Quantum Tests

Abstract

2-1. What is a quantum system? A quantum system is a useful abstraction, which frequently appears in the literature, but does not really exist in nature. In general, a quantum system is defined by an equivalence class of preparations. (Recall that "preparations" and "tests" are the primitive notions of quantum theory. Their meaning is the set of instructions to be followed by an experimenter.) For example, there are many equivalent macroscopic procedures for producing what we call a photon, or a free hydrogen atom, etc. The equivalence of different preparation procedures should be verifiable by suitable tests. The ambiguity of these notions emerges as soon as we think of concrete examples. Is a hydrogen atom in a 2p state the same system as one in a 1s state? Or is it the same system as a hydrogen atom in a 1s state accompanied by a photon? The answer depends on the problem in which we are interested: energy levels or transition rates. In a Stern-Gerlach experiment, we have seen (page 17) that the "quantum system" is not a complete silver atom. It is only the magnetic moment µ of that atom, because the goal of the Stern-Gerlach test is to determine a component of µ. The center of mass of the atom can be treated classically. These examples show that we must be content with a vague "definition": A quantum system is whatever admits a closed dynamical description within quantum theory. While quantum systems are somewhat elusive, quantum states can be given a clear operational definition, based on the notion of test. Consider a given preparation and a set of tests, among which some are mutually incompatible, as in Fig. 1.6. If these tests are performed many times, after identical preparations, we find that the statistical distribution of outcomes of each test tends to a limit. Each outcome has a definite probability. We can then define a state as follows: A state is characterized by the probabilities of the various outcomes of every conceivable test. This definition is highly redundant. We shall soon see that these probabilities are not independent. One can specify-in many different ways-a restricted set 24