A New Method for Constructing
Squeezed States for the Isotropic 2D
Harmonic Oscillator
James Moran and Véronique Hussin
Abstract We introduce a new method for constructing squeezed states for the 2D
isotropic harmonic oscillator. Based on the construction SU (2) coherent states, we
define a new set of ladder operators for the 2D system as a linear combination of the
x and y ladder operators and construct the SU (2) coherent states. The new ladder
operators are used for generalizing the squeezing operator to 2D and the SU (2)
coherent states play the role of the Fock states in the expansion of the 2D squeezed
states. We discuss some properties of the 2D squeezed states.
Keywords Coherent states · Squeezed states · Harmonic oscillator · SU (2)
coherent states · 2D coherent states · 2D squeezed states · Uncertainty principle
1 Introduction
Degeneracy in the spectrum of the Hamiltonian is one of the first problems we
encounter when trying to define a new type of coherent states for the 2D oscillator.
As a continuation of the work in [1] we produce a non-degenerate number basis
(SU (2) coherent states) for the 2D isotropic harmonic oscillator with accompanying
generalized creation and annihilation operators. The squeezed states for the 2D
isotropic harmonic oscillator are then defined in terms of the SU (2) coherent states
and generalized ladder operators.
J. Moran ()
Centre de Recherches Mathématiques, Université de Montréal, Montréal, QC, Canada
Département de Physique, Université de Montréal, Montréal, QC, Canada
e-mail: james.moran@umontreal.ca
V. Hussin
Centre de Recherches Mathématiques, Université de Montréal, Montréal, QC, Canada
Département de Mathématiques et de Statistique, Université de Montréal, Montréal, QC, Canada
e-mail: hussin@dms.umontreal.ca
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_24
255
Squeezed States for the Isotropic 2D
Harmonic Oscillator
James Moran and Véronique Hussin
Abstract We introduce a new method for constructing squeezed states for the 2D
isotropic harmonic oscillator. Based on the construction SU (2) coherent states, we
define a new set of ladder operators for the 2D system as a linear combination of the
x and y ladder operators and construct the SU (2) coherent states. The new ladder
operators are used for generalizing the squeezing operator to 2D and the SU (2)
coherent states play the role of the Fock states in the expansion of the 2D squeezed
states. We discuss some properties of the 2D squeezed states.
Keywords Coherent states · Squeezed states · Harmonic oscillator · SU (2)
coherent states · 2D coherent states · 2D squeezed states · Uncertainty principle
1 Introduction
Degeneracy in the spectrum of the Hamiltonian is one of the first problems we
encounter when trying to define a new type of coherent states for the 2D oscillator.
As a continuation of the work in [1] we produce a non-degenerate number basis
(SU (2) coherent states) for the 2D isotropic harmonic oscillator with accompanying
generalized creation and annihilation operators. The squeezed states for the 2D
isotropic harmonic oscillator are then defined in terms of the SU (2) coherent states
and generalized ladder operators.
J. Moran ()
Centre de Recherches Mathématiques, Université de Montréal, Montréal, QC, Canada
Département de Physique, Université de Montréal, Montréal, QC, Canada
e-mail: james.moran@umontreal.ca
V. Hussin
Centre de Recherches Mathématiques, Université de Montréal, Montréal, QC, Canada
Département de Mathématiques et de Statistique, Université de Montréal, Montréal, QC, Canada
e-mail: hussin@dms.umontreal.ca
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_24
255