How is equation 4.61 derived from n dot s in the Arbitrary Spin Operator?

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Discussion Overview

The discussion centers around the derivation of the matrix definition for the spin operator \( S_n \) as presented in equation 4.61 from the expression \( n \cdot s \). Participants are seeking clarification on the steps involved in this derivation, particularly in relation to the Pauli matrices and their application in quantum mechanics.

Discussion Character

  • Technical explanation, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant shares a link to a document that contains the relevant equation but expresses uncertainty about the derivation process from \( n \cdot s \) to the matrix form of \( S_n \).
  • Another participant provides the definition of the radial versor \( n \) and mentions the use of Pauli matrices, suggesting that the scalar product should be taken into account.
  • A third participant questions the notation used, suggesting that the lowercase \( s \) should be an uppercase \( S \), referencing an earlier equation that defines the spin operator.
  • A later reply acknowledges a misunderstanding regarding the tables and matrices involved, indicating a realization of a previous error.

Areas of Agreement / Disagreement

Participants express varying levels of understanding regarding the notation and derivation process. There is no clear consensus on the steps involved in the derivation, and some confusion remains about the definitions and components of the spin operator.

Contextual Notes

There are references to specific equations and tables, but the discussion does not clarify the assumptions or definitions that might be necessary for a complete understanding of the derivation process.

nateHI
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http://www.tampa.phys.ucl.ac.uk/~tania/QM4226/SEC4B.pdf

At the above link, I'm not quite sure how the instructor got to the matrix definition for Sn(equation 4.61 on page 4) from n dot s. Does someone know of a link that doesn't skip that step?
 
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nateHI said:
At the above link, I'm not quite sure how the instructor got to the matrix definition for Sn(equation 4.61 on page 4) from n dot s. Does someone know of a link that doesn't skip that step?

Well, n = (sin(theta) cos(phi), sin(theta) sin(phi), cos(theta)) is the radial versor, and the Pauli matrices (sigmax, sigmay, sigmaz) are the usual ones... take the scalar product... what don't you understand exactly?
 
I'm not sure what the lower case s is, I think it should be upper case, i.e. the operator defined in eqn 4.52. The components of S are the pauli matrices times hbar/2 as in 4.55.
 
OK I get it. It was a silly mistake. I was getting table 4.60 mixed up with the pauli matrices. Thanks.
 

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