Renormalisation of the Schrodinger equation

[spaghetti3451]

Hi, I am a senior year Physics undergraduate and my current understanding of quantum mechanics stands at the level of the Griffiths textbook.

I am trying to understand what it means to renormalise the Schrödinger Equation. I know that it's not possible to understand the detailed mathematics of it all from my present standpoint, so I'd appreciate it if you could provide a simple explanation of this idea.

 

[dextercioby]

Do you know what renormalization is?

 

[spaghetti3451]

I know (in a very vague manner) only the following:

When we are given a) the parameters that define the long-range behaviour of a system and b) a sample set of observations for the short-range behaviour of the system, then we can model the system using a particular approach called renormalisation. The theory that results is called an effective field theory, because it is not the true theory, but still produces correct predictions for the short-range observations.

Also, sometimes when we perform a pertubative calculation to find the energy eigenvalues and eigenfunctions of a potential. In some cases, the corresponding results diverge to infinity. There also we apply the technique of renormalisation.

That's all I know.

 

[bhobba]

The following will likely help:
http://arxiv.org/pdf/hep-th/0212049.pdf

Thanks
Bill

 

[spaghetti3451]

I got that one. I have actually downloaded several journal articles which focus on anomalous symmetry breaking in quantum mechanics. I've been studying this topic lately for my coursework.

Anomalous symmetry breaking (in the context of ordinary quantum mechanics) is not covered in textbooks as it is a very novel topic, so I'm having to get my hands on journal articles. However, I think the math is fairly accessible to anyone with a good grounding in Griffiths-level Quantum Mechanics, so that saves me a lot of hassle.

 

[atyy]

You might like to look at chapter 23 of Schwartz's lectures notes: http://isites.harvard.edu/fs/docs/icb.topic521209.files/QFT-Schwartz.pdf.

Schwartz discusses the relationship between the Schroedinger and Dirac equations. However, it is very hard (impossible?) to make sense of the Dirac equation except as a field theory. So it is useful to be aware that the Schroedinger equation for many identical particles can be rewritten as a quantum field theory: http://hitoshi.berkeley.edu/221b/QFT.pdf.

 

[atyy]

I googled "anomalous symmetry breaking" and one of the things that came up was the Efimov effect. Is that what you were asking about in the OP? In the renormalization group picture it seems to be (not certain?) related to a limit cycle.

http://www.int.washington.edu/talks/REU/2004/REU_04_talks/People/Swingle_B/swingletalk.pdf
http://arxiv.org/abs/quant-ph/0503074
On the limit cycle for the 1/r^2 potential in momentum space
H.-W. Hammer (INT), Brian G. Swingle

http://arxiv.org/abs/1102.3789
Efimov physics from a renormalization group perspective
Hans-Werner Hammer, Lucas Platter