On Measurement, Probabilities and "Sub-ensembles"

G.R.Dixon, 10/01/2007

Let us begin by supposing that we have a mole of Hydrogen atoms in a suitable size container. The sample is at room temperature. Each atom can have any of a large range of discrete energies, E_{o}, E_{1}, etc., all of which are well known. The energy of an atom is *increased* from energy E_{k} to energy E_{k+n} (n=1,2,…) when the atom *absorbs* a quantum of energy from its environment. And it is *decreased* from energy E_{k} to energy E_{k-n} when it *emits* a quantum of energy. (Atoms with energy E_{o} can only absorb energy. We shall only consider cases where k>>0.)

Now the atoms at energy E_{k} are distributed throughout the volume, as are those at energies E_{k+1}, E_{k+2}, etc. Thus if we consider a randomly selected atom it could be at any of the discrete energies. In order to determine which energy the atom is at we must make a *measurement*. We propose to do this by isolating the atom and then noting what frequency radiation it emits. (We know experimentally that, at all energies other than E_{o}, the atom will emit a photon of energy in a short period of time.) The energy of the emitted photon is related to the frequency of the emitted radiation by E=hf.

Now the atom’s energy levels are not evenly spaced. That is,

(1)

Thus our experiment yields up *two* pieces of information: (1) the initial energy just prior to the emission of a photon, and (2) the final energy right after the emission of a photon.

Although we have no idea what these initial and final energies will be, quantum theory provides a formula for the *probabilities* that the various frequency photons will be emitted. Let us assume a general tenet of the theory is that, if an atom belongs to an ensemble whose wave function coincides with a *particular energy eigenfunction*, then the atom will *certainly* emit a photon with that initial energy. For example, let us say that an atom’s "parent" ensemble’s wave coincides with eigenfunction e_{k}. Then according to the theory it is a virtual certainty that the atom will emit a photon of energy E_{k}-E_{k-1} or E_{k}-E_{k-2} or … In other words, it is a virtual certainty that the atom’s initial energy will be measured to be E_{k}. Of course since atoms of various energies are all mixed together, we do not definitely know what the energies (initial and final) for our selected atom will turn out to be. At least this is true for the *first* emitted photon.

But what if we keep the atom isolated and wait for a *second* photon to be emitted? According to the theory, we *know* what the initial energy in this second measurement will be: *it will be the final energy of the first measurement*. Since this is a virtual certainty it would seem that, following the first measurement, the atom belongs to a "sub-ensemble" whose wave function coincides with a particular energy eigenfunction.

From the point of view of ensembles, then, the above discussion suggests the following scenario. Our mole sample has a wave function (or Hilbert space state vector) Y. This wave function is a *linear combination* of any complete, orthonormal set of basis vectors {a_{i}}:

. (2)

In particular the basis set can be the set of energy eigenvectors {e_{i}}.

Now at least *logically* we can divide our initial ensemble (the mole of gas) into *sub*-ensembles, one for each energy eigenvalue. And all the members of the k’th sub-ensemble have their "sub-Y" equal to a_{k} (times a complex factor of modulus one). At the risk of being redundant, prior to the first measurement we can only know the *probability* that our atom will belong to the k’th sub-ensemble. But after the first measurement we know with certainty (1) which sub-ensemble the atom initially belonged to, and (2) which sub-ensemble the atom transferred to upon emitting the photon. Since we know which energy eigenfunction the sub-Y coincides with following the first emission, we know with certainty what the initial energy will be in the second emission.

One of the practical difficulties of the above-sketched experiment is keeping our atom isolated from the other atoms long enough to obtain the second measurement. For the atoms are constantly exchanging quanta among themselves; i.e., each atom is jumping from sub-ensemble to sub-ensemble. In brief, we can predict with certainty what the initial energy of our second measurement will be only if we make the measurement before the atom can interact with one or more other atoms (or before it can absorb a stray photon).

In conclusion, the very form of Eq. 2 seems to beg the logical partitioning of a sample (or ensemble) into sub-ensembles, each with its own sub-Y coinciding with a particular energy eigenfunction. By doing so we know, after the fact, which sub-ensemble an atom belonged to prior to emitting a first photon. Furthermore, provided a second photon is emitted before the atom leaves its *new* sub-ensemble, we can predict with certainty what the pre-emission energy for the second photon will be.