Free Electron Model in One Dimension

FEM is a quantum mechanical treatment of the electron in a box. The repulsive Coulomb interaction between electrons is assumed to be cancelled by a uniform positive background rather than by positive atomic nuclear point charges. This approximation is possible because we are only interested in the loosely bound valence electrons, among which we find the conduction electrons. 

We now instead let the electron move on a circular track, like a skier in a training tunnel. We have L = 2tt R, where R is the radius. The relevant boundary condition is that the skier continues with the same speed when she arrives at the end of the tunnel, from where she started. The unnormalized eigenfunctions are 

Since the energy is kinetic, we also have E = p /2m, which implies p = /ik. The momentum operator. Equation 1.13, is 

From this, it follows that only in the circular case are the eigenfunctions of energy also eigenfunctions of momentum. In the box case, we may define an expectation value over part of the phase space, but the expectation value over aU space is equal to zero, since the particle moves back and forth. In the circular motion, the momentum 

The circular motion and the linear motion for L c are analoguous. It is of great value to be able to describe metals in momentum space. Electrons are described as infinitely extended waves with a fixed momentum. Conductivity at T 0 is hindered by scattering between electron waves and localized vibrational states. 

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