[Paper Review] Localization properties of squeezed quantum states in nanoscale space domains

А. С. Трушечкин, И. В. Волович|arXiv (Cornell University)|Apr 23, 2013

Quantum Mechanics and Non-Hermitian Physics41 references6 citations

TL;DR

This paper constructs families of squeezed quantum states on bounded intervals—such as a circle or infinite square well—to analyze their localization properties in nanoscale domains. By generalizing coherent and squeezed states using theta functions and truncated Gaussians, the authors derive quantitative estimates showing that position dispersions can reach ~0.1 nm and momentum dispersions ~10⁻²⁴ kg·m/s, demonstrating feasible quantum particle localization for nanotechnology applications.

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.

Motivation & Objective

• To investigate the possibility of high-precision localization of quantum particles in nanoscale space domains, such as quantum dots or quantum wires.
• To extend the concept of squeezed states to bounded one-dimensional systems like the infinite square well and the circle, where standard uncertainty relations do not directly apply.
• To provide rigorous estimates for position and momentum dispersions in such states, ensuring they satisfy physical constraints like finite energy dispersion.
• To clarify the physical significance of the domain of a self-adjoint operator in quantum mechanics, linking it to the finiteness of observable dispersions.

Proposed method

• Constructing squeezed states on an interval using truncated Gaussian wave functions, with normalization and dispersion analysis via integration over finite domains.
• Introducing a family of squeezed states based on Jacobi theta functions, generalizing known circular coherent and squeezed states to bounded geometries.
• Defining a third family of states through discretization of continuous momentum probability distributions, enabling flexible control over momentum spread.
• Applying asymptotic analysis in the large interval limit and semiclassical regime to validate consistency with standard quantum mechanics.
• Using self-adjoint operator theory to establish that a physical observable has finite dispersion if and only if the wave function lies in the operator’s domain.
• Deriving bounds on position and momentum dispersions using Fourier analysis and trigonometric sum estimates, including bounds on partial sums of cosine series.

Experimental results

Research questions

• RQ1 Can quantum particles be localized with sub-nanometer precision in bounded nanoscale domains such as quantum wells or rings?
• RQ2 How do the uncertainty relations for position and momentum differ in bounded intervals compared to the infinite line?
• RQ3 What are the conditions under which energy and momentum dispersions remain finite in squeezed states confined to an infinite square well?
• RQ4 What is the physical meaning of the domain of a self-adjoint operator in quantum mechanics, particularly regarding observable finiteness?
• RQ5 Can squeezed states on a circle or interval be constructed such that they asymptotically saturate the standard Heisenberg uncertainty relation on the line?

Key findings

• For a 100 nm interval, the paper constructs wave packets with position standard deviation Δx ≈ 0.1 nm and momentum standard deviation Δp ≈ 10⁻²⁴ kg·m/s, demonstrating sub-nanometer localization.
• The family of squeezed states based on the theta function generalizes known circular coherent and squeezed states, enabling high-precision localization on compact manifolds.
• The discretization-based construction (Theorem 3) guarantees Δx ≤ 0.1 nm when Δp ≈ 10⁻²⁰ kg·m/s, sufficient for nanoscale localization despite coarser estimates.
• Finite energy dispersion in the infinite well is achievable only if the wave function lies in the domain of the Hamiltonian operator, which is shown to be a necessary and sufficient condition.
• An arbitrary physical observable has finite dispersion if and only if the wave function belongs to the domain of the corresponding self-adjoint operator, establishing a direct physical interpretation of this mathematical concept.
• In the large interval limit, the squeezed states on the interval asymptotically approach the standard coherent and squeezed states on the real line, confirming consistency with standard quantum mechanics.