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.
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.
Classical sample space and event algebra Probability theory is based upon the concept of a sample space of mutually exclusive possibilities, one and only one of which actually occurs, or is true, in any given situation. The elements of the sample space are sometimes called points or elements or events . In classical and quantum mechanics the sample space usually consists of various possible states or properties of some physical system. For example, if a coin is tossed, there are two possible outcomes: H (heads) or T (tails), and the sample space S is { H, T }. If a die is rolled, the sample space S consists of six possible outcomes: s = 1, 2, 3, 4, 5, 6. If two individuals A and B share an office, the occupancy sample space consists of four possibilities: an empty office, A present, B present, or both A and B present. Associated with a sample space S is an event algebra B consisting of subsets of elements of the sample space. In the case of a die, “ s is even” is an event in the event algebra. So are “ s is odd”, “ s is less than 4”, and “ s is equal to 2.” It is sometimes useful to distinguish events which are elements of the sample space, such as s = 2 in the previous example, and those which correspond to more than one element of the sample space, such as “ s is even”. We shall refer to the former as elementary events and to the latter as compound events.
The concept of the physical state of a system is ubiquitous in physics but is usually presented in terms of specific cases. For example, the state of a point particle of mass m is completely characterized by its position and momentum. There is a tendency to consider such states as "real", i.e., as physical properties of a system. This rarely causes problems in classical physics but the notion of real quantum states has contributed mightily to the philosophical conundrums associated with quantum mechanics. The Einstein-Rosen-Podolsky paradox is a prime example. In fact, quantum states are not physical properties of a system but rather subjective descriptions that depend on the information available to a particular observer. This realization goes a long way toward resolving such dilemmas as Schrödinger's cat, wave function collapse, quantum non-locality, and parallel universes.
Mature theories of physical science associate states with systems, for instance with particles or waves in classical physics. The quantities whose values comprise states in classical mechanics, namely position and momentum, are chosen, first of all, for predictive purposes. Everything that one would want to say about classical systems can, it seems, be said in terms of properties: specification of the properties of a system gives a complete specification of its state, and conversely. Quantum mechanics associates states with physical systems; whatever else these might be, systems are the particulars which bear physical properties. The former is non-reduced state, the quantum state proper, while the latter is the reduced state, the class of quantitative properties possessed by the system. The state therefore determines which quantities and which properties are instantiated. The non-reductive account of states was initially appealing because it seemed to promise some additional explanatory scope.
Quantum mechanics is one of the pillars of modern physics, however rather difficult to teach at the introductory level due to the conceptual difficulties and the required advanced mathematics. Nevertheless, attempts to identify relevant features of quantum mechanics and to put forward concepts of how to teach it have been proposed.1–8 Here we present an approach to quantum physics based on the simplest quantum mechanical system—the quantum bit (qubit).1 Like its classical counterpart—the bit—a qubit corresponds to a two-level system, i.e., some system with a physical property that can admit two possible values. While typically a physical system has more than just one property or the property can admit more than just two values, in many situations most degrees of freedom can be considered to be fixed or frozen. Hence a variety of systems can be effectively described as a qubit. For instance, one may consider the spin of an electron or atom, with spin up and spin down as two possible values, and where other properties of the particle such as its mass or its position are fixed. Further examples include the polarization degree of freedom of a photon (horizontal and vertical polarization), two electronic degrees of freedom (i.e., two energy levels) of an atom, or the position of an atom in a double well potential (atom in left or right well). In all cases, only two states are relevant to describe the system.
Ensembles and expectation values Equations of motion provide predictions of time-evolving variables of a dynamical system, and thereby enable us to predict the changing behavior. These equations are characteristic of a particular type of system, say a classical harmonic oscillator or a two-state quantum system. To complete the description of behavior one must specify initial conditions that select, from the many possible solutions to the equations of motion, a particular realization. In the example of a classical point particle the equation is second order in time, and so we need two initial conditions, of position and velocity, to specify a particular solution. These two numbers, taken with the classical equation of motion, provide a complete description of the behavior of a specific harmonic oscillator. Similarly the two-state quantum system requires two initial values to complete its definition. The initial conditions are never defined with infinite precision, even for classical mechanics. More generally we deal with an ensemble, that is, a collection of similar dynamical systems (e.g. oscillators) that differ in their preparation. The individual cases share some common attributes, but not necessarily all. Thus we have only incomplete information about any single system. We might, for example, consider planetary motion in which we know only the mean energy and the orbital plane. The loci of positions of the planets then form ellipses that make up the ensemble. Although it is possible to prepare a single two-state quantum system in which the initial state is known (apart from an overall phase), often one deals with ensembles of such systems, each of which may differ in some uncontrollable property.
The properties of a noncanonical 3D Wigner quantum oscillator, whose position and momentum operators generate the Lie superalgebra sl(1|3), are further investigated. Within each state space W(p), p=1,2,..., the energy E_q, q=0,1,2,3, takes no more than 4 different values. If the oscillator is in a stationary state ψ_q \in W(p) then measurements of the non-commuting Cartesian coordinates of the particle are such that their allowed values are consistent with it being found at a finite number of sites, called ``nests''. These lie on a sphere centered on the origin of fixed, finite radius \varrho_q. The nests themselves are at the vertices of a rectangular parallelepiped. In the typical cases (p>2) the number of nests is 8 for q=0 and 3, and varies from 8 to 24, depending on the state, for q=1 and 2. The number of nests is less in the atypical cases (p=1,2), but it is never less than two. In certain states in W(2) (resp. in W(1)) the oscillator is ``polarized'' so that all the nests lie on a plane (resp. on a line). The particle cannot be localized in any one of the available nests alone since the coordinates do not commute. The probabilities of measuring particular values of the coordinates are discussed. The mean trajectories and the standard deviations of the coordinates and momenta are computed, and conclusions are drawn about uncertainty relations. The rotational invariance of the system is also discussed.