Here we consider three fascinating problems in particle physics that can be approximated as two-level systems with a somewhat phenomenological Hamiltonian. Two of them involve neutrinos and one involves neutral K-mesons. Together they provide a remarkable success story of the applications of simple quantum-mechanical principles.

Neutrinos

Neutrinos are spin ½ particles with no charge and a minuscule mass. They interact with other elementary particles only through the so-called weak interaction. Neutrinos come in three different species (called flavors): electron-neutrino (νe), muon-neutrino (νµ), and tau-neutrino (ντ) and form a part of the so-called lepton family. They are neutral accompaniments to the charged leptons: electrons (e), muons (µ), and tau leptons (τ). In the following two sections we will ignore ντ and discuss the solutions of some rather fundamental problems in neutrino physics within the framework of two-level systems.

The solar neutrino puzzle

To understand the solar neutrino puzzle we need first to note that the energy that is radiated from the solar surface comes from intense nuclear reactions that produce fusion of different nuclei in the interior of the sun. Among the by-products of these reactions are photons, electrons, and neutrinos. In the interior it is mostly electron-neutrinos, νe's, that are produced. The shell of the sun is extraordinarily dense, so that the electrons and photons are absorbed. However, because neutrinos undergo only weak interactions they are able to escape from the solar surface and reach the earth.

An electric current in a normal conductor can be thought of as a fluid made up of electrons flowing across lattices made up of heavy ions and constantly colliding with them. The kinetic energy of the electrons decreases with each collision, effectively being converted into the vibrational energy of the ions. This dissipation of energy then corresponds to electrical resistivity. It is found that the resistivity decreases as the temperature is decreased but it never completely vanishes even at absolute zero.

In a conventional superconductor, however, the electrons occur in pairs, called Cooper pairs, because of the attractive force generated by the exchange of phonons. If one looks at the energy spectrum of these pairs, there is an energy gap that is the minimum of energy needed to excite the pair. If the thermal energy (kT) of the electrons is less than the gap energy, then the Cooper pairs will act as individual entities and travel without undergoing any scattering with the ions. Therefore, there will be no resistivity. Thus, in a superconductor the resistance drops abruptly to zero below a certain temperature, called the “critical temperature.” An electric current flowing in a loop of wire consisting of a superconductor then flows indefinitely with no resistance and without the help of any power source. Below, we briefly describe the mechanism that gives rise to this superconductivity.

Many-body system of half-integer spins

We consider a many-body system consisting of identical fermions that group themselves in pairs like quasiparticles where each pair consists of electrons that are degenerate in energy but have opposite linear momenta, p and −p, as well as opposite spin directions.

While writing this book I was reminded at times of what Professor Francis Low used to say when I took his class on undergraduate electromagnetism at the University of Illinois, Urbana-Champaign. “Be sure to understand the subject thoroughly,” he said, “otherwise, your only other chance will be when you have to teach it.” Knowing now what I know by having written this book, I would add that, if at that point one still does not understand the subject, there will be yet another opportunity when writing a book on it. That was certainly the case with me and this book.

For the last twenty years or so I have taught a one-year graduate course in quantum mechanics at the University of California, Riverside. I have used several books, including the text by Schiff which also happens to be the text I used when I was taking my graduate courses at the University of California, Berkeley (along with my class notes from Professor Eyvind Wichmann who taught the quantum electrodynamics course). However, it became clear to me that I would need to expand the subject matter considerably if I wanted the book not only to be as thorough and up-to-date as possible but also organized so that one subject followed the other in a logical sequence. I hope I have succeeded.