What is the speed of light doing in the spin half hamiltonian?

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Discussion Overview

The discussion revolves around the presence of the speed of light (c) in the spin half Hamiltonian as presented in Sakurai's 'Modern Quantum Mechanics'. Participants explore the implications of this term within the context of quantum mechanics, particularly in relation to classical energy expressions for magnetic moments and the role of the Dirac equation.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions the inclusion of c in the Hamiltonian, noting the classical expression for energy of a magnetic moment does not include it.
  • Another participant suggests that the term is related to spin-orbit coupling, which is a relativistic effect, and connects it to the Dirac equation.
  • A different participant challenges the assertion that the term is related to spin-orbit coupling by asking about the absence of angular momentum (L) in the discussion.
  • One participant mentions that the magnetic moment of the electron, as derived from the Dirac equation, includes c in its formulation.
  • A later reply indicates a reconsideration of the previous comments, suggesting that the presence of c may ultimately be a matter of convention, similar to the differences between CGS and MKS systems in electromagnetism.

Areas of Agreement / Disagreement

Participants express differing views on the significance and necessity of the speed of light in the Hamiltonian, indicating that multiple competing interpretations exist regarding its role and implications.

Contextual Notes

There are unresolved questions regarding the relationship between the spin half Hamiltonian and classical formulations, as well as the implications of relativistic effects and conventions in physical equations.

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I'm currently reading Sakurai's 'Modern Quantum Mechanics' (Revised Edition) and at page 76 he introduces a spin half hamiltonian

[tex]H = - (\frac{e}{mc}) \vec S \cdot \vec B.[/tex]

But what is c doing in this hamiltonian? Clasically the energy of a magnetic moment in a magnetic field is

[tex]E = - \vec \mu \cdot \vec B = - \frac{e}{2m} \vec S \cdot \vec B[/tex]

and as far as i know what one does to go to QM is to introduce the g-factor. What am I missing here?
 
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Standard question, standard answer: the spin-orbit coupling is a specially relativistic effect, even though the spin notion itself can be derived without special relativity. The <c> comes from the Dirac equation.
 
Say what? If this term is spin-orbit coupling, where is L??

The c comes about because according to Dirac the magnetic moment μ of the electron is -eħ/2mc, or -e/mc times the spin ħ/2.
 
ah, thanks. I guess I have to be reading up on that then.
 
Sorry, not paying attention. Disregard my post above.

If the <c> is there or not must be in the end a matter of convention, just like the as the case of CGS vs MKS in electromagnetism.
 
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