What could not be in Feynman lectures: Bell inequalities

Lecture notes written in the spirit of Feynman lectures on quantum mechanics. Feynman extensively uses Stern-Gerlach filters to describe basic principles of QM using thought experiments with polarized electron beams. We add the discussion on the EPR paradox, hidden parameters and Bell inequalities. This is not meant as an introductory text from quantum mechanics. It is rather a supplement to be read by students after they learned the basic course of quantum mechanics and in particular the formalism of spin-1/2 particles. However it may be useful also for students who are interested in, say, quantum computers and the logical aspects of the problems of measurements in quantum mechanics. They should then rely on the mathematical formalism and accept the fact that some pieces of ”physical argumentation” will not be completely clear to them. 1 Stern Gerlach filters In this section we briefly repeat what was (at least ”in spirit”) written in the Feynman lectures. We shall speak on beams of electrons passing through various Stern Gerlach filters. This kind of language has only symbolical meaning. We do not care whether the described experiments are really technically feasible. We use the philosophy of the thought experiments heavily exploited by Einstein, Bohr and Feynman. Thought experiments are performed virtually, just in our imagination. There might be technical problems if we tried to perform them in reality. However we carefully check that on a principal level there is nothing to prevent them working. The principal components used in our experiments will be an electron gun producing a beam of unpolarized electrons, Stern Gerlach filters and electron detectors which detect electrons with ideal efficiency irrespective of their polarization. Electrons are fermions with spin 1/2. Their spin states are described by vectors in two-dimensional Hilbert space. A base in the Hilbert space can be chosen arbitrarily: most often one chooses the base formed by the eigenstates of the operator of the projection of the spin to the z-axis. W shall denote these base vectors as |↑〉 and |↓〉 In the state |↑〉 the projection of the electron’s angular momentum on the z-axis is h̄/2 while in the state |↓〉 it is −h̄/2. To simplify the discussion we shall not consider real electrons but certain imaginary particles. What concerns their spatial motion we shall consider just a (quasi-classical) uniform motion in the direction of the y-axis. What concerns polarization, we shall consider only polarizations perpendicular to the beam axis. Our ”electrons” will be polarized in the xz-plane and we shall completely neglect the polarization in the direction of the y-axis (which is possible in real world). So we shall consider particles living in a one dimensional space along the y-axis with a two-dimensional space of ”internal degrees of freedom” given by the plane xz. Mathematically it will mean that we shall take into account not all the vectors from the Hilbert space spanned on the two basis vectors |↑〉 and |↓〉 but only those superpositions c↑ |↑〉+ c↓ |↓〉 (1) where the coefficients c↑ and c↓ are real and satisfying the normalization condition |c↑| + |c↓| = 1 The Stern Gerlach Filter (SGF) is extensively discussed in the ”Feynman lectures on physics” so we skip even a schematic description of its internal construction here and we just describe its function as a black box. SGF is device used to filter electrons in the electron beam. Each SGF is characterized by the direction and orientation of its ”principal axis” (denoted by an arrow on our figures) which is perpendicular to the direction of the electron