What Are the Permissible (L,S) Pairs for J=1 in a System of Two Spin-1 Bosons?

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SUMMARY

The discussion focuses on determining the permissible (L,S) pairs for J=1 in a system of two indistinguishable spin-1 bosons. It establishes that the total angular momentum J is defined as the sum of the relative orbital angular momentum L and the total spin S. The participant successfully demonstrated that for states S=0 and S=2, the wavefunction remains symmetric under particle interchange, while for S=1, it becomes antisymmetric. The significance of the center of mass frame and the implications of Bose-Einstein statistics on the overall wavefunction symmetry were also highlighted.

PREREQUISITES
  • Understanding of total angular momentum in quantum mechanics
  • Familiarity with spin-1 bosons and their statistical behavior
  • Knowledge of symmetric and antisymmetric wavefunctions
  • Concept of center of mass frame in quantum systems
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  • Study the properties of spin-1 particles in quantum mechanics
  • Learn about the implications of Bose-Einstein statistics on wavefunctions
  • Explore the mathematical formulation of angular momentum coupling
  • Investigate the significance of the center of mass frame in quantum systems
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Quantum physicists, students studying quantum mechanics, and researchers focusing on the behavior of indistinguishable particles and angular momentum in bosonic systems.

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Homework Statement



We have a system of 2 indistinguishable spin-1 bosons. We shall adopt the center of mass frame.

Let
S = total spin
L = relative orbital angular momentum
J = L+S = total angular momentum

Prove that J = 2m where m is an integer.
If given that J=1, what are the permissible (L,S) pairs?

Homework Equations


Bose-Einstein Stats?

The Attempt at a Solution


I am lost with this. I have managed to show that for the states S=0, S=2 interchanging particles 1,2 is symmetric, whereas it is antisymmetric for S=1. However, I don't know how to use this here.
What is the significance of the CoM frame choice?
Also, due to Bose-Einstein stats, the overall wavefunction should be symmetric under the interchange of the 2 bosons. However, I don't know how even and odd wavefunctions for orbital angular momentum behave. Anyone please?
 
Last edited:
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No worries. Problem resolved. Thanks for reading anyways.
 

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