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This paper asks an experimental question about photon absorption and emission in a two-state electronic system where the incident frequency of exciting light is below the mean frequency of the emission line but still within the line width. Since it is well known that the absorption spectrum and the emission spectrum are identical [1]-[3], then quantum mechanics as well as astronomical observations say that excitations will still occur. Once a quantum state excites, then it is expected to decay back with a typical time constant and the usual emission spectrum. The problem is that the mean frequency of that emission spectrum will then be higher than the frequency of the exciting photon, suggesting that more energy is emitted than being received on average. A search of quantum mechanical references finds no discussion of this issue, so this paper considers the options available to resolve this inconsistency, including an experiment.

It is well known that an excited electronic state emits a spectrum of light with bandwidth proportional to 1/τ, where τ = the decay time constant of the excited state [

A heavy two state electronic system (such as used in Einstein’s 1917 paper on stimulated emission) [_{o} of the Lorentz spectrum corresponds to emitting a photon whose energy exactly equals the energy change in the quantum electronic transition. Incident photons with frequencies below that peak have less energy than the electronic state and those above that frequency have more since the energy of a photon equals hν.

To explain the range of frequencies of light emitted from a fixed state transition, quantum theory tells us that the finite decay time of the excited state creates an uncertainty in the energy of the state, resulting in a spread of energies in the emitted and absorbed photons. Mathematically, the Fourier transform squared of the exponential decay amplitude produces the Lorentz line shape-commonly observed. [

This paper proposes an experiment shown in

1) If they must be equal, then a precisely determined excitation energy should imply a precisely determined emission energy from the decaying state. This will change the resulting line shape to be much narrower than usual, resulting in much longer coherence length for the emitted light. This can easily be measured in a scanning interferometer as shown in

2) The other option is that the resulting emission after monochromatic excitation is unchanged compared to any other emission. This would mean that any excitation, no matter what frequency input, results in the same type of excited state-much simpler from that point of view. In this case the line shape of the emitted light will be the usual one determined by its standard decay time. However, if we consistently excite with a frequency ν_{ex} below the mean Lorentz frequency ν_{o} while getting a mean frequency of ν_{o} re-emitted, then we must ask about energy conservation. How can we, on average, get h(ν_{o} − ν_{ex}) more energy emitted out than we put in?

In this experiment we irradiate a cold gas with a wavelength tuned to a narrow line of frequency ν_{1}, which is below the mean frequency ν_{o} of the Lorentz line but within the width of the Lorentz line as shown in

We note that this experimental configuration is very similar to Doppler cooling of atoms [

A simple question has been asked about the interaction between how an electron is excited and its resulting decay. Quantum mechanics does not contain such an interaction at present, but the absence of such an interaction leads to non-conservation of energy or an unknown process to balance the input and output energy. Since experiments always have the final word, it would be useful to know which path to follow.

This discussion was intended to motivate an experimental investigation of the excitation and emission of excited states. Does the emission depend at all on the excitation process? If not, how can we maintain conservation of energy? We could just as easily have focused on excitation energies larger than the mean value of the Lorentz spectrum and asked where the extra energy went. Answers of uncertainty will not suffice here, because the energy of an exciting photon of frequency ν_{ex} is known to be exactly hν_{ex}, and the mean energy of the emission back to the ground state is also accurately measured by the interferometer. By irradiating the atom with that known energy and measuring the frequency distribution of the emitted light, we can precisely determine energy balance or its lack. Either outcome will provide new insight into electronic transitions. Our hope is that an experimentalist with appropriate skills and equipment will provide the answer.