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Total Angular Momentum

Explore the physics of total angular momentum, addition of angular momentum, and fine structure in quantum mechanics. Understand the implications of spin-orbit interactions and the two-body problem. Learn spectroscopic notations and transition rules.

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Total Angular Momentum

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  1. Total Angular Momentum • Since J is an angular momentum like L and S, its algebra is identical to that for orbital and spin angular momentum

  2. Total Angular Momentum • We can determine the possible values of j by looking at the z components • The maximum value of j is the maximum value of mj is the maximum value of ml and ms is l+s • The minimum value of j can be found using the vector inequality • Thus j runs from

  3. Addition of Angular Momentum • These rule hold true for the addition of any two types of angular momenta • Example • What are the possible s and ms values for the combination of two spin 1/2 particles? • What are the possible l and ml values for two electrons having l1=1 and l2=2? • What are the possible j and mj values for one electron with l=2 and s=1/2?

  4. Fine Structure • For the hydrogen atom, we found that when one includes the spin-orbit interaction n, l, ml, s, ms are no longer “good” quantum numbers • Instead we must use n, l, s, j, mj • We usually use spectroscopic notation to describe these states • We’ll use this notation for multielectron states as well

  5. Fine Structure • Examples • What are first three states (ground and first two excited states) for hydrogen? • What are l, s, j values for 3F2?

  6. Fine Structure • The potential energy associated with the spin-orbit interaction was calculated as • Additionally we noted • Hence

  7. Fine Structure • Furthermore, the energy eigenvalues are just the expectation values of H and with time you could show • The point is that the spin-orbit interaction lifts the 2l+1 degeneracy in l

  8. Fine Structure • Hydrogen fine structure

  9. Fine Structure • The selection rules for transitions between states must be updated as well

  10. Fine Structure • Transitions with spin-orbit interaction

  11. Two Body Problem • Two body systems (proton and electron, two electrons, …) are obviously important in quantum mechanics • A two body problem can be reduced to an equivalent one body problem if the potential energy one depends on the separation of the two particles

  12. Two Body Problem

  13. Two Body Problem • This equation is now separable • The two equations represent • A free particle equation for the center-of-mass • A one body equation using the reduced mass and relative coordinates

  14. Two Body Problem • In the case of a central potential problem (like He) with m1=m2=m • The point is the following: the interchange of two particles in central potential simply means

  15. Two Body Problem • Particle interchange in the central potential problem means • This means • Under particle interchange of 1 and 2 • The radial wave function is unchanged • The symmetry of the angular wave function depends on l

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