Application of the "OR" Rule
(On the Time-Dependence of an Ensemble’sY when the Temperature is Greater than Absolute Zero)
Let an ensemble be defined to be a set of N identically prepared, single-electron systems, where N is arbitrarily large. By "identically prepared" we shall mean that there are N widely separated, inertial coordinate systems, and that system i is prepared in coordinate system Ki at time t=0, system j in Kj at t=0, etc. The identical parameters will be the position and the momentum. For convenience we consider positions on the x-axis and momenta in the x-direction. If the electrons are subject to a conservative force, then presumably the potential energy is an identical function of x in all N systems.
A tenet of quantum theory (and a distinct break with classical theory) is that x and px cannot be known with absolute precision. Even if the systems are (in our opinion) identically prepared, there will be a nonzero uncertainty in x (denoted Dx) and a nonzero uncertainty in p (denoted Dp). The least value for DxDp is specified by the Heisenberg Uncertainty Principle.
Associated with the ensemble, and containing all of our a priori knowledge of the system, is a wave function Y. The only way we can improve our knowledge of x or p in a given case is to remove that particle from the ensemble by interacting with it. The wave function in every case is governed by the Schroedinger equation, and is generally complex.
Perhaps the simplest ensemble is a set of free electrons (not subject to any external force). At time t=0 the electrons’ x-positions are centered on the Origin(s) and their momenta are centered on, say, some positive value. The uncertainty in momenta equates to an uncertainty in (kinetic) energies. And by the deBroglie condition E=hw/2p, the set of energies equates to a set of wave function frequencies.
Although it is not necessary to do so, the individual Yi can conveniently be modeled as spinning spirals whose threads propagate in the positive x-direction. If the Yi are all in phase at x=0 and at t=0, then the resultant Y (i.e., the sum of all the constituent Yi) is a wave packet or group that moves in the positive x-direction at the most probable electron velocity.
The fundamental idea behind summing the individual Yi is that any given particle in the ensemble could have energy Ei OR Ej OR …. And a priori we have no way of narrowing our knowledge in this regard without interacting with the particle.
Another celebrated example where this "OR Rule" applies is the double slit experiment. If a "monochromatic" beam of electrons is directed at a double slit, then a priori we have no idea which slit a given particle passes through. Consequently we must sum the Yi’s from each slit, in order to obtain a resultant Y at some array of detectors.
The general rule seems to be that if a given particle’s energy (or position or …) could be this OR that OR …, then the Yi’s for the various possibilities must be summed in order to get the ensemble (resultant) Y. The result that a given particle will be found to have a particular position (for example) is statistically predicted by YY*.
The simple, free-particle and the double slit experiment deal with unbound particles (zero potential energy). More often than not we are concerned with bound particles. Perhaps the simplest example of a bound particle is one trapped in an infinite, square (potential energy) well. Plausible boundary conditions on the Yi dictate that only certain discrete energies can persist in such a well. Figs. 1 – 3 show YY* for the lowest 3 possible energies. (Eo generally denotes the lowest possible, "ground state" energy.)
YY* vs. x, E = Eo
YY* vs. x, E = E1
YY* vs. x, E = E2
An interesting feature of Figs. 1 – 3 is that the plots do not change with time. The reason for this time independence is that, although in the spinning spiral model the individual Yi’s spin "jump rope style" around the x-axis, their magnitude at any given x does not vary in time. Such "stationary states" are important features in solving Schroedinger’s equation when an ensemble can be assumed to be populated by a single energy state.
Most often, however, the reality is that any ensemble of "real" square wells will contain electrons at many energies. The fractions at energies Eo, E1, etc. will be functions of the ensemble’s absolute temperature, T. In thermal equilibrium the ensemble temperature will equal the temperature of the ensemble’s environment.
If we apply the "OR" rule in the infinite well case, then we find a resultant ensemble YY* that is not independent of time (i.e. is not "stationary"). Indeed as demonstrated in a previous article, summing all the constituent Yi’s produces an initial wave group at, say, the left side of the well. In contrast to free particles, however, in the case of the well the Yi are reflected off the well walls and the initial wave group does not elegantly propagate across to the right wall (where it is reflected), etc. The actual form of YY*, at 10 time epochs in a half period, is shown in a previous article. In brief, YY* for multi-energy ensembles of bound particles is not generally time independent.
There are of course other bound systems of at least equally practical interest … the Hydrogen atom, an electron subject to a spring force, etc. Here again the only ensemble with time-independent YY* would be one at a temperature of absolute zero. In all other cases the resultant Y for the ensemble must be obtained by summing the Yi (one for each Ei) prior to computing YY*.
The "OR" rule seems to be closely connected to this phenomenon of ensembles at T>0. For the idea of an ensemble of Hydrogen atoms, with some of the atoms having a stationary YY* appropriate to E = Eo etc., suffers from the difficulty that such stationary states will absorb/emit quanta only if we suppose in ad hoc fashion that they must (since experiment indicates that they do). The time varying YY* does not require such ad hoc suppositions, as it already absorbs and emits quanta. (At least this is to be expected if YY* maps to a time-varying charge density.)
In conclusion, in the case of a typical ensemble at a temperature greater than absolute zero, the ensemble YY* does not correspond to any particular energy; it reflects the presence of all possible energies. As such, it is not time-independent. In summing the constituent Yi’s to obtain an ensemble Y, the weighting factors for the various Ei are a function of ensemble temperature. At very low temperatures the lowest energy weighting factors are dominant, etc.