Wave function when there is coupling between spin and position

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Discussion Overview

The discussion revolves around the form of the wave function in quantum mechanics when there is coupling between spin and position. Participants explore the implications of this coupling on the symmetry properties of the wave function components.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions why the general state must have a specific symmetric form in the presence of coupling, suggesting that the coefficients of the spinors should exhibit symmetry.
  • Another participant challenges the necessity of symmetry and antisymmetry in the wave function, indicating a need for more context and expressing uncertainty about the implications of changing the orientation of the z-axis.
  • A third participant proposes that the spatial wave function associated with the spin-up spinor does not need to be symmetric, reflecting on the potential misinterpretation of the notation used by Griffiths in the context of coupling.
  • A later reply acknowledges the clarification regarding the interpretation of the wave function components.

Areas of Agreement / Disagreement

Participants express differing views on the necessity of symmetry in the wave function components due to coupling, indicating that the discussion remains unresolved with multiple competing interpretations.

Contextual Notes

There is ambiguity regarding the definitions of symmetric and antisymmetric wave functions in the context of the specific coupling discussed, as well as the notation used in Griffiths' text, which may lead to confusion.

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Why can't the general state, in the presence of coupling, take the form $$\psi_-(r)\chi_++\psi_+(r)\chi_-$$ where ##\psi_+(r)## and ##\psi_-(r)## are respectively the symmetric and anti-symmetric part of the wave function, and ##\chi_+## and ##\chi_-## are respectively the spinors representing spin up and spin down? In other words, why must the "coefficient" of the spin-up spinor be symmetric, in the presence of coupling?

Reference: Intro to QM, David J Griffiths, p210
 
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You just changed the orientation of the z axis (by swapping what up and down is). Why would it be symmetric and antisymmetric? I don't know, needs more context, I don't have Griffiths around.
 
The part of the spatial wave function associated/coupled with ##\chi_+## does not need to be symmetric, right?

On second thought, I think ##\psi_+## does not represent a symmetric wave function, but it’s just to denote the part of the spatial wave function associated with ##\chi_+##. Throughout the book, Griffiths has been using ##\psi_+## to mean a symmetric wave function, but I think it does not mean that in this particular case, hence the confusion.
 
Last edited:
That makes more sense.
 

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