U(1):?:SU(3) 1D:2D:3D harmonic oscillator.

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SUMMARY

The one-dimensional harmonic oscillator is linked to the group U(1), while the three-dimensional harmonic oscillator corresponds to the group SU(3). The two-dimensional harmonic oscillator is associated with the group SU(2), which is a simply connected special unitary group that is larger than U(1) but smaller than SU(3). This relationship is confirmed by references to Goldstein's "Classical Mechanics," which also discusses the SO(4) symmetry related to the 1/r potential.

PREREQUISITES
  • Understanding of harmonic oscillators in quantum mechanics
  • Familiarity with group theory, particularly unitary groups
  • Knowledge of SU(2) and SU(3) group properties
  • Basic concepts from classical mechanics, specifically from Goldstein's work
NEXT STEPS
  • Study the properties of SU(2) and its applications in quantum mechanics
  • Explore the relationship between harmonic oscillators and group theory
  • Read Goldstein's "Classical Mechanics" for insights on SO(4) symmetry
  • Investigate the implications of group theory in quantum field theory
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Physicists, mathematicians, and students of quantum mechanics interested in the connections between harmonic oscillators and group theory.

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The one dimensional harmonic oscillator is associated with the group U(1) and the three dimensional harmonic oscillator is associated with the group SU(3). Is their a group associated with the two dimensional harmonic oscillator?

Thank you for any thoughts.
 
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I was given the hint by Igor, thoovler@excite.com:

"What simply connected special unitary group is bigger
than U(1), but smaller than SU(3)?"

My guess was SU(2), a Google search seems to confirm this.
 
Yes, I think you are correct. If you have a copy of Goldstein (Classical Mechanics), they discuss this, as well as the SO(4) symmetry of the 1/r potential.
 

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