Speaker
Description
Let me now tell you about this project. This work is mainly on group theory and perturbation theory and symmetry and I enjoyed it very very much . In this work, I studied about the Laplace-Runge-Lenz vector(A) and built it’s quantum mechanical analogue and its commutation and Lie algebra. This vector commutes with the unperturbed coulombic Hamiltonian of the Hydrogen atom and hence is conserved. The algebra is SO(4) and hence we need 6 generators ie. Three components of the Laplace-Runge-Lenz vector(A_i) and three components of ordinary angular momentum (Li). I have also shown how here S0(4) is isomorphic to SO(3)xSO(3) and we can have two different species of compatible angular momenta to describe the SO(4) structure.
Now when we have a perturbation, we will have to check whether that perturbed hamiltonian still commutes with the original generators or not. If it does then the original symmetry group will be preserved but most of the time it doesn’t ! If it doesn't then the new symmetry group becomes a subgroup of the unperturbed symmetry group and the new algebra will be a subalgebra of the original. I then went on to add perturbations from relativistic 1st order correction(which reduces the symmetry group to a smaller and less rich SO(3)), Strong field Zeeman effect (which reduces it further to SO(2) and it’s irreducible representations are 1-dimensional, and hence the degeneracy is completely lifted in this case) and so on and tried to give a general theory on how symmetry breaking works from a group theoretical point of view. This helps us to naturally see that why the energy depends on more and more quantum numbers as perturbations break the degeneracy. Thus, whenever we have a non-abelian symmetry group associated with a given quantum system, then this leads to degeneracy. Then I went on to show that actually degeneracy is the dimensionality of a given representation and is not accidental at all. The calculations of the commutation relations were very tedious yet satisfying and I am planning to get a paper out of this work in the coming days. This work helps us see ultimately how symmetry considerations can become guiding principles for understanding a physical system better.
Other Fields of Research (Track) for Abstract
Group theory, Representation theory, perturbation theory, Lie Algebra, Symmetry breaking
| Nature of work | Theoretical |
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