# Is the Schroedinger equation not fundamental in BCS theory?

1. Aug 6, 2008

### Maaneli

Can the Schroedinger equation be derived from first principles in BCS theory? Zapperz suggests the Madelung and Schroedinger equations are not fundamental in BCS theory. If such is the case, how exactly does such a derivation work?

2. Aug 6, 2008

### ZapperZ

Staff Emeritus
Er... re-read what had transpired. I never claim such a thing, but rather I asked you if this is the consequences of what you claimed. In other words, if you equate the Madelung equations as being on par with the Ginzburg-Landau or London equations in superconductivity (remember them?), then it is YOU who are then designating them as not being fundamental, not me.

How one could make a statement about "Medelung and Schrodinger equations .... in BCS theory" is a mystery to me. Whoever started with "schrodinger" equations superconductivity anyway?

Zz.

3. Aug 6, 2008

### Maaneli

.. and that is exactly my point. I can derive such "hydrodynamical" equation using CONVENTIONAL field-theoretic method. I then arrive at these "hydrodynamical equations", which you claim are "BQM equations". What have I proven? That BQM equations are NOT fundamental, per your admission that you never claim that such a thing can be derive from First Principles.

If you can derive such "hydrodynamical equation using CONVENTIONAL field-theoretic method", then you are also claiming you can derive the Schroedinger equation because those hydrodynamical equations are MATHEMATICALLY EQUIVALENT TO THE SCHROEDINGER EQUATION. DUH!

Whoever started with "schrodinger" equations superconductivity anyway?

If you really are a CM physicist, then this should be well-known and elementary to you that there is a Schroedinger equation description of a superconducting fluid. More precisely, in the dilute gas approximation, one can describe a Bose gas through a quantum field psi satisfying a nonlinear Schroedinger equation:

http://relativity.livingreviews.org/Articles/lrr-2005-12/

Then, from a mean-field approximation, you get the Gross-Pitaevski equations, which can be rewritten as the hydrodynamical Madelung equations, AKA the equations of "BQM".

If you say you don't know this or disagree with this, well, then I think you're outright lying. Feynman also discusses it in his lectures - are you saying Feynman was wrong or that you don't know basic superconductivity theory?

Last edited: Aug 6, 2008
4. Aug 6, 2008

### ZapperZ

Staff Emeritus
But that was based on YOUR response to what I asked you regarding those "hydrodynamical" equations. The London equations are such "hydrodynamical" equations in superconductivity, because that's all it describes, the dynamics of the system! That's why I brought it up! Such equations are NOT fundamental, and it is via the association you implied that I argued that if this what you are claiming, that you have implied that they are not fundamental!

Read the BCS paper. Where did you think the GROUND STATE BCS wavefunction came from? The "Schrodinger Equation"? When they used the variational method to get the $u_k$ and $v_k$, the wavefunction for the singlet state incorporating the Cooper pairing state have already been constructed.

You still haven't looked at Tinkham yet, have you?

And I keep asking you WHERE, in Tinkham or even the BCS paper, this was used. It seems as if you are arguing that there's a parallel theory of superconductivity separate from BCS!

I'm sorry, but this is getting awfully tiring and boring. Don't you have anything better to do?

Zz.

5. Aug 6, 2008

### Maaneli

The London equations are such "hydrodynamical" equations in superconductivity, because that's all it describes, the dynamics of the system! That's why I brought it up! Such equations are NOT fundamental, and it is via the association you implied that I argued that if this what you are claiming, that you have implied that they are not fundamental!

You must think that saying something a thousand times makes it true. The Madelung equations ARE NOT the London equations. They are a mathematically equivalent representation of the Gross-Pitaevski equations. You should already know that. Oh and BTW, I did look at Tinkham, and he doesn't say either that the London equations are the Gross-Pitaevski equations. So the real question is, have you looked at Tinkham yet?

And I keep asking you WHERE, in Tinkham or even the BCS paper, this was used. It seems as if you are arguing that there's a parallel theory of superconductivity separate from BCS!

Red herrings yet again. That's quite convenient to ignore any reputable reference I give you, just because it is not Tinkham (which I'm still not convinced you yourself have looked at). And even when I suggest exactly the chapter of Tinkham to look at, you still don't bother. Have you ever admitted to not knowing something or making a mistake in your life? It's OK, you can tell me.

Read the BCS paper. Where did you think the GROUND STATE BCS wavefunction came from? The "Schrodinger Equation"? When they used the variational method to get the $u_k$ and $v_k$, the wavefunction for the singlet state incorporating the Cooper pairing state have already been constructed.

Thanks! I'll look into it.

I'm sorry, but this is getting awfully tiring and boring. Don't you have anything better to do?

Oh absolutely! That's why I'm always talking to other more reasonable people on this forum. I just thought I'd give you a chance to redeem some of your credibility by confronting and resolving this issue. But I guess you're implying I should have known better, eh?

Last edited: Aug 6, 2008
6. Aug 6, 2008

### ZapperZ

Staff Emeritus
I never said they were! I said that the HYDRODYNAMICAL equation that you keep saying is equivalent to the London equations in superconductivity. It describes the DYNAMICS of the system. This is what I call to be not fundamental! Oy vey!

Er.. I used Tinkham as a textbook!

It is you who is spewing out the red herrings here, because you continue to not be able to point out exactly where the "Madelung equations" are used in the BCS theory. Remember, you continue to insist that they are responsible for condensed matter physics.

Now this makes no sense IF you have looked at Tinkham, which you claimed you have. Tinkham derived the BCS properties EXACTLY the way BCS did - he even said so on page 53 (Second Edition) when he started with the variational method. By then, the BCS N-particle ground state are already fully developed. And it certainly didn't come out of some "Schrodinger Equation"!

I have none to redeem. I'd rather stop associating with a thread that was going around in circles simply to entertain you.

Edit: As expected, this is getting tiring. You can go ahead and continue this yourself, but I'm bailing out because I can see the same thing happening. Not that I expected anything different.

Zz.

Last edited: Aug 6, 2008
7. Aug 6, 2008

### Maaneli

You can bail out if you want. But just to correct some distortions you made for everyone else:

I said that the HYDRODYNAMICAL equation that you keep saying is equivalent to the London equations in superconductivity.

I NEVER said the hydrodynamical equation is equivalent to the London equations in superconductivity. In fact I just said in my last post that they are not, despite YOUR claim the contrary. And for you to twist things around like that is alarming. Reminds me a little of Karl Rove!

It is you who is spewing out the red herrings here, because you continue to not be able to point out exactly where the "Madelung equations" are used in the BCS theory.

Er.. I was never supposed to show you where the Madelung equations are used in BCS theory. I asked you to justify your claim that you could derive these equations from BCS theory. So, again, you've twisted the issue around.

Now this makes no sense IF you have looked at Tinkham, which you claimed you have.

I looked at Tinkham's section on superconductivity and superfluidity to confirm the difference between Gross-Pitaevski and London equations. I did not look yet at the BCS section, because I wanted to ask YOU to explain exactly where BCS derives the Madelung and Gross-Pitaevski equations.

I have none to redeem.

You're right, you don't.

I'd rather stop associating with a thread that was going around in circles simply to entertain you.

This certainly was not to entertain; but because of your unwillingness to get the question and the facts straight, it showed myself and others how tricky and dodgy you can really be.

8. Aug 6, 2008

### ZapperZ

Staff Emeritus
1. You brought up the "hydrodynamical equations"

2. *I* said that "hydrodynamical equations" in superconductivity is something like the London equations. What else is there? You didn't say that. *I* did.

3. I then said that this hydrodynamical equations represented by London equations are not fundamental.

4. That is why I wanted you to explain where such a thing comes in, because it CAN'T be just "hydrodynamical", because these can easily be phenomenological. I asked this way in the beginning when I responded to your post. I asked this repeatedly. To say that something is "hydrodynamical" in superconductivity means it isn't fundamental, because it isn't the starting point!

I didn't! Read above! I was the one who wanted you to explain how it actually can derive superconductivity. I believe Reilly also asked you the same thing.

Notice that I was not the one who claim that all of condensed matter physics based based on such "hydrodynamical" equations. My first question to you want to seek such clarification. It never came, even till now. And it is strange how one can look at "section" on superconductivity in Tinkham and not look at BCS section. All you did was look at the phenomenological models then, which are not fundamental.

BCS doesn't derive Madelung equations. BCS doesn't even USE such a thing (did you find any reference to Madelung equation in Tinkham at all? What about in Mahan?). So as I asked in the very first question, where did this come in many-body physics and superconductivity? In other words, we are back to the very first square.

I'm done!

Zz.

Last edited: Aug 6, 2008
9. Aug 6, 2008

### Maaneli

I'm done!

That's fine. Then you won't mind if I give the last words.

2. *I* said that "hydrodynamical equations" in superconductivity is something like the London equations. What else is there? You didn't say that. *I* did.

Thank you, I'm happy you corrected yourself. But actually, you said the hydrodynamical equations "ARE" the London equations, not just that they are "something like" the London equations. I could easily quote you. You seemed to entirely miss the fact that I have ALWAYS been talking about the the Madelung equations, when I say "hydrodynamical". So HYDRODYNAMICAL = MADELUNG EQUATIONS, AND MADELUNG EQUATIONS = NONLINEAR SCHROEDINGER (GROSS-PITAEVSKI) EQUATIONS.

3. I then said that this hydrodynamical equations represented by London equations are not fundamental.

That is why I wanted you to explain where such a thing comes in, because it CAN'T be just "hydrodynamical", because these can easily be phenomenological. I asked this way in the beginning when I responded to your post. I asked this repeatedly.

But then I said I was NEVER claiming these Madelung equations were "fundamental" in the sense of BCS theory. And you then said you could derive these hydrodynamical equations from first principles, and therefore that if these are the equations of BQM, then the equations of BQM cannot be fundamental. So I then asked you to justify how you could derive the Madelung equations (and therefore the nonlinear Schroedinger equation they are equivalent to). Now maybe you were referrring to the London equations, but then that means you weren't listening to me carefully in the beginning, as I was ALWAYS talking about the Madelung equations as the hydrodynamical equations, and the equations of BQM.

I believe Reilly also asked you the same thing.

Actually he did not. He asked me 4 entirely different questions, which I addressed, and he eventually conceded in private.

Notice that I was not the one who claim that all of condensed matter physics based based on such "hydrodynamical" equations. My first question to you want to seek such clarification. It never came, even till now.

Actually that clarification did come. In case you forgot, let me quote myself from the electrons thread:

No, the way you quote me brought it out of context. It sounds like I'm saying all of condensed matter owes its success to the equations of BQM, which is not what I said. I said it owes much of its practical and methodological success to the equations of QM. But I should have been more specific, namely, the theory of superconductiivyt and superfluidity owes much of its practical success to the equations of BQM. And indeed that is definitely true as Feynman, Likharev, Visser et al., show.

All you did was look at the phenomenological models then, which are not fundamental.

Yes, because I wanted to hear from you first how the Madelung and Schroedinger equations, which are according to you phenomenological, could be derived from first principles.

BCS doesn't derive Madelung equations. BCS doesn't even USE such a thing (did you find any reference to Madelung equation in Tinkham at all? What about in Mahan?).

OK, if BCS doesn't derive or use the Madelung equations, then what you said earlier about the equations of BQM not being fundamental, is not true. I referenced in the electrons thread the section in Tinkham's book to look at. I was the superconductivity and superfluidity chapter (I forget which chapter number).

So as I asked in the very first question, where did this come in many-body physics and superconductivity? In other words, we are back to the very first square.

I'll quote Feynman once again:

"Schroedinger's equation for the electron pairs in a superconductor gives us the equations of motion of an electrically charged ideal fluid. Superconductivity is the same as the problem of the hydrodynamics of a charged liquid. If you want to solve any problem about superconductors you take these equations for the fluid adn combine them with the Maxwell equations to get the fields" p. 21-13/14.

This applies, by the way, for N particles.

Cheers,
Maaneli