Examples in Physics

Diederik Aerts, Thomas Durt, Bruno Van Bogaert

We present an explicit physical example of an experimental situation on a physical entity that gives rise to a fuzzy set. The fuzziness in the example is due to ∞uctuations of the experimental apparatus, and not to an indeterminacy about the states of the physical entity, and is described by a varying parameter †. For zero value of the parameter (no ∞uctuations), the example reduces to a classical mechanics situation, and the corresponding fuzzy set is a quasi-crisp set. For maximal value (maximal ∞uctuations), the example gives rise to a quantum fuzzy set, more precisely a spin- model. In between, we have a continuum of fuzzy situations, neither classical, nor quantum. We believe that the example can make us understand the nature of the quantum mechanical fuzziness and probability, and how these are related to the classical situation.

Travis Norsen

Arkady Bolotin

Unlike mathematics, in which the notion of truth might be abstract, in physics, the emphasis must be placed on algorithmic procedures for obtaining numerical results subject to the experimental verifiability. For, a physical science is exactly that: algorithmic procedures (built on a certain mathematical formalism) for obtaining verifiable conclusions from a set of basic hypotheses. By admitting non-constructivist statements, a physical theory loses its concrete applicability and thus verifiability of its predictions. Accordingly, the requirement of constructivism must be indispensable to any physical theory. Nevertheless, in at least some physical theories, and especially in quantum mechanics, one can find examples of non-constructive statements. The present paper demonstrates a couple of such examples dealing with macroscopic quantum states (i.e., with the applicability of the standard quantum formalism to macroscopic systems). As it is shown, in these examples the proofs of the existence of macroscopic quantum states are based on logical principles allowing one to decide the truth of predicates over an infinite number of things.

S. Brandt, Hans Dieter Dahmen

Robert B. Griffiths

Classical and quantum properties We shall use the term physical property to refer to something which can be said to be either true or false for a particular physical system. Thus “the energy is between 10 and 12 μJ” or “the particle is between x 1 and x 2 ” are examples of physical properties. One must distinguish between a physical property and a physical variable , such as the position or energy or momentum of a particle. A physical variable can take on different numerical values, depending upon the state of the system, whereas a physical property is either a true or a false description of a particular physical system at a particular time. A physical variable taking on a particular value, or lying in some range of values, is an example of a physical property. In the case of a classical mechanical system, a physical property is always associated with some subset of points in its phase space. Consider, for example, a harmonic oscillator whose phase space is the x, p plane shown in Fig. 2.1 on page 12. The property that its energy is equal to some value E 0 > 0 is associated with a set of points on an ellipse centered at the origin. The property that the energy is less than E 0 is associated with the set of points inside this ellipse. The property that the position x lies between x 1 and x 2 corresponds to the set of points inside a vertical band shown cross-hatched in this figure, and so forth.