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, E_{m} and E_{n}, with E_{n} > E_{m}. Assuming spherical symmetry, we can focus upon points along the x-axis. The individual wave functions are

, (1a)

, (1b)

where y_{m} and y_{n} can be real. With both energies simultaneously occupied, the net wave function is then

, (2)

where k_{m} and k_{n} are weighting factors (and theoretically functions of the ambient temperature). "Squaring" Y_{m,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 k_{m}k_{n} = b), we can rewrite Eq. 3 as

(5)

where

, (6a)

. (6b)

Or, defining w_{m,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 q_{e} is the electron’s charge, then when energy states E_{m} and E_{n} are simultaneously occupied, Eq. 10 indicates that part of r (say r_{t}) 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, r_{t}(-x,t) does __not__ equal r_{t}(+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 r_{t}(-x,t) = r_{t}(+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 sample’s 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.