Spin vector and operator transformations query

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SUMMARY

The discussion centers on the confusion surrounding spin transformations in quantum mechanics, specifically using Sakurai's method for changing basis vectors. The user attempts to apply the transformation Sz' = Udagger Sz U but encounters discrepancies, yielding Sz' = Sx instead of the expected Sz' = Sy. The user later finds that using Sz' = U Sz Udagger produces the correct result, indicating a misunderstanding of the operator transformation process. This highlights the importance of correctly applying the transformation rules in quantum mechanics.

PREREQUISITES
  • Understanding of quantum mechanics principles, particularly spin operators.
  • Familiarity with linear algebra concepts, especially basis transformations.
  • Knowledge of operator algebra in quantum mechanics.
  • Experience with Sakurai's "Modern Quantum Mechanics" methodology.
NEXT STEPS
  • Study the derivation of spin operator transformations in quantum mechanics.
  • Learn about the implications of unitary transformations on quantum states.
  • Review the mathematical framework of basis changes in quantum systems.
  • Explore common pitfalls in operator transformations and their resolutions.
USEFUL FOR

Students of quantum mechanics, physicists working with spin systems, and anyone seeking to deepen their understanding of operator transformations in quantum theory.

deneve
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I am struggling to understand spin transformations and have used Sakurai's method of
|new basis> = U |old basis> to change basis vectors and hence should have

Sz' = Udagger Sz U

to transform the operator. I thought this should give Sz' = Sy in the workings (see attachment below) but it gives Sx instead. If I use
Sz' = U Sz Udagger then I do get Sz'=Sy but according to the texts I have, the first arrangement is the correct form. So I'm really confused. Can anyone see what I'm doing wrong here.


2. Homework Equations

I would be really grateful for any help with this. Thank you
 

Attachments

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. 3. The Attempt at a Solution (The attachments are not letting me paste the math here so I have posted the link to my calculation instead). My Calculation: https://ibb.co/n6nC9x
 

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