Moments in QM: When Can I Say JLS^2 Approx Equal JL+S?

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Discussion Overview

The discussion centers on the conditions under which the squared angular momentum operators can be approximated by their eigenvalue expressions in quantum mechanics, specifically regarding the total angular momentum \(\vec{J}\), orbital momentum \(\vec{L}\), and spin momentum \(\vec{S}\). The scope includes theoretical considerations of angular momentum in quantum states.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants suggest that the approximations \(J^2 \approx J(J+1)\), \(L^2 \approx L(L+1)\), and \(S^2 \approx S(S+1)\) hold for large enough values of \(J\), \(L\), and \(S\).
  • One participant clarifies that the left-hand side of the equations represents operators, while the right-hand side represents eigenvalues, indicating that the equations hold for angular momentum states of pure \(j\), \(l\), and \(s\).
  • Another participant questions the applicability of the formula \(g_J = \frac{J(J+1) + L(L+1) - S(S+1)}{2J(J+1)}\), suggesting it may not be limited to large \(J\), \(L\), and \(S\).
  • It is noted that while \(\vec{J}^2 |jm\rangle = j(j+1) |jm\rangle\) is valid, the equations are only valid when \(J\), \(L\), and \(S\) are negligible compared to their squared values.
  • Another participant reiterates that the formula for \(g_J\) is applicable for all eigenstates of \(J\), \(L\), and \(S\), not just for large values.

Areas of Agreement / Disagreement

Participants express differing views on the conditions under which the approximations and formulas apply, indicating that the discussion remains unresolved with multiple competing perspectives.

Contextual Notes

Participants highlight limitations regarding the assumptions of large angular momentum values and the definitions of eigenvalues versus operators, which are not fully resolved in the discussion.

Petar Mali
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\vec{J} - mechanical moment
\vec{L} - orbital moment
\vec{S} - spin moment

\vec{J}=\vec{L}+\vec{S}

When can I say
J^2\approx J(J+1)
L^2\approx L(L+1)
S^2\approx S(S+1)?
 
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For large enough J, L and S :)
 
In your equations the J on the left is an operator, the j on the right should be an eigenvalue (a number). Then your equations always hold for angular momentum states of pure j,l,s.
 
J - eigen-value

I'm asking you because the formula

g_J=\frac{J(J+1)+L(L+1)-S(S+1)}{2J(J+1)}

I think that that formula is in the game not just for very large J,L,S.
 
You always have \vec J\thinspace ^2|jm\rangle=j(j+1)|jm\rangle, but if the left-hand sides of your equations are eigenvalues too, then the equations are obviously only valid when J,L,S are negligible compared to J²,L²,S².
 
Petar Mali said:
J - eigen-value

I'm asking you because the formula

g_J=\frac{J(J+1)+L(L+1)-S(S+1)}{2J(J+1)}

I think that that formula is in the game not just for very large J,L,S.
That formula is true for all eigenstates of J,L,S.
 

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