Calmness, finally

Things are finally settling down around here. Yesterday was my second statistical mechanics midterm: it was easy, but I am very disappointed at myself. I worked very slowly and only had time to work half of the first problem.

There were two problems. First we were given a 2-D electron ideal gas and were asked to find all thermodynamic characteristics for the case of zero temperature and temperatures very small compare to the Fermi energy. The other problem was to discuss an aparent contradiction with Bose-Einstein condensates. Below the critical temperature we know that the chemical potential is zero. This would imply that the Gibbs energy is zero. Generally the entropy of a system can be found by differentiation of the Gibbs energy with respect to temperature while fixing the volume and number of particles. Since the Gibbs energy is zero, then the entropy will also be zero. But in class the professor found an expression for the specific heat of the Bose gas as a function of temperature below the critical temperature. This specific heat can be integrated to yield the entropy, which in this case will be a non-trivial function of temperature. WTF?

I am disappointed at myself. I spent to much time organizing my thoughts during the test. First I quickly started calculating the energy and occupation number for the electron gas when the temperature was zero. Then I found the other potentials, but not the specific heats nor the entropy. All I found were things in terms of the area, surface tension and number of particles, so I could not take any derivatives, of course, since the temperature was fixed. It turns out that for any temperature you can calculate the number of particles in the system exactly since the integral can be evaluated analytically. But Melvin here just stared at the paper for a long time trying to remember how to integrate such things as one-over-e-to-the-x-plus-one.

In the end I do not think all the work I handed in is worth 50 points (out of the 100 points the first problem was worth). I did not even worked on the case for temperatures below the Fermi energy. I presume that some sort of series expansion was needed. I have realized that I am not good with series expansions and I need a lot of time to set everything up. This makes me very sad because I believe that being able to work with series expansions is very important for physics. I should settle down and learn them right once and for all, before this goes to far... if it does...

For the second problem I just wrote down some garbage on how I believe the Gibbs energy was not zero dealing with the fact that there are two phases present below the critical temperature (the condensate and the gas). Now that I think of it, well no. The entropy when the temperature tends to zero should tend to zero too. Also in phase equilibrium both phases have the same value for the chemical potential. Anyways, the whole point is that the test made me feel really stupid.

Yesterday at the electrodynamics recitation the professor stated the following problem. Consider a charged particle at rest in a region of constant electric and magnetic field, with the fields perpendicular to one another and having the same magnitude (in Gaussian units!). What is the trajectory of the particle?

We fiddled with the problem by writing down the Lorentz force law. Because of the Lorentz-gamma factor, the differential equations for the velocity or momentum components look kinda nasty. The professor believed that there was an analytical solution, but he said he was to lazy to find it so he would prefer to solve this problem numerically. This was just for fun, but I kept on thinking about it and decided to work on it during the night. I discretized the two equations and found a nice system of couple differential equations. It is not very complicated, but I just suck at debugging code, so it was not until this morning when I was able to make the program run properly. I found what was expected: the magnitude of the velocity tends to the speed of light, and the trajectory just blasts off to infinity with slight distortions. I felt happy, since I had accomplished something that appeared to be correct. So I emailed the professor some of my results and went to my class.

It turns out that the professor had found an analytical solution for this problem, so I seem not to pay much attention to my results. My ego was just demolished :-(.

And it has been like this during the past weeks. I am feeling like a forgotten piece of crap, very useless. On my previous SM midterm I got something like an 84 out of 150, on the QM midterm I did slightly better with 14.8 out of 20, the average being around 14 so I guess I am above average. But I have not been doing that well in the homeworks, there is alway a bloody problem where I do not have a bloody clue on what to do. A more positive picture is on EM where I was able to score a 95/100 in the midterm, and my homeworks are not that bad. But in SM... oh boy. Two homeworks ago we had to calculate the third virial coefficient for the hard-sphere model of a non-ideal gas. It was way more complicated than I thought, so I only got half credit. In fact, only two students got the correct result, most of the class got half-credit. Then on the previous homework we had to find an expression for the vapor pressure of the van der Waals model. Again, I did not worked this problem completely and expect to do very poorly. All this added to not doing well on this second midterm may imply that I could fail this class. Ahh!

I do not even want to think about it. Currently I have been over worried thinking about next semester. I am planning on taking Quantum Field Theory and Relativity and hopefully a breadth course. At the same time I should attend more theory seminars and maybe the Strings course.