Where Can I Find Resources on Bogoliubov Transformations for Quantum Mechanics?

[hidetsugu]

hello

Let me start by saying that is my 1st time posting on the forum an so I'm not sure if I should post this here or on the homework/coursework section. This is technically coursework related but doesn't seem to fit the "model" used in that section (it is not specific enough), so if this is the wrong place to post this, I do apologise.

So, in my quantum mechanics class I have been given the option to write a small paper in the upcoming weeks about bogoliubov transformations for extra credit. This is not part of the class curriculum (hence being extra credit) and I was hoping someone could point me in the right direction regarding useful literature keeping in mind my relatively low understanding of the subject (that would mean: relatively new familiarity with bra-ket notation and creation/destruction operators).

As I started my research I currently have Fetter & Walecka's Quantum theory of many-particle systems in my lap (bedside reading) and Kittel's Quantum theory of solids (that I haven't really looked at yet).

Any books or revisions papers that you might suggest on the topic would be very welcome.

thanks in advance for any help you may provide

 

[vanhees71]

Wikipedia is a good starting point, and Fetter-Walecka is a great standard reference for many-body theory.

http://en.wikipedia.org/wiki/Bogoliubov_transformation

 

[strangerep]

So, in my quantum mechanics class I have been given the option to write a small paper in the upcoming weeks about bogoliubov transformations for extra credit.

Try:

Greiner, "Quantum Mechanics -- Special Chapters",
Springer, ISBN 3-540-60073-6

He talks about why B-transforms are useful in Bose-Einstein condensates, and also in superfluidity, iirc.
Greiner's textbooks are good basic introductions because he doesn't skip steps in calculations.

On a (much) more difficult level, B-transforms enter the picture in advanced QFT. But you said "QM class", so maybe you haven't studied QFT yet? (I could give further QFT-relevant references, but there's no point if you're not at that stage yet.)

 

[hidetsugu]

Wikipedia is a good starting point, and Fetter-Walecka is a great standard reference for many-body theory.

http://en.wikipedia.org/wiki/Bogoliubov_transformation

Hum... would have never though of that...

Try:

Greiner, "Quantum Mechanics -- Special Chapters",
Springer, ISBN 3-540-60073-6

He talks about why B-transforms are useful in Bose-Einstein condensates, and also in superfluidity, iirc.
Greiner's textbooks are good basic introductions because he doesn't skip steps in calculations.

On a (much) more difficult level, B-transforms enter the picture in advanced QFT. But you said "QM class", so maybe you haven't studied QFT yet? (I could give further QFT-relevant references, but there's no point if you're not at that stage yet.)

Thank you, I'll take a look.

And no I have not studied QFT... yet. up until last week I haven't even heard of bogoliubov's transformations. I am definably a little out of depth... but I think that's the point of this assignment .

Since you mentioned "B-transforms" I would also ask about common notation. In most sources I found so far the creation/destruction operators are defined as $a^{\dagger},a$ and we define the bogoliubov operators as

$b=ua+va^{\dagger}$
$b^{\dagger}=u^{\dagger}a^{\dagger}+v^Ta$

At fetter's however, right at the start the n particles Hamiltonian is defined using $b_k^{\dagger},b_{k'}$ which obey

$[b_k,b_{k'}]=[b^{\dagger},b^{\dagger}]=0$
$[b_k,b_k^{\dagger}]=δ_{kk'}$

Am I correct to assume that these "b's" are the usual "a's"? or do the bogoliubov's operators follow the same rule? bogoliobovs transforms are not mentioned until a later chapter (where diferent notation is used) so I'm not sure if if they where used ad hoc to define the Hamiltonian at start or if this is just a notation option

 

[strangerep]

Since you mentioned "B-transforms" I would also ask about common notation. In most sources I found so far the creation/destruction operators are defined as $a^{\dagger},a$ and we define the bogoliubov operators as

$b=ua+va^{\dagger}$
$b^{\dagger}=u^{\dagger}a^{\dagger}+v^Ta$

At fetter's however, right at the start the n particles Hamiltonian is defined using $b_k^{\dagger},b_{k'}$ which obey

$[b_k,b_{k'}]=[b^{\dagger},b^{\dagger}]=0$
$[b_k,b_k^{\dagger}]=δ_{kk'}$

Am I correct to assume that these "b's" are the usual "a's"? or do the bogoliubov's operators follow the same rule? bogoliobovs transforms are not mentioned until a later chapter (where diferent notation is used) so I'm not sure if if they where used ad hoc to define the Hamiltonian at start or if this is just a notation option

I'm not very familiar with Fetter -- I only took a quick look at it on Amazon when Hendrik (vanhees71) recommended it. I've always found his recommendations helpful.

But it's the commutation relations themselves that matter. B-transforms are just a case of "canonical transformation" (check a classical mechanics book about Hamiltonian dynamics if you haven't heard that term). I.e., they preserve the important algebraic structure. The only difference is which vacuum state the annihilation operators... annihilate. In general, after a B-transform, although the commutation relations have been preserved, the new Hilbert space is inequivalent to the original one (at least in the case of QFT with infinite degrees of freedom).

So it doesn't really matter whether you use "a" or "b". Indeed, other authors might express the transformation as something like:
$$
\widetilde{a} = ua+va^{\dagger} ~,~~~~~~ [ \cdots ] ~.
$$
Rather, the important thing is the vacuum (state of lowest energy) associated with the operators you're using.

HTH.

 

[hidetsugu]

well, somewhere from the initial chapters of the book until the canonical transformations section (300 pages in) seems to be a notation change, so I guess I'll have to go figure out where and why that is. That is the price of trying to leap ahead ;)

edit: oh, one last question: how are b-transforms relevant hawking radiation? I mean, I obviously have a very limited understanding on how hawking radiation works but it is an area that I'm interested in so I was curious (I've done some re...eeerrr I served as "code monkey" to someone doing research in the area).

 

[strangerep]

how are b-transforms relevant hawking radiation?

Oh, geez.

Well,... ordinary QFT is formulated on a background of flat spacetime. The free fields are (Fourier-)decomposed into "modes", and these modes are used in constructing a basis for the Hilbert space.

But in curved spacetime, it turns out that any given observer can only construct such a basis locally. In general, these bases of modes which seem natural to different observers are in fact unitarily inequivalent. I.e., different observers do not agree in general on what the vacuum state is. B-transforms map between these representations.

A similar thing happens with the Unruh effect, btw, if you've heard of that.

I'd point you to the textbook of Birrel & Davies, but since you haven't studied ordinary QFT, there's not much value in that, I guess.

 

[hidetsugu]

Oh, geez.

thank you