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 x1 and x2” 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 E0 > 0 is associated with a set of points on an ellipse centered at the origin. The property that the energy is less than E0 is associated with the set of points inside this ellipse. The property that the position x lies between x1 and x2 corresponds to the set of points inside a vertical band shown cross-hatched in this figure, and so forth.