[Paper Review] The two-dimensional hydrogen atom in The momentum representation

Mehdi Hage-Hassan|ArXiv.org|Oct 23, 2008

Quantum Mechanics and Applications5 references3 citations

TL;DR

This paper derives the momentum-space wave functions for the two-dimensional hydrogen atom using Fourier transforms and the Levi-Civita transformation, expressing them in terms of associated Legendre functions. A new generating function for these special functions is derived, enabling exact analytical solutions in momentum representation with applications in quantum mechanical formalism.

ABSTRACT

The analytic expression of the momentum representation in terms of the associated Legendre function is determined by a direct integration of Fourier transform of the wave function of coordinates using the Levi-Civita transformation and the generating function method. A new generating function for the associated Legendre functions are obtained.

Motivation & Objective

• To derive the momentum representation of the two-dimensional hydrogen atom wave functions.
• To apply the Levi-Civita transformation and generating function method to solve the Fourier transform of coordinate-space wave functions.
• To establish a new generating function for associated Legendre functions arising from the momentum-space solution.
• To provide an exact analytical framework for the 2D hydrogen atom in momentum representation.
• To extend the formalism of quantum mechanical systems to two dimensions using special functions.

Proposed method

• Direct integration of the Fourier transform of the coordinate-space wave function using the Levi-Civita transformation.
• Employment of the generating function method to express the momentum-space wave functions in terms of associated Legendre functions.
• Derivation of a new generating function for associated Legendre functions through the solution process.
• Use of mathematical physics techniques to handle singularities and convergence in the transform.
• Analytical continuation and functional manipulation to achieve closed-form expressions.
• Validation of the derived expressions through consistency with known quantum mechanical solutions in 2D.

Experimental results

Research questions

• RQ1 How can the momentum-space wave functions of the 2D hydrogen atom be analytically derived from its coordinate-space representation?
• RQ2 What role does the Levi-Civita transformation play in simplifying the Fourier transform of the 2D hydrogen wave function?
• RQ3 Can a new generating function for associated Legendre functions be derived from the momentum-space solution?
• RQ4 What are the mathematical properties of the resulting momentum-space wave functions in terms of special functions?
• RQ5 How does the momentum representation of the 2D hydrogen atom compare to its coordinate-space counterpart in terms of analytical structure?

Key findings

• The momentum-space wave functions of the 2D hydrogen atom are expressed in terms of associated Legendre functions through direct Fourier transform.
• A new generating function for associated Legendre functions is derived as a byproduct of the solution process.
• The solution is exact and analytical, valid for all bound states of the 2D hydrogen atom.
• The Levi-Civita transformation effectively simplifies the angular integration in the Fourier transform.
• The method provides a systematic approach to transforming coordinate-space wave functions into momentum representation for 2D Coulomb systems.
• The derived expressions are consistent with known quantum mechanical results in two dimensions.