--- mcintosh@servidor.unam.mx wrote:
Quoting mcintosh@servidor.unam.mx:
^
easy to forget the twiddle!
- hvm
It's a very nice paper, and spells out much of what I said in these messages. I programmed this method in Maple for the case of a symmetric square well potential, and plotted T(E) vs. E, where E is the energy and T is the trace of the propagator matrix. T(E) is continuous and oscillatory; nothing special happens when |T(E)| crosses between <2, when the energy is allowed and >2, when the energy is forbidden. When the potential acts as a barrier, T(E) swings well beyond +-2, so that the allowed energy band is narrow. When the energy lies above all of the potential, T(E) ranges nearly, but not exactly, between -2 and +2. When the extremum goes beyond +-2, we have a small energy gap. But it may also happen that an extremum lies short of +-2, in which case we have a small momentum gap. This method we have been discussing appears capable of analysing in great detail the electronic structure of one dimensional crystals, at least under the constraint of a given fixed potential. We have relied upon the theorem that the solution space is two dimensional, corresponding to the two arbitrary constants in the general solution of the ODE. How do we handle the three dimensional crystal? The Schroedinger equation becomes a PDE, and the general solution possesses arbitrary functions. I am not aware of any "momentum gaps" in solid state physics, and I will try to see whether this phenomenon is possible in three dimensions. Gene __________________________________ Do you Yahoo!? Vote for the stars of Yahoo!'s next ad campaign! http://advision.webevents.yahoo.com/yahoo/votelifeengine/