Constants of motion, ladder operators and supersymmetry of the two-dimensional isotropic harmonic oscillator

2002 ◽  
Vol 35 (12) ◽  
pp. 2979-2984 ◽  
Author(s):  
R D Mota ◽  
V D Granados ◽  
A Queijeiro ◽  
J García
2004 ◽  
Vol 82 (10) ◽  
pp. 767-773 ◽  
Author(s):  
R D Mota ◽  
M A Xicoténcatl ◽  
V D Granados

We show that the well-known Stokes operators, defined as elements of the Jordan–Schwinger map with the Pauli matrices of two independent bosons, are equal to the constants of motion of the two-dimensional isotropic harmonic oscillator. Taking the expectation value of the Stokes operators in a two-mode coherent state, we obtain the corresponding classical Stokes parameters. We show that this classical limit of the Stokes operators, the 2 × 2 unit matrix and the Pauli matrices may be used to expand the polarization matrix. Finally, by means of the constants of motion of the classical two-dimensional isotropic harmonic oscillator, we describe the geometric properties of the polarization ellipse. Our study is restricted to the case of a monochromatic quantized-plane electromagnetic wave that propagates along the z axis.PACS Nos.: 42.50.–p, 42.25.Ja, 11.30.–j


1998 ◽  
Vol 13 (19) ◽  
pp. 3347-3360
Author(s):  
LUIS J. BOYA ◽  
ERIC CHISOLM ◽  
S. M. MAHAJAN ◽  
E. C. G. SUDARSHAN

We contruct and study a continuous family of representations of the N-dimensional isotropic harmonic oscillator (N≥2) which are not unitarily equivalent to the standard one. We explain why such representations exist and we investigate their simpler properties: the spectrum of the Hamiltonian (which contains nonstandard values), the form of the energy eigenfunctions, and their behavior under the ladder operators. Various symmetry and dynamical groups (e.g. the rotation group) which are valid on the standard representation are not implemented on the new ones. We comment very briefly on the prospects of observing these representations experimentally.


Author(s):  
M. R. Dennis ◽  
M. A. Alonso

The connection between Poincaré spheres for polarization and Gaussian beams is explored, focusing on the interpretation of elliptic polarization in terms of the isotropic two-dimensional harmonic oscillator in Hamiltonian mechanics, its canonical quantization and semiclassical interpretation. This leads to the interpretation of structured Gaussian modes, the Hermite–Gaussian, Laguerre–Gaussian and generalized Hermite–Laguerre–Gaussian modes as eigenfunctions of operators corresponding to the classical constants of motion of the two-dimensional oscillator, which acquire an extra significance as families of classical ellipses upon semiclassical quantization. This article is part of the themed issue ‘Optical orbital angular momentum’.


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