Monday, July 30, 2012

A correspondence with a colleague on our nonlinear topology/geometry project

We are not resting on our laurels, but rather are continuing to work on new ideas and extending our work to more general cases.  In the process, we call get confused about the physics of the system, and try to find ways to picture what is going on that gives insights into how a system may behave under various conditions.

In working on this project, I realize that I spend lots of time writing emails to colleagues and students about various aspects of the work.  An email may come in at 10:30 at night because my collaborator is hot to figure something out and wants insights.  I too write emails at weird hours asking students for more details on something they observed in an effort to test my newest crazy ideas.  This is not a line of work for those who want to sleep soundly.  Being excited about physics is the best stimulant.

Anyway, today I took a two-hour break to play floor hockey and found a series of emails from a collaborator who is very excited (as am I) on the new direction of our work and the new physics that is implied.   Since we are moving into new territories, many things are not well mapped out, so we have to navigate partially by instinct.  I occasionally like to post excerpts from emails to give readers a sense of the kinds of exchanges that we have in the process of working on a project.  Here is my response to an exchange about a new quantum graph that we are working on, as shown above.

An excerpt from my email response (edited for typos and names removed):

First, I think there is a big difference in taking the limit as the prong length goes to zero and actually having a zero prong length.  In the former, I think that you get a broken vertex, because as the wire gets shorter, the end of the wire is getting closer to the vertex, so indeed the wavefunction on the prong must get vanishing small as the prong length goes to zero.  Then, this will look identically to a break in the loop.  Of course, when there is no prong, you get the traveling wave solutions.  So it is interesting that the limiting case yields a very different result than the case without the prong - again a statement about topology.  As long as the prong is there, no matter how short, the topology is of a loop-star.  Get rid of a prong, and then it is just a loop!

I am a little confused about what you are saying with regards to the ground state energy.  So, let me just make a point that may not be related at all to what you are saying.  If I have a prong with zero wavefunction, the loop cannot have a constant wavefunction, i.e. the wavefunction with n=0.  By continuity, the amplitude of the constant wavefunction will be zero, so there is no particle in the system.  So, I believe that in a loop, you have the zero-energy wavefunction but as long as a prong is there, you will not have a zero-energy wave function.

However, you bring up an interesting possibility of a wavefunction in which there is zero wavefunction in the prong and a standing wave in the loop in which there is a node at the prong.  This seems to be alright in terms of the continuity of the wavefunctions, and probability current is certainly conserved.  So unless there is another constraint that I am not seeing, this looks reasonable.

The problem is that this may be a legitimate wavefunction, but not an energy eigenfunction and is therefore not a stationary state.  Forcing this kind of state would be like having a particle in a box wave function, let's say with multiple nodes.  Having the wavefunction oscillate to he left of the node, then zeroing it to the right of the node obeys all the boundary conditions, but, this is not an energy eigenstate.

So, I think I will stick with my original assessment, though now I have thought about this for an additional 10 minutes for a total of 15.  I am not confident in my view because I have not done anything with paper and pencil, just picturing things in my head - a very dangerous activity!

These kinds of discussions do not come out of the graduate students until the point that they are getting ready to graduate.  Only then does a light bulb turn on, which makes them get it.  Once they become very useful and great fun in terms of being intellectually stimulating, they leave for greener pastures.  Then they can start to challenge and stimulate their new advisers.

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