Dirac ket-bra simplification over here.

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Discussion Overview

The discussion revolves around the manipulation of Dirac notation, specifically the simplification of expressions involving bras and kets as presented in Section 4.4 of Sakurai's text. Participants are exploring the implications of orthogonality and duality in the context of double summations.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant expresses difficulty in manipulating bras and kets due to a double summation.
  • Another participant suggests using the orthogonality condition as a potential solution.
  • A different participant mentions the identity \( U^{\dagger}U = I \) as relevant to the discussion.
  • Concerns are raised about reaching a zero result, questioning the implications of the orthogonality condition.
  • One participant considers the concept of duality but finds it unhelpful in this context.
  • Another participant clarifies that the inner product \( \langle a''|a' \rangle \) equals 1 when \( a = a'' \) and 0 otherwise, indicating that the summation behaves accordingly.
  • A follow-up question prompts participants to consider if additional information is needed to complete the simplification from line two to line three in the attachment.

Areas of Agreement / Disagreement

Participants express differing views on the implications of the orthogonality condition and the handling of duality, indicating that the discussion remains unresolved with multiple competing perspectives.

Contextual Notes

There are limitations regarding the assumptions made about the orthogonality condition and the specific definitions of the states involved, which may affect the conclusions drawn from the mathematical manipulations.

M. next
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I usually know how to manipulate bras and kets. But I probably found difficulty because of the double summation. How did Sakurai manage to go from the second line to the ird line in the attachment?

SECTION 4.4 in Sakurai.
 

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Using the orthogonality condition
 
Last edited:
How come? If so I will end up with a zero, no?
 
I was thinking about taking duality but that didn't seem to work.
 
M. next said:
How come? If so I will end up with a zero, no?
No. What is [itex]\langle a''|a' \rangle[/itex] equal to and what does this imply for [itex]\Sigma_{a''} \langle a''|a' \rangle[/itex]?
 
It is equal to 1 if a=a'' and 0 otherwise if we take orthogonality into consideration.. It implies that the summation = 1 when a'' = a' and 0 for all the other summations. Right?
 
Yes. If you put Bill's comment and this together, do you still miss something to go from line two to line three?
 

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