A quantum mechanical proof of the fourier inversion formula

Authors

  • Nelson Castro Universidade Federal da Paraíba.
  • Ramón Mendoza Universidade Federal de Pernambuco.
  • Jacqueline F. Rojas Universidade Federal da Paraíba.

DOI:

https://doi.org/10.4067/S0716-09172011000300010

Keywords:

Fourier transform, Position operator, Momentum operator, , Extension.

Abstract

The translation of the observable, position and momentum, of a given particle in the real line, at a certain time t, from Classical Mechanics, into the operators, position and momentum, in Quantum Mechanics, gives us the inspiration to make a proof of the existence of the Fourier's Inverse Transform, using algebraic relations involving these operators (position and momentum), a few of Linear Algebra and Analysis, without resorting to the classical technics like Fubini's Theorem and Lebesgue's Dominated Convergence Theorem.

References

[1] P. Cordaro and A. Kawano, O Delta de Dirac. Livraria da Física Editora, (2002).

[2] D. G. Figueiredo, Análise de Fourier e Equacoes Diferenciais Parciais. Projeto Euclides, IMPA, (2000).

[3] J. Glimm and A. Jaffe, Quantum Physics, A Functional Integral Point of View. Springer, 2nd edition, (1987).

[4] J. Hounie, Teoria Elementar das Distribuicoes. IMPA, Rio de Janeiro, (1979).

[5] E. L. Lima, Algebra Linear. IMPA, Segunda Edicao, (1996).

[6] E. H. Lieb and M.Loss, Analysis. AMS, (1997).

[7] A. H. Zemanian, Distribution Theory and Transform Analysis: An Introduction to Generalized Functions, with Applications. Dover Publications, (2010).

Published

2011-12-10

How to Cite

[1]
N. Castro, R. Mendoza, and J. F. Rojas, “A quantum mechanical proof of the fourier inversion formula”, Proyecciones (Antofagasta, On line), vol. 30, no. 3, pp. 441-457, Dec. 2011.

Issue

Section

Artículos