta function. It is a generalization of the known coherent and squeezed states on the circle. Also we construct a family of squeezed states based on truncated Gaussian functions and a family of wave packets based on the discretization of an arbitrary continuous momentum probability distribution. The problem of finiteness of the energy dispersion for the squeezed states in the infinite well is discussed. Finally, we perform the limit of large interval length and the semiclassical limit. As a supplementary general result, we show that an arbitrary physical quantity has a finite dispersion if and only if the wave function of a quantum system belongs to the domain of the corresponding self-adjoint operator. This can be regarded as a physical meaning of the domain of a self-adjoint operator.

DOI
10.48550/arxiv.1304.6277

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quant-pharXiv:2601.12739•Jan 20, 2026
We construct families of squeezed quantum states on an interval (depending on boundary conditions, we interpret the interval as a circle or as the infinite square potential well) and obtain estimates of position and momentum dispersions for these states. A particular attention is paid to the possibility of proper localization of a particle in nanoscale space domains. One of the constructed family of squeezed states is based on the theta function. It is a generalization of the known coherent and squeezed states on the circle. Also we construct a family of squeezed states based on truncated Gaussian functions and a family of wave packets based on the discretization of an arbitrary continuous momentum probability distribution. The problem of finiteness of the energy dispersion for the squeezed states in the infinite well is discussed. Finally, we perform the limit of large interval length and the semiclassical limit. As a supplementary general result, we show that an arbitrary physical quantity has a finite dispersion if and only if the wave function of a quantum system belongs to the domain of the corresponding self-adjoint operator. This can be regarded as a physical meaning of the domain of a self-adjoint operator.
We show unexpected behaviour in simulations of generalized squeezing performed with finite-dimensional truncations of the Fock space: even for extremely large dimension of the state space, the results depend on whether the truncation dimension is even or odd. This situation raises the question whether the simulation results are physically meaningful. We demonstrate that, in fact, the two truncation schemes correspond to two well-defined, distinct unitary evolutions whose generators are defined on different subsets of the infinite-dimensional Fock space. This is a consequence of the fact that the generalized squeezing Hamiltonian is not self-adjoint on states with finite excitations, but possesses multiple self-adjoint extensions. Furthermore, we present results on the spectrum of the squeezing Hamiltonians corresponding to even and odd truncation size that elucidate the properties of the two different self-adjoint extensions corresponding to the even and odd truncation scheme. To make the squeezing operator applicable to a physical system, we must regularize it by other terms that depend on the specifics of the experimental implementation. We show that the addition of a Kerr interaction term in the Hamiltonian leads to uniquely converging simulations, with no dependence on the parity of the truncation size, and demonstrate that the Kerr term indeed renders the Hamiltonian self-adjoint and thus physically interpretable.
A careful study of the physical properties of a family of coherent states on the circle, introduced some years ago by de Bièvre and González (in 1992 Semiclassical behaviour of the Weyl correspondence on the circle Group Theoretical Methods in Physics vol I (Madrid: Ciemat)), is carried out. They were obtained from the Weyl-Heisenberg coherent states in by means of the Weil-Brezin-Zak transformation, they are labelled by the points of the cylinder , and they provide a realization of by entire functions (similar to the well known Fock-Bargmann construction). In particular, we compute the expectation values of the position and momentum operators on the circle and we discuss the Heisenberg uncertainty relation.
We introduce a general formalism, based on the stochastic formulation of quantum mechanics, to obtain localized quasi-classical wave packets as dynamically controlled systems, for arbitrary anharmonic potentials. The control is in general linear, and it amounts to introduce additional quadratic and linear time-dependent terms to the given potential. In this way one can construct for general systems either coherent packets moving with constant dispersion, or dynamically squeezed packets whose spreading remains bounded for all times. In the standard operatorial framework our scheme corresponds to a suitable generalization of the displacement and scaling operators that generate the coherent and squeezed states of the harmonic oscillator.
This research note proposes an experimental-setting interpretation of self-adjoint realization for boundary-sensitive quantum systems. The starting point is the standard operator-theoretic fact that a formal differential expression does not by itself define quantum dynamics. A Hamiltonian becomes a physical generator of unitary evolution only after a Hilbert space, an operator domain, and a self-adjoint realization have been specified. The note extends the ordinary half-line problem in quantum mechanics to a multidimensional setting in which the coordinate axes act as axial endpoint or boundary sets. A weighted Hilbert space is introduced, and the local radial behavior near an axis is reduced to an effective inverse-square potential problem. The endpoint classification is then described in terms of the usual limit-circle/limit-point distinction. In particular, the phrase “spectral inaccessibility” is used only as an interpretive term; the rigorous statement is endpoint classification, such as limit-point behavior. The main conceptual contribution is the proposal that, in boundary-sensitive effective systems, an experimental setup may participate not only in selecting a state, but also in defining the effective subsystem itself. In this viewpoint, the setup may help select the relevant effective Hilbert space, operator domain, and self-adjoint realization before a subsystem wave function is assigned. Schematically, the proposed viewpoint is Ψtot⟶(Hsys,Heff(θapp),ψsys).\Psi_{\rm tot}\quad\longrightarrow\quad\left(\mathcal H_{\rm sys},H_{\rm eff}^{(\theta_{\rm app})},\psi_{\rm sys}\right).Ψtot​⟶(Hsys​,Heff(θapp​)​,ψsys​). Here θapp\theta_{\rm app}θapp​ is not asserted to be determined by a universal theorem. It is regarded as a model-dependent effective parameter that may arise from the apparatus, boundary region, contact interaction, or environment after reduction to a low-energy subsystem description. The note also clarifies that the statement “no effective wave function has yet been defined” does not deny the possible existence of a total quantum state. Rather, it means that the reduced subsystem wave function is meaningful only after the effective subsystem, its Hilbert space, and its self-adjoint Hamiltonian have been selected. Finally, the note identifies an open problem: the concrete derivation of \[H_{\rm tot}\quad\longrightarrow\quadH_{\rm eff}^{(\theta_{\rm app})}\] from a full self-adjoint total Hamiltonian. This reduction is interpreted as a state-preparation and effective Hamiltonian-engineering problem, rather than merely a measurement-outcome problem. A concrete model of this reduction is left for future work.
We analyze the problem of a free 1D Klein-Fock-Gordon-Majorana (KFGM) particle in an interval. By free, we mean that there is no potential within the interval and that its walls are penetrable; hence, the pertinent energy current density does not vanish at the walls. Certainly, quantization in an interval is not trivial because certain restrictions imposed by the domains of the operators involved arise. Here, our objective is to obtain the Hamiltonian for these particles. In practice, the Feshbach-Villars (FV)--free Hamiltonian is the proper operator for characterizing them and is a function of the momentum operator. Additionally, a Majorana condition must also be imposed on the wavefunctions on which these two operators can act. Thus, we start by calculating the pseudo self-adjoint momentum operator. A three-parameter set of boundary conditions (BCs) constitutes its domain. Up to this point, the domain of the Hamiltonian is induced by the domain of the momentum operator; however, we ensure that only the BCs for which the energy current density has the same value at each end of the interval are in its domain. All these BCs essentially belong to a one-parameter set of BCs. Moreover, because the FV equation is invariant under the operation of parity, the parity-transformed wavefunction is also a solution of this equation, which further restricts the domain of the free FV Hamiltonian. Finally, knowing the most general three-parameter set of BCs for the pseudo self-adjoint FV Hamiltonian for a 1D KFGM particle in an interval, we find that only two BCs can remain within the domain of the FV--free Hamiltonian: the periodic BC and the antiperiodic BC. These BCs are satisfied by both the two-component FV wavefunction, with these components being related, and the one-component KFG wavefunction, which can be real or imaginary.