Hilbert space

The prototype qubit systems of the last chapter are very simple, but they can be generalized to more complicated versions. We can send a photon through an interferometer with three, four or more distinct beams. We can perform experiments on particles with higher intrinsic angular momentum than the spin-½ particles we have discussed. And we can analyze atomic systems in situations that involve more than two different energy levels. For these cases and others, we will need a more general version of quantum theory.

That theory will include two pieces. First, we will have a general mathematical structure that is applicable to many kinds of system. Here the qubit case will be our guide, since many of the basic concepts for other quantum systems are already present in the qubit case. Second, we will have to describe how to apply the quantum formalism to specific physical situations. Though the quantum systems we discuss will appear quite various, they share strong family resemblances that are expressed in the common mathematical framework. Keeping the framework in mind will help us understand specific examples; keeping the examples in mind will help us understand the framework.

The states of a quantum system are described by kets |ψ〉, which obey the principle of superposition. This means that the kets are elements of an abstract vector space ℋ called a Hilbert space.

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.