A question on momenta of electrons

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Discussion Overview

The discussion revolves around the significance of the relationship between the angular momentum vectors of electrons, specifically the total angular momentum \(\vec{J}\), orbital angular momentum \(\vec{L}\), and spin angular momentum \(\vec{S}\). Participants explore the implications of these relationships in the context of quantum mechanics, particularly focusing on whether \(\vec{L}\) and \(\vec{S}\) being in-phase or out-of-phase has any physical significance.

Discussion Character

  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions the physical significance of the conditions | \(\vec{J}\) | = | \(\vec{L}\) | + | \(\vec{S}\) | and | \(\vec{J}\) | = | \(\vec{L}\) | - | \(\vec{S}\) |, particularly in relation to the phase relationship between \(\vec{L}\) and \(\vec{S}\).
  • Another participant asserts that the total angular momentum is defined as \(\vec{J} = \vec{L} + \vec{S}\) and mentions that the rule pertains to the allowed values of the \(J^2\) quantum number.
  • A request for further explanation is made, indicating a desire for deeper understanding of the initial claims.
  • A participant expresses a lack of time to elaborate further but suggests that others may contribute and provides a link for additional reference.

Areas of Agreement / Disagreement

The discussion remains unresolved, with participants expressing differing levels of understanding and interest in the implications of the relationships between angular momentum vectors. No consensus is reached on the significance of the phase relationship.

Contextual Notes

Participants assume simple electronic configurations, which may limit the applicability of their claims. The discussion does not resolve the implications of the phase relationship or the conditions stated.

rushil
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Is there any special significance of the fact that [tex]\vec{L}[/tex] and [tex]\vec{S}[/tex] of an electron are in-phase or out-of-phase at all times?

i.e. is there any special physical significance of the fact that

(i) [tex]| \vec{J} | = | \vec{L} | + | \vec{S} |[/tex]
(ii) [tex]| \vec{J} | = | \vec{L} | - | \vec{S} |[/tex]
I am referring to magnitudes above!
Also, is there any significance of the fact that [tex]\vec{L}[/tex] and [tex]\vec{S}[/tex] are NOT in phase at any time?
We are obviously assuming simple electronic configurations! :-p :biggrin:
 
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The total angular momentum is always [tex]\vec{J} = \vec{L} + \vec{S}[/tex], the rule you refer to has to do with the allowed values of the [tex]J^2[/tex] quantum number.
 
Can you please explain a bit more!
 

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