Fermi's golden rule derivation a bit hazy

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Discussion Overview

The discussion revolves around the derivation of Fermi's golden rule, focusing on the mathematical steps involved and the potential errors in the application of hermiticity and adjoint operations within quantum mechanics. Participants are seeking clarification on specific transitions between states and the implications of their calculations.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant expresses confusion regarding the derivation, noting that their calculations yield ##\omega = E_m + E_n## instead of the expected ##E_m - E_n##.
  • Another participant questions whether the first participant might have miscalculated to arrive at ##\omega = -\left(E_m + E_n\right)##, suggesting a need to consider the properties of bra-ket notation and the conjugate linearity involved.
  • A participant suggests focusing on the term ##e^{i H_0 t}## and its transformation into ##e^{-i E_m t}## as a critical point in the derivation.
  • There is a discussion about the correctness of the adjoint operation applied to the state transformations, with one participant affirming that the steps presented are indeed correct.

Areas of Agreement / Disagreement

Participants express differing views on the correct interpretation of the mathematical steps in the derivation, indicating that multiple competing views remain. While some affirm the correctness of certain transformations, others are still uncertain about specific aspects of the derivation.

Contextual Notes

Limitations include potential misunderstandings of the hermiticity and adjoint operations, as well as unresolved steps in the derivation that may affect the conclusions drawn by participants.

Replusz
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I am stuck between line 1 and 2 in the derivation below. I have attached what I tried
1586805944414.png

1586806504002.png

1586806124910.png
My thought is following:
1586806435700.png

However this would give me w=E_m + E_n instead of E_m-E_n
I guess something relating to hermiticity or adjointing something has gnoe wrong in my version.
Can someone point me in the right direction please? :)
Thank you in advance!
 
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Replusz said:
However this would give me w=E_m + E_n instead of E_m-E_n

Are you sure you didn't get ##\omega = -\left(E_m + E_n \right)##? Then I would have replied that, in Dirac notation, the bra - ket is conjugate linear in the bra, i.e., if you pull a scalar outside of the bra, you have to take the complex conjugate of the scalar.

I do get the required result.
 
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Replusz said:
Summary:: I am stuck between line 1 and 2 in the derivation below. I have attached what I tried

View attachment 260581
View attachment 260585
View attachment 260582My thought is following:
View attachment 260584
However this would give me w=E_m + E_n instead of E_m-E_n
I guess something relating to hermiticity or adjointing something has gnoe wrong in my version.
Can someone point me in the right direction please? :)
Thank you in advance!
I agree with George. Focus on your ##e^{i H_0 t}## which you turned into ##e^{-i E_m t}##.
 
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nrqed said:
I agree with George. Focus on your ##e^{i H_0 t}## which you turned into ##e^{-i E_m t}##.

So ##<m|e^{i H_0 t} = adjoint(e^{-i H_0 t} |m> ) = adjoint(e^{-i E_m t} |m> ) = e^{ i E_m t} <m| ##

Is this correct?
(if this is correct then everything makes sense)
 
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Replusz said:
So ##<m|e^{i H_0 t} = adjoint(e^{-i H_0 t} |m> ) = adjoint(e^{-i E_m t} |m> ) = e^{ i E_m t} <m| ##

Is this correct?
(if this is correct then everything makes sense)
Yes, this is correct.
 
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