A Quantum Field Theory for the Gifted Amateur: Fourier Transforms & Excitations

[anklimekruk]

Dear Sir,
P 25 in quantum field theory for the gifted amateur One makes Fourier transforms from the position to the frequency space for the system of linear chain of N atoms.
How can I see that in the frequency space the excitations are uncoupled .

I also don’t understand equation 2.50

 

[ddd123]

Dear Sir,
P 25 in quantum field theory for the gifted amateur One makes Fourier transforms from the position to the frequency space for the system of linear chain of N atoms.
How can I see that in the frequency space the excitations are uncoupled .

Write the equations of motion for the Hamiltonians 2.45 and 2.59.

I also don’t understand equation 2.50

If m=0 you are simply counting all the j's, which means it equals N. If m ≠ 0 summing all terms they cancel out symmetrically.

 

[anklimekruk]

1)I am sorry but I don't understand why the comparaison of 2.45 with 2.59 means uncoupled frequenties.
The book says to compare 2.66 with 2.41 where it becomes clear that the HO's are uncoupled.
My question nevertheless is : can you see that before making all this calculations?
I have the principle: first understand the concept then prove it mathematicaly.
2)I understand when k = 0 i am counting N .I thought if m is not 0 the kronicker delta gave 0.I don't see the symmetric canceling out.

 

[ddd123]

1)I am sorry but I don't understand why the comparaison of 2.45 with 2.59 means uncoupled frequenties.

Write the equations of motion which you've learned from Hamiltonian mechanics. 2.45 is coupled, 2.59 is decoupled. It means that the resulting differential equations are dependent on each other for 2.45 and independent for 2.59.

My question nevertheless is : can you see that before making all this calculations?

You don't have to solve the equations of motion. Just by writing them you see that for the coupled system they form a system of differential equations that is difficult to solve. Of course, if you've clever enough you don't even need to write the equations of motion but just look at the Hamiltonians, but clearly you're not seeing this so you have to write the equations of motion.

2)I understand when k = 0 i am counting N .I thought if m is not 0 the kronicker delta gave 0.I don't see the symmetric canceling out.

Edit: never mind on that, I'm missing something too.

 

[anklimekruk]

I am going to do what you suggest.It will take some time,but i hope to understand.
Thank's a lot.

 

[anklimekruk]

Q 2) became clear.
Q1) I have studied both volumes of Susskind's the theoretical minimum. I hope to find there how to write the equations of motion for both H's and see that the dif eq's are coupled for 2.45 and uncoupled for 2.59.

 

[ddd123]

Q1) I have studied both volumes of Susskind's the theoretical minimum. I hope to find there how to write the equations of motion for both H's and see that the dif eq's are coupled for 2.45 and uncoupled for 2.59.

Hamilton's equations are

$\dot{x_i} = \frac{\partial H}{\partial p_i}$
$\dot{p_i} = -\frac{\partial H}{\partial x_i}$

 

[anklimekruk]

https://www.physicsforums.com/cid:0BBDC2FB-5CF0-954B-85E1-F2214146A7C5.png
https://www.physicsforums.com/cid:6FE7B597-2587-FD41-89BF-3670D418254F.png
https://www.physicsforums.com/cid:0D0F0552-CB7C-4245-80C5-E5C7467E7251.png

 

[anklimekruk]

I have tried to take the partial derivatives of both hamiltonian's with respect to p and x according to hamilton's equations but failed.
I don't understand how to do this particular problem.I think it is because of the indices.
Something else: I notice that in 2.45 & 2.59 we have operators.Not in hamilton's equations.
I very much appreciate your help.

 

[ddd123]

I can't see the images.

I have tried to take the partial derivatives of both hamiltonian's with respect to p and x according to hamilton's equations but failed.
I don't understand how to do this particular problem.I think it is because of the indices.

The main point is that, in one case, to find the solution at i you need the solution at i-1, so you have all equations dependent one another. In the other case, you can have independent equations to be solved separately, because the variables have been decoupled.

I suggest you google linear coupled oscillators and find a treatment that suits you.

Something else: I notice that in 2.45 & 2.59 we have operators.Not in hamilton's equations.

That only complicates the maths. That same decoupling argument works in classical mechanics so you can trust that.

 

[anklimekruk]

I understand your explanation of the dependence of i on i-1
In Gilbert Strangs Linear Algebra p 319 one can find how to take derivatives of operators in the exponential
I go to google following your advice
Thanks a lot !

 

[anklimekruk]

In QFT for the gifted amateur P 26 eq 2.58 is treated similarly to 2.56
I suppose i should substitute p in 2.56 by {(x_j+1) - (x_j)} and get the RHS of the first line of 2.58.
But {(x_j+1) - (x_j)} contains two terms one with index j+1 and one with index j and so i don't understand how to do it

 

[ddd123]

In QFT for the gifted amateur P 26 eq 2.58 is treated similarly to 2.56
I suppose i should substitute p in 2.56 by {(x_j+1) - (x_j)} and get the RHS of the first line of 2.58.
But {(x_j+1) - (x_j)} contains two terms one with index j+1 and one with index j and so i don't understand how to do it

Similarly means using 2.46 instead of 2.47.

 

[anklimekruk]

In my calculation I will encounter a summation over the term 2(X_j+1)(X_j).
I think it is the same as 2 (X_j)^2.
Than I can make the calculation
Is it correct?

 

[ddd123]

In my calculation I will encounter a summation over the term 2(X_j+1)(X_j).
I think it is the same as 2 (X_j)^2.
Than I can make the calculation
Is it correct?

No.

$\sum_k \tilde{x}_k e^{i k (j+1) a} - \sum_m \tilde{x}_m e^{i m j a} = \sum_k \tilde{x}_k e^{i k j a} ( e^{i k a} - 1)$

 

[anklimekruk]

I think it's clear
Thank you !

 

[anklimekruk]

https://www.physicsforums.com/cid:17989CD4-99C1-5842-ADFA-0587E697EEF5.png
I don't understand why from the definitions we automatically have... (P26 QFT for the gifted amateur)

 

[anklimekruk]

Dear Sir,
I hope you understand what is in the in my last mail.
Many mathematical signs are missing on my keybord.
I tried to use the app OneNote where i can use an electronic pencil and then copy and past it on my mails to You but it failed.
Is there an other option?
Thank you

 

[vanhees71]

Yes, in the forum you can use LaTeX, which you have to learn anyway to be able to type readable papers :-). You find an intro to how to use it in the forums here:

https://www.physicsforums.com/help/latexhelp/

 

[anklimekruk]

Thanks a lot I am going to do it later or tomorrow
Was it impossible to understand my question? The book uses the argument of hermitian matrices to assert that p_k is = to p_k_
I don't understand it

 

[anklimekruk]

Page 26 it says "with the definition we automaticaly have $p\dagger_k=p_\-k$ " I do see a minus sign in the exponential of $p\dagger$ because of the Fourier transform.
Is this the reason for the - sign?
But then it says:because $p\dagger$ and p are hermitian which means that if I take the complex conjugate and transpose I get the same matrix.
Complex conjugation does not yield a - sign so I don't understand it.
I must notice that this text is confusing!

 

[vanhees71]

It is really impossible to answer your questions without enough context. I don't own the book. So I can't help.

 

[ddd123]

I must notice that this text is confusing!

It's not confusing... You just have to calmly sit down and do the algebra. You seem like in a rush (or, no offense, lazy to do the algebra). The definition it talks about is of course 2.49:

$\tilde{p}^\dagger_k=\frac {1} {\sqrt{N}}\sum_j p^\dagger_j e^{i k j a}=\frac {1} {\sqrt{N}}\sum_j p_j e^{i k j a}=\tilde{p}_{-k}$

because $\hat{p}_j$ is hermitian.

 

[anklimekruk]

It is really impossible to answer your questions without enough context. I don't own the book. So I can't help.

It will be my pleasure to offer you the book as a present.
I guess you just have to give me the adress where i can tell amazone to send it

 

[anklimekruk]

I want to ask if i understand it well:
In the first equality the - sign becomes + because it is $p\tilde\dagger$. In the second equality $p\dagger$ becomes $p$ because $p$ is hermitian and the third equality is definition again.
Is it correct?

 

[ddd123]

Yes.

 

[anklimekruk]

Thanks a lot

 

[anklimekruk]

In QFT for the gifted amateur p 26
I square 2.62 and 2.63 then i plug these values into 2.59 to get 2.64
Is it correct?

 

[ddd123]

No. Your questions, except for the first that was legitimate, are all solved by trivial plugging in. I don't know why you keep getting stuck at that level. It's going to get a lot harder than this, you know.

 

[anklimekruk]

Indeed I see I can trivialy plug in.But then I have a product of $\rm{p_k}$ with $\rm{p_k}$ .
Thats why I said "I square 2.63 and plug it in 2.59".And the same for 2.62.
In my humble opinion this yields the same result.
As for the fact that it is going to get much harder,I don't intend to give up.
I am passionated by the subject and will do my best to understand.I have allready overcome some difficult points with your help, and i am gratefull to you.

 

[anklimekruk]

I want to say that i suppose i am wrong because you don't agree and would very much like to understand why?

 

[anklimekruk]

In eq's 3.17 and 3.18 p 33 QFT for the gifted amateur
Are the constants of proportionality $\sqrt (n_i+1) and \sqrt n_i$ explained somewhere in the book?

 

[ddd123]

I want to say that i suppose i am wrong because you don't agree and would very much like to understand why?

If I understand what you meant, then it's the same I guess, but you should only ask if you're stuck somewhere. These problems are easy to figure out, we aren't being paid as tutors so you should only ask a question when you've tried everything first.

In eq's 3.17 and 3.18 p 33 QFT for the gifted amateur
Are the constants of proportionality $\sqrt (n_i+1) and \sqrt n_i$ explained somewhere in the book?

Page 22, bottom.

 

[anklimekruk]

Thanks!

If I understand what you meant, then it's the same I guess, but you should only ask if you're stuck somewhere. These problems are easy to figure out, we aren't being paid as tutors so you should only ask a question when you've tried everything first.
Page 22, bottom.

thanks!

 

[anklimekruk]

I am really doing my best to understand before I ask a question!
May I ask if it is necessary to mention each time "QFT for the gifted amateur"
Thanks again for helping!

 

[ddd123]

I am really doing my best to understand before I ask a question!

Then work on why you couldn't figure it out even if it was your best. You may need a change in method rather than effort. But anyway, you read less than 25 pages and you already missed the normalization of creation and annihilation operators which was clearly stated. It's not like you lost a minor detail after 250 pages of study...

As I said, your FIRST post asked legitimate questions, all the others were really trivial and they seem to be getting more and more trivial. I kept answering because your first questions were good questions, but it has to stop at some point. If your next question will have another obvious answer you'll have to hope someone else comes along, I can't walk you through a whole book, I have things to do...

 

[anklimekruk]

I am just a fascinated selfstudent.
I understand your time is precious and I fully respect it!
It's also a chalenge to understand things by myself.
These are two excellent reasons to do my very best to understand before asking a question.
Anyhow i am gratefull to you for all your answers.They have been helpful and instructive as well.
Thanks to you i reached p 33.
It's clear you expect a higher level but I hope you can trust me to improve with time.
If not it will be like failing the admission exams.What can i do? That's life...

 

[ddd123]

It's clear you expect a higher level but I hope you can trust me to improve with time.
If not it will be like failing the admission exams.What can i do? That's life...

No, you have it all wrong, I don't expect anything from you because I'm not a university. Posting a conceptual question is accepted in the quantum section of physics forums, but if you're going to keep on like this you should be posting here instead: https://www.physicsforums.com/forums/advanced-physics-homework.154/ . Now the misunderstanding is clear so this is my last post in this thread, the homework section is where you should post.

 

[anklimekruk]

No, you have it all wrong, I don't expect anything from you because I'm not a university. Posting a conceptual question is accepted in the quantum section of physics forums, but if you're going to keep on like this you should be posting here instead: https://www.physicsforums.com/forums/advanced-physics-homework.154/ . Now the misunderstanding is clear so this is my last post in this thread, the homework section is where you should post.

I thank you for guiding me further to "homework PF"
By the way i am p 35.Thanks to your legitimate critics i made an extra effort and understood by my self the mathematics of the previous pages.
I found it relatively easy.So i am surely not as you say "going to keep on like this." I hope to satify the requirements necessary for getting the help i will need.Maybe some day i will be able to help others on my turn.