Gerry and Knight solution to single-mode Maxwell equation

[Kekeedme]

TL;DR

I have two questions concerning treatment of the single-mode solution to Maxwell's equation presented in Gerry and Knight's. Namely, where does the expression for the amplitude of the electric field comes from, and secondly how did they handle the integral over dV in the classical E-M Hamiltonian that contains the V^-1 term, without it blowing up.

In the book, they present the solution to the maxwell equation for a wave propagating in 1-D. My only issue is determining where the specific expression for the amplitude of the electric field comes from. Their solution is:

Even if we use the classical energy in the field and equate it to the single-photon energy (ignoring zero point) as such:

we'd get 𝜔 inside the parentheses not 𝜔^2? Can someone help shed some light please?


Secondly, when we go from:

to:

I understand what they are trying to get, the only thing that trips me up is I don't see how they handled the integral, seeing that you have to integrate over d𝑉/𝑉 because of the coefficient in front (mentioned above in the amplitude ). What am I missing please?

Thank you!

 

[Roberto Pavani]

I'll try to answer as best I can.

For the first part, the "missing ω" is effectively hidden inside the definition of q(t). In the quantized theory,
$q=\sqrt{\frac{\hbar}{2\omega}}(a+a^\dagger),$
so q already contains a factor $1/\sqrt{\omega}$.

For the second part, it seems to me that the notation is somewhat unfortunate because the same symbol V is used both for the total volume and in $dV$ for the volume element. Then the factor $1/V$ is not problematic because V is the total cavity (quantization) volume, i.e. a constant, not the integration variable. More explicitly, one could write the quantization volume as $\mathcal V$, so that
$\int_{\mathcal V}\frac{dV}{\mathcal V} = \frac{1}{\mathcal V}\int_{\mathcal V} dV = 1.$
Also, if I'm not mistaken, for the cavity mode
$\int_V \sin^2(kz)\,dV = \frac{V}{2},$
which exactly cancels the explicit $1/V$ factor in the field normalization.

 

[Kekeedme]

Thank you so much for the reply.
So at first, does it mean that we should indeed get this expression by ignoring zero-point energy and equation the classical field energy to that of the single-photon?

For the V portion, that is very helpful, and also very unfortunate that they have this confusion in there. But I think I see what you mean. Thank you so much for this! Sorry for the late reply

 

[bhobba]

For those interested in the gory details of quantising the EM field, see the paper I often reference:
https://digitalcommons.usu.edu/cgi/viewcontent.cgi?article=3211&context=physics_facpub

Thanks
Bill

 

[Kekeedme]

This seems amazing! Thank you so much for this resource

 

[Kekeedme]

For those interested in the gory details of quantising the EM field, see the paper I often reference:
https://digitalcommons.usu.edu/cgi/viewcontent.cgi?article=3211&context=physics_facpub

Thanks
Bill

Hi Bill, I have been working on this paper for a while, and it is really helpful! Thank you!
I get stuck on deriving the momentum field operator (equation 5.88 on page 73). I was able to get exactly what he got for the energy operator, but for the momentum part I tried a lot of different things and I get stuck. I have tried using convolution in the fourier space, but maybe I am doing it wrong. I have tried distributing my cross products and using BAC-CAB but no dice, among other things. Can you please help me with the details? I have hard time moving on when I cannot convince myself I understand where one gets certain expressions. Thank you!

 

[bhobba]

After thinking about this for a while and considering that I am in respite care ATM, I think it's better to post this in the Advanced Homework Help.

Thanks
Bill

 

[Matterwave]

I have tried using convolution in the fourier space, but maybe I am doing it wrong.

You don't need to do anything fancy, it's basically just a bunch of plug and chug and being very careful with every term.

Take $\vec{E},\vec{B},\vec{\mathcal{A}}$ from 5.74, 5.75, 5.78 and plug and chug. The integral over $d^3x$ combined with the $e^{i\vec{k}\cdot\vec{x}}$ terms in $\vec{E},\vec{B}$ should get you a delta function (make sure to integrate over two different dummy variables) collapsing you to that single $d^3k$ integral. Then use some symmetry properties to get rid of some terms and BAC-CAB and 5.79 to simplify the triple cross products that remain.

It's a lot of algebra admittedly, but nothing more than plug and chug (and a single delta function).

 

[Kekeedme]

After thinking about this for a while and considering that I am in respite care ATM, I think it's better to post this in the Advanced Homework Help.

Thanks
Bill

I am sorry to hear that Bill. But thank you again for all your suggestions.

 

[Kekeedme]

You don't need to do anything fancy, it's basically just a bunch of plug and chug and being very careful with every term.

Take $\vec{E},\vec{B},\vec{\mathcal{A}}$ from 5.74, 5.75, 5.78 and plug and chug. The integral over $d^3x$ combined with the $e^{i\vec{k}\cdot\vec{x}}$ terms in $\vec{E},\vec{B}$ should get you a delta function (make sure to integrate over two different dummy variables) collapsing you to that single $d^3k$ integral. Then use some symmetry properties to get rid of some terms and BAC-CAB and 5.79 to simplify the triple cross products that remain.

It's a lot of algebra admittedly, but nothing more than plug and chug (and a single delta function).

Hi Matterwave,
thank you for your reply. This is how I did it at first, plug and chug, used $dˆ3k$ variable for say $\vec{E}$ and $dˆ3l$ variable for $\vec{B}$ and then got an $e^{i\vec{k}\cdot\vec{x}}$ term and also an $e^{i\vec{k}\cdot\vec{l}}$ term, which gives me a delta function over the $dˆ3x$ integral when $l=-k$ and then I took each term did the BAC-CAB with that $-k$ but I get $e^{=\pm 2i\omega t}$ terms I can't get rid of. I am trying to see where I might have gone wrong. What you are saying helps me because it shows my initial attempts were on the right track.

 

[Matterwave]

but I get $e^{=\pm 2i\omega t}$ terms I can't get rid of. I am trying to see where I might have gone wrong.

Check the symmetry properties of those terms relative to your domain of integration.

 

[Kekeedme]

Check the symmetry properties of those terms relative to your domain of integration.

hmm, I will go back and check. I am not sure which symmetry properties are helpful here, but I might be overlooking.

 

[Kekeedme]

Hi @Matterwave
I am not making any real headway, would you mind taking a look please? I attached the part I am working with now after using the delta function. I have tried to use the fact that the $\vec{A}(\vec{k},t)^*= \vec{A}(\vec{-k},t)$ and see that since I am doing integrals over positive and negative values of k I could get rid of some terms, but I am not sure how to do it properly. Can you please help?
Thank you!