Finding Expression for Emitted Field: A Confused Student's Journey

[MaestroBach]

Homework Statement

Find the frequency spectrum of the radiated power in spontaneous emission and use it to find the DOS.

Relevant Equations

Desired result below

All I'm reallly confused on this problem is what the expression for the emitted field is. As long as I've got that, I'm good to go, but I just don't know what to use. I've tried looking for an expression for the emitted field but I've had no luck. Would appreciate any ideas or someone telling me I'm missing something obvious.

(I'm told to find the Fourier transform of the field and go from there, which is why I'm trying to find the expression for the field)

 

[DrClaude]

You have to give some more context.

Are you starting from the Einstein coefficients?

 

[MaestroBach]

You have to give some more context.

Are you starting from the Einstein coefficients?

No, I'm being told to start by taking the Fourier transform of the emitted field, and as far as I'm understanding, I won't even be using the einstein coefficients.

I'm being told to calculate the Fourier transform of the emitted field to find the frequency spectrum of the radiated power, and then use that to find the DOS.

I just have absolutely no idea what the emitted field expression is, other than that it decays with the form $e^{-t/\tau}$ (Sorry I should have included that info in the original post, but now it seems like I can't even edit it).

 

[DrClaude]

No, I'm being told to start by taking the Fourier transform of the emitted field, and as far as I'm understanding, I won't even be using the einstein coefficients.

Then I don't know what $A_{21}$ is.

I'm being told to calculate the Fourier transform of the emitted field to find the frequency spectrum of the radiated power, and then use that to find the DOS.

I just have absolutely no idea what the emitted field expression is, other than that it decays with the form $e^{-t/\tau}$ (Sorry I should have included that info in the original post, but now it seems like I can't even edit it).

Do you have the full text of the question?

 

[MaestroBach]

Then I don't know what $A_{21}$ is.Do you have the full text of the question?

Yeah, it is:

where eq 14.68 is what I have in the original post.

 

[DrClaude]

So the problem tells you that the field is $\propto e^{-t/2 \tau}$, i.e., $f(t) = C e^{-t/2 \tau}$, with C a constant. That is your starting point.

 

[MaestroBach]

So the problem tells you that the field is $\propto e^{-t/2 \tau}$, i.e., $f(t) = C e^{-t/2 \tau}$, with C a constant. That is your starting point.

Fair enough. For some reason I thought there would be another term that also has some other kind of dependence, along with $f(t) = C e^{-t/2 \tau}$. How in the world do I go about finding C though?

 

[MaestroBach]

So I've gone and tried $f(t) = Ce^{\frac{-t}{2\tau}}$, but that does not give me the correct answer. It actually comes close, but I get $E^2$ instead of $(E - \hbar \omega_{21})^2$ like my answer is supposed to be (correct answer in OP), which makes me suspect my electric field expression is still incorrect. If anyone has any insight I'd appreciate it.