2D H.O., graph phase space in R^2?

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SUMMARY

The discussion focuses on graphing the phase space of a 2D harmonic oscillator in R^2, utilizing vectors to represent position and momentum. It establishes that these vectors can be combined to form a single vector in R^4 without losing information. The relationship between the 2D harmonic oscillator and the group SU(2) is highlighted, particularly in the context of spinor rotations. The user seeks clarification on how the orbits of the 2D harmonic oscillator correspond to two-component spinors.

PREREQUISITES
  • Understanding of 2D harmonic oscillators
  • Familiarity with phase space concepts
  • Knowledge of SU(2) group theory
  • Basic grasp of spinors and their rotations
NEXT STEPS
  • Explore the mathematical representation of phase space in classical mechanics
  • Study the relationship between harmonic oscillators and SU(N) symmetry
  • Investigate the implications of spinor rotations in quantum mechanics
  • Review the provided lecture notes on symmetries in quantum mechanics
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Physicists, mathematicians, and students studying quantum mechanics, particularly those interested in the interplay between classical systems and quantum symmetries.

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Can I graph the phase space of a 2D harmonic oscillator in R^2 in the following way?

Let one vector in R^2 represent for position of the point mass and let another vector represent momentum. Together these two vectors in R^2 can represent a single vector in R^4? Do we loose any "information" in such a representation?

The 2D harmonic oscillator and the group SU(2) are related. If we rotate a spinor by 4∏ we come back to where we started, What with the 2D harmonic oscillator, if anything, corresponds to the double rotation with spinors?

Thanks for any help!

Edit, to make sense maybe the momentum vector is attached to the end of the position vector.
 
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Should we be able to map the different orbits of the 2D H.O. to the set of two component spinors?
 

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