Fermi's golden rule derivation a bit hazy

[Replusz]

TL;DR

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

My thought is following:

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!

 

[George Jones]

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.

 

[nrqed]

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 $e^{-i E_m t}$.

 

[Replusz]

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)

 

[nrqed]

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.