[Paper Review] Fisher information for quasi-one-dimensional hydrogen atom

Aparna Saha, B. Talukdar+1 more|arXiv (Cornell University)|Mar 10, 2017

Quantum Mechanics and Non-Hermitian Physics2 references3 citations

TL;DR

This paper resolves the challenge of constructing a physically valid momentum-space wave function for quasi-one-dimensional hydrogen atoms by rigorously analyzing the Fourier transform of the coordinate-space wave function. It identifies the imaginary part of the momentum-space wave function as the correct physical representation and derives analytical expressions for position- and momentum-space Fisher information in terms of the principal quantum number and energy eigenvalue, confirming the information-theoretic uncertainty relation.

ABSTRACT

The coordinate-space wave function $ψ(x)$ of quasi-one-dimensional atoms is defined in the $x\geq 0$ region only. This poses a typical problem to write a physically acceptable momentum-space wave function $ϕ(p)$ from the Fourier transform of $ψ(x)$. We resolve the problem with special attention to the behavior of real and imaginary parts of the complex-valued function $ϕ(p)$ as a function of $p$ and confirm that $ϕ_i(p)$ (the imaginary part of $ϕ(p)$) represents the correct momentum-space wave function. We make use of the results for $ψ(x)$ and $ϕ_i(p)$ to express the position- and momentum-space Fisher information in terms of the principal quantum number and energy eigen value of the system and provide some useful checks on the result presented with particular attention on the information theoretic uncertainty relation.

Motivation & Objective

• To address the challenge of defining a physically consistent momentum-space wave function for quasi-one-dimensional hydrogen atoms due to the half-space domain (x ≥ 0).
• To resolve the ambiguity in the Fourier transform of the coordinate-space wave function ψ(x) by analyzing the real and imaginary parts of the resulting momentum-space function φ(p).
• To identify the imaginary part φ_i(p) as the correct physical momentum-space wave function.
• To compute position- and momentum-space Fisher information in terms of the principal quantum number and energy eigenvalue.
• To verify the information-theoretic uncertainty relation using the derived Fisher information expressions.

Proposed method

• Fourier transforming the coordinate-space wave function ψ(x) defined on x ≥ 0 to obtain the complex-valued momentum-space wave function φ(p).
• Analyzing the behavior of the real and imaginary parts of φ(p) as functions of momentum p to identify the physically acceptable component.
• Selecting the imaginary part φ_i(p) as the correct momentum-space wave function based on physical consistency and normalization criteria.
• Deriving analytical expressions for Fisher information in position space I_x and momentum space I_p using ψ(x) and φ_i(p).
• Expressing I_x and I_p explicitly in terms of the principal quantum number n and the energy eigenvalue E_n.
• Verifying the information-theoretic uncertainty relation I_x × I_p ≥ 4 using the derived expressions.

Experimental results

Research questions

• RQ1 How can a physically consistent momentum-space wave function be constructed for a quasi-one-dimensional hydrogen atom with ψ(x) defined only on x ≥ 0?
• RQ2 What is the physical significance of the real and imaginary parts of the Fourier transform φ(p) of ψ(x)?
• RQ3 Which component of φ(p) corresponds to the correct momentum-space wave function?
• RQ4 What are the analytical expressions for position- and momentum-space Fisher information in this system?
• RQ5 Do the derived Fisher information values satisfy the information-theoretic uncertainty relation?

Key findings

• The imaginary part φ_i(p) of the Fourier transform φ(p) is identified as the correct physical momentum-space wave function for the quasi-one-dimensional hydrogen atom.
• Analytical expressions for position-space Fisher information I_x and momentum-space Fisher information I_p are derived in terms of the principal quantum number n and the energy eigenvalue E_n.
• The derived Fisher information values satisfy the information-theoretic uncertainty relation I_x × I_p ≥ 4.
• The results confirm the consistency of the wave function reconstruction method and the physical validity of φ_i(p).
• The study provides a systematic framework for computing Fisher information in systems with restricted spatial domains.
• The derived expressions are exact and explicitly dependent on quantum numbers, enabling quantitative analysis of information content in quasi-one-dimensional quantum systems.