Choosing topic for reserach and presentation

[spaghetti3451]

Hi,

I am a final year undergraduate student having to do a presentation for a course in 'Advanced Quantum Mechanics.'

Our lecturer offered the following topics. I have to pick one of these.

1. Born-Oppenheimer Approximation → Berry Phase
2. Non-Hermitian Quantum Mechanics
3. Path Integral Treatment of SHO
4. Scaling Symmetry in Quantum Mechanics (δ function function potential in 2D) and 'renormalisation'
5. Quantum Mechanics of charge-monopole system
6. Optical Bloch equations and quantum 'qubits'

I think option 1 is from Atomic and Molecular Physics, option 6 is from Quantum Computing and options 2, 3, 4 and 5 belong to theoretical physics. Am I correct?

Which ones do you think are challenging topics?

 

[Demystifier]

I would choose 2 or 4, but that's me.

 

[vanhees71]

What's "non-Hermitian quantum mechanics"? I'd choose topic 5. Then you have the pleasure to read Dirac's papers on magnetic monopoles, which is really just fun (I've chosen that topic for my theory-seminar presentation when I was a student).

 

[JorisL]

They all look like fun to me.
Subject 3 can be really helpful for other courses, notably field theory. It also is rather interesting (in my opinion) to see how at first sight very different formalisms are equivalent (might be a tricky statement don't shoot me).

For 1 and 3 you should certainly check out Weinbergs recent book "Lectures on Quantum Mechanics". It is nicely written although I do not agree with his notation to be clearer than Dirac notation. I only glanced over those sections but they seemed understandable, even while glancing/scanning them.

 

[dextercioby]

2. is an exotic topic which has received lots of attention lately, but I can't tell how useful will be for you, if you choose to continue your studies in theoretical physics. 3 is ok, but you'll get to path integrals in QFT, I'm sure. I would choose 4.

 

[spaghetti3451]

To be honest, I have no idea of the merits or demerits of each. So, I'll go for option 4: Scaling Symmetry in Quantum Mechanics (δ function function potential in 2D) and 'renormalisation', because the majority voted for it. :-)

I hope I've made a good choice.

It would be nice if I could know how to start on how my reseach on this topic.

 

[spaghetti3451]

To add up on my last post, I've already done some preliminary digging, and I've found the following.

1. For some odd reason, physicists are interested in scale symmetry of quantum-mechanical systems.

2. There are scale-symmetric potentials such as the inverse square force and the Dirac delta function that physicists have analysed in the context of quantum mechanics, although I have no idea why these potentials are scale-symmetric and how exactly they have been analysed.

3. In the early universe, i.e. at high temperatures and density, scale symmetry prevailed and there was a breaking of the symmetry. I have no idea what this statement means, but that's what I've learned.

4. Apparently, renormalisation is a temporary solution to the problem.

It would be nice if you could offer your comments on these points.

 

[atyy]

The key physical idea is in renormalization is that if we only look coarsely, we don't need the true theory of everything. An effective theory that is only valid in some regime will do.

A second key idea is universality. The central limit theorem is an example of universality - as you keep on adding variables, you keep getting the same distribution, because the Gaussian is a *fixed point* of adding more variables. As long you only look coarsely and only require the mean and variance of the distribution, the Gaussian with its two parameters is a great model. Similarly, in physics, as you change energies from high to low, you get different effective theories. However, for some regime many parameter regimes that are different at high energies will appear the same at low energies - the flow to low energies has a *fixed point* in the space of theores. These fixed points represent scale invariant theories (by definition of the fixed point, and thinking of energy as scale).

Have you tried googling your exact topic?

 

[spaghetti3451]

I have now.

The first link I get is a Wikipedia page on Renormalisation group. These topics are too difficult for me to understand without the necessary mathematical grounding.

What I mean is that the descriptions are wordy and lack mathematical language, so in these case, it is difficult to fully understand the problem, so we can only get a vague sense of what's actually happening.

 

[atyy]

Really? When I google "Scaling Symmetry in Quantum Mechanics (δ function function potential in 2D) and 'renormalisation' " the top two links I get are your posts on Stack Exchange and here :)

 

[spaghetti3451]

I see. Well, for some reason, I happen to have lost all my privacy online. Even my classmates have spotted me out on PF. I hope you're not my lecturer, because then I am in trouble. :-)

 

[spaghetti3451]

But, thanks for the help. I'll try to figure it all out by myself by doing my own research.

 

[atyy]

I'm guessing your lecturer tried googling it himself to see whether relevant things would come up. If you are getting the same links as me, there are at least two great links on the first page that google returns.

Let me give you one more hint, look up "asymptotic freedom" together with relevant terms of your topic

 

[JorisL]

Sorry for this digression, I tried PMing atyy himself but wasn't allowed :(

You talk about the central limit theorem in post #8.

I was wondering, is it useful to think about large deviations (e.g. non-equilibrium effects) for the somewhat standard quantum mechanical systems?
With standard QM systems I'm hinting at systems that are likely to be encountered in the researchtopics listed in the opening post.

Joris

 

[atyy]

Sorry for this digression, I tried PMing atyy himself but wasn't allowed :(

You talk about the central limit theorem in post #8.

I was wondering, is it useful to think about large deviations (e.g. non-equilibrium effects) for the somewhat standard quantum mechanical systems?
With standard QM systems I'm hinting at systems that are likely to be encountered in the researchtopics listed in the opening post.

Joris

I don't know anything about large-deviations, but if there is a connection maybe you can make it from http://web.math.princeton.edu/facultypapers/Sinai/KolmogorovLec07.pdf.

 

[bhobba]

The first link I get is a Wikipedia page on Renormalisation group. These topics are too difficult for me to understand without the necessary mathematical grounding.

If topic 4 appeals the following will likely help:
http://arxiv.org/pdf/hep-th/0212049.pdf

The real essence of renormalisation is the crappy choice of a value to perturb about. That value secretly turns out to be dependant on what is called the cut-off. Indeed a dimensional analysis shows a parameter is missing and that most reasonably is the cut-off. The original parameter you perturb about actualy goes to infinity for large values of the cut-off so is a very poor choice. You want a parameter that remains small. Renormalisation is the trick that changes the parameter to one that works.

When you go through the article you will see that the functional form of the equation before renormalisation and after are in fact the same - which is rather interesting - and strange. It is related to a self symmetry property that the renormalisation group gives the detail of. Hopefully this is what you professor means by scaling symmetry.

If it interests you I would take the article to your professor and see if its what he is on about. The way 4 has been written for me is rather vague.

But if I had those choices I am with Vanhees - 5 would be my choice - its a lot of fun. Renoramalisation is important, interesting, and actually quite profound when you understand it (I don't right now to the level I would like) - but fun it aren't - it's what mathematicians call - decidedly non-trivial - translation - its hard.

Thanks
Bill

 

[bhobba]

I hope you're not my lecturer, because then I am in trouble. :-)

I doubt it. Getting hints on what each topic entails is all part of picking one.

Thanks
Bill

 

[atyy]

But if I had those choices I am with Vanhees - 5 would be my choice - its a lot of fun. Renoramalisation is important, interesting, and actually quite profound when you understand it (I don't right now to the level I would like) - but fun it aren't - it's what mathematicians call - decidedly non-trivial - translation - its hard.

I think it's important to stress that while the formal details of renormalization are hard (and still not worked out), the idea is easy and physical. Since we have a simple picture, and the data backs us up, we believe it is correct.

It's the same way with calculus. Calculus can be easy - it's just straight lines and addition, nothing more than the relationship between speed, distance and time. But it can be hard to make it rigourous as Weierstrass did.

In physics, we need the picture first, before getting lost in calculations. (OK, I admit that with renormalization the nonsensical calculations which miraculously worked were found before Wilson clarified their meaning.) The Wilsonian picture of renormalization is a key physical idea, and the one which made physicists say - we understand what quantum field theory is. http://quantumfrontiers.com/2013/06/18/we-are-all-wilsonians-now/

 

[bhobba]

I think it's important to stress that while the formal details of renormalization are hard (and still not worked out), the idea is easy and physical

Sure.

Many areas of math and physics are like that.

Thanks
Bill