Graduate Course: Quantum Physics, since Spring Break...(2001-3)

The first set of topics were lumped under the title Consequences of Spin First we considered many particle systems, and within them, the special case of two identical particles. Since identical particles cannot be distinguished, this leads to consideration of the exchange operator P12(r), with eigenvalues +1 or -1. We then showed that with time evolution, these two types of states evolve independently, since P12 commutes with the Hamiltonian. Thus systems with eigenvalue -1 are Fermions, which are identified experimentally as particles with half integral spin (1/2, 3/2, 5/2, etc). Those with integral spin (0, 1, 2, etc) are Bosons. The Pauli principle applies to Fermions, where no more than two particles (e.g. electrons) can occupy the same spatial state, and these two particles must have the opposite (+1/2 and -1/2) spin states. Thus the wavefunction of such a system has to be properly symmetrized, and in the case of Fermions, this anti-symmetry can be expressed as a Slater determinant. Degeneracy, the filling of energy states up to the Fermi level, occurs readily for Fermions, with examples taken from solid state, nuclear and astrophysics. Antisymmetry makes the particles tend to avoid each other in real space, i.e. they experience an effective repulsion. On the other hand Bosons attract each other, in that the particles can be in the same place, and when the particle density gets high, and the temperature low, we get Bose Einstein Condensation (BEC), which is the phenomenon behind superconductivity, superfluidity in liquid helium, and more recently observed in atoms and ions trapped by laser beams and magnetic fields. The similarities and differences between Fermi-Dirac and Bose-Einstein distributions form the central core of Statistical Mechanics (PHY 541...). Note for future reference that the Exchange interaction arises from the symmetry of the wavefunction; there is an additional Coulomb interaction if the particles are charged, as for example in the Helium atom. This means that most real problems involve considerable approximation to approach them within QM.

The second set of topics were considered under the title Atoms and Spectroscopy First we reviewed the structure of the Periodic Table of the elements, and the configuration of atoms. Then we discussed how such information has been obtained, via Atomic and Molecular Spectroscopy. Term schemes and term notation was established, both using older (Herzberg) and more recent (Heckmann and Träbert) sources, including real experimental examples. This provided a list of phenomena that Quantum Mechanics needs to explain. We went quickly through the extension of the Schroedinger equation to 3-dimensional, radially symmetric potentials, building on what we had done previously on the angular and spin functions. This scheme accounts for the 'Gross' Energy Levels of atoms, quantitatively for hydrogen. The qualititive aspects of the radial and angular solutions, and the types of nodes which occur, were emphasized. The third sub-topic under this heading was Selection Rules for allowed transitions, showing that dipole transitions are associated with time varying electric fields, which are correlated with angular and spin states and the polarization of the emitted or absorbed radiation. This gives meaning to 'creation' and 'annihilation' operators, A and A+, which create or annihilate a photon. Finally we looked at Fine Structure, which is dependent on terms which are split by S.S and L.S interactions. Some of the operator algebra was performed to show that this splitting can account for singlet and triplet states, and hence the spectroscopic classification of atomic He, Ca and other such cases. Other examples, such as the S = 1 ground state of the deuteron, and the Heisenberg model of ferromagnetism, were mentioned as suitable topics for projects.