Harmonic oscillator position eigenstates via application of an operator on the vacuum

Francisco Soto-Eguibar and Héctor Manuel Moya-Cessa

Instituto Nacional de Astrofísica, Óptica y Electrónica, Luis Enrique Erro 1, Santa María Tonantzintla, San Andrés Cholula, Puebla, 72840 México.

Received 21 August 2013 ]]> Accepted 17 October 2013

Abstract

Harmonic oscillator squeezed states are states of minimum uncertainty, but unlike coherent states, in which the uncertainty in position and momentum are equal, squeezed states have the uncertainty reduced, either in position or in momentum, while still minimizing the uncertainty principle. It seems that this property of squeezed states would allow to obtain the position eigenstates as a limiting case, by doing null the uncertainty in position and infinite in momentum. However, there are two equivalent ways to define squeezed states, that lead to different expressions for the limiting states. In this work, we analyze both definitions and show the advantages and disadvantages of using them in order to find position eigenstates. With this in mind, but leaving aside the definitions of squeezed states, we find an operator that applied to the vacuum gives position eigenstates. We also analyze some properties of the squeezed states, based on the new expressions obtained for the eigenstates of the position.

Keywords: Position eigenstates; harmonic oscillator; squeezed states; minimum uncertainty states; squeeze operator.

PACS: 03.65.-w

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