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Simultaneous Energy States and Non-Radiating and Radiating Atoms

G.R.Dixon, 10/16/2006

As briefly mentioned in a previous article, the belief that any system has one and only one energy at any instant in time can be considered to be part and parcel of the classical paradigm. For reasons already discussed, there is little if any real justification for imposing such a proposition on microscopic systems.

The apparent alternative would appear to be that the system occupies a set of accessible energies simultaneously. In quantum mechanical terms, this implies a sum of weighted wave functions, one for each accessible energy state. Since atoms are microscopic systems that do not constantly radiate, it becomes a matter of interest what conditions must exist when an atom does (and does not) radiate.

Let us consider two simultaneous Hydrogen atom energy states, Em and En, with En > Em. Assuming spherical symmetry, we can focus upon points along the x-axis. The individual wave functions are

, (1a)

, (1b)

where ym and yn can be real. With both energies simultaneously occupied, the net wave function is then

, (2)

where km and kn are weighting factors (and theoretically functions of the ambient temperature). "Squaring" Ym,n produces

(3)

Since the fixed, positive nucleus potential is time-independent, it is generally true that

. (4)

Hence the first two terms in Eq. 3 are independent of time. Abbreviating their sum as "a," and assuming the weighting factors are constant (say kmkn = b), we can rewrite Eq. 3 as

(5)

where

, (6a)

. (6b)

Or, defining wm,n to be

, (7)

we can say that

. (8)

But

, (9a)

. (9b)

Thus

. (10)

Now if, in a Hydrogen atom, the negative charge density is

, (11)

where qe is the electronís charge, then when energy states Em and En are simultaneously occupied, Eq. 10 indicates that part of r (say rt) is time-dependent:

. (12)

The question is, under what conditions does such a time-varying part of the total charge density not radiate, and under what other conditions does it radiate?

Let us first consider a case where

. (13)

This condition is satisfied if the points of maximum negative charge density alternately move away from and toward the origin (or nucleus). In 3 dimensions such a time-varying distribution could be modeled as a spherical shell whose radius oscillates between minimum and maximum values. Since the B field of such a pulsating shell is constantly zero, there is no radiation.

If, on the other hand, rt(-x,t) does not equal rt(+x,t), then the combined (positive) nucleus and the distributed electronic charge could be modeled as an oscillating electric dipole. And oscillating dipoles do radiate.

The above discussion can be generalized to more than two simultaneously occupied energy states. In general, so long as rt(-x,t) = rt(+x,t), there will be no radiation. But if this equality is perturbed, then the atom will emit (or absorb) radiant energy.

It should perhaps be borne in mind that, given a multi-atom sample, the weighting factors are common to all the atoms in the sample. If the sample is in thermal equilibrium with its environment, then the factors do not vary in time, notwithstanding the fact that the constituent atoms may be exchanging quanta of energy. But if the environment temperature exceeds the sampleís temperature, so that the sample is absorbing energy from the environment, then the weighting factors are skewed upward. (Conversely, they are skewed downward if the sample is hotter than its environment.)

For present purposes, the manner whereby a spherically symmetric electronic charge distribution might be perturbed will remain an open issue. But it seems clear from the Franck-Hertz experiment, for example, that certain perturbation thresholds must be met or exceeded in order for energy emission/absorption to occur. Quite as a given electron orbital mode must meet discrete criteria first enunciated by Bohr et al, the energy emission/absorption situation does not constitute a continuum. Evidently either (a) perturbations falling short of the "quantum threshold" do not result in energy emission/absorption, or (b) inadequate environmental stimuli are ignored by the atom, and result in no perturbation at all.