Absorption Paradox: Continuous Photon Energies and Quantized Molecule Energies

[p.tryon]

The possible energies of photons is continuous and therefore a photon could have an infinite number of possible numerical values for energy. In contrast all forms of molecular energy are quantized (electronic, rotational, vibrational and translational). Surely the probability of any given photon having EXACTLY the right energy to be absorbed by a molecule should be negligibly small (effectively zero) since there is a finite number of possible molecular energies and an infinite number of possible photon energies. Any finite number divided by infinity is zero.

I know my reasoning must be incorrect in some way but as yet no one has given me a good explanation of why so I though I would ask here.

 

[meopemuk]

The possible energies of photons is continuous and therefore a photon could have an infinite number of possible numerical values for energy. In contrast all forms of molecular energy are quantized (electronic, rotational, vibrational and translational). Surely the probability of any given photon having EXACTLY the right energy to be absorbed by a molecule should be negligibly small (effectively zero) since there is a finite number of possible molecular energies and an infinite number of possible photon energies. Any finite number divided by infinity is zero.

I know my reasoning must be incorrect in some way but as yet no one has given me a good explanation of why so I though I would ask here.

Energy levels in atoms, molecules, etc. are not truly discrete. There are several mechanisms for the level broadening, e.g., temperature. But even a single atom at T=0 does not have discrete energy levels. All excited levels have "natural widths" which are inversely proportional to their lifetimes. If a level had a sharp energy (the width is zero), then its lifetime would be infinite and the probability of emitting/absorbing a photon from/to this level would be zero, just as you said.

 

[p.tryon]

Ah right. Thank you.

 

[DeShark]

Energy levels in atoms, molecules, etc. are not truly discrete. There are several mechanisms for the level broadening, e.g., temperature. But even a single atom at T=0 does not have discrete energy levels. All excited levels have "natural widths" which are inversely proportional to their lifetimes. If a level had a sharp energy (the width is zero), then its lifetime would be infinite and the probability of emitting/absorbing a photon from/to this level would be zero, just as you said.

I know the OP was ok with this reply, but I was thinking the same thing about a year ago and decided to drop it, with arguments made to myself about "It must be some relativistic effect or something that doesn't come into shroedinger equation solutions". So then, is this increase in temperature (i.e. increase in energy, both translational and internal) responsible for a higher velocity of the molecule or atom, which in turn will doppler shift the energies? And at T=0, I presume that there's still some translational motion going on, which explain why you can never freeze out the "blurryness" of the energy levels. I also don't understand why a sharply defined energy means that the lifetime would be infinite. Obviously a stationary state has an infinite lifetime, but that's because the potential is constant in time. A purturbation to the potential (like a photon) would break the stationaryness of the state and prevent that from being the case.

 

[meopemuk]

I know the OP was ok with this reply, but I was thinking the same thing about a year ago and decided to drop it, with arguments made to myself about "It must be some relativistic effect or something that doesn't come into shroedinger equation solutions". So then, is this increase in temperature (i.e. increase in energy, both translational and internal) responsible for a higher velocity of the molecule or atom, which in turn will doppler shift the energies? And at T=0, I presume that there's still some translational motion going on, which explain why you can never freeze out the "blurryness" of the energy levels.

Yes, Doppler effect is one factor for the temperature level broadening. I am not sure if this effect reduces to zero at T=0, but I can believe that. Also note that in macroscopic bodies (e.g., crystals) energy levels are not discrete, but form continuous bands, so that photons of various energies can be absorbed there even at T=0.

I also don't understand why a sharply defined energy means that the lifetime would be infinite. Obviously a stationary state has an infinite lifetime, but that's because the potential is constant in time. A purturbation to the potential (like a photon) would break the stationaryness of the state and prevent that from being the case.

Only the ground state of any isolated stable system has sharply defined energy. All other (excited) levels are unstable and eventually emit photons and drop onto the ground level. This instability is the result of ever present interaction with the photon subsystem. As for the relationship between the lifetime and the linewidth, search for the "Breit-Wigner formula".

 

[Joskoplas]

Then there's the uncertainty principle itself. When you translate it from position and momentum to energy and time, you get

$$\Delta E \Delta t \ge \frac{\hbar}{2}$$

So, if we could force the energy level to be more precisely defined, delta E would go down. But since it has to multiply by delta t to be greater than a constant value, that means delta t must increase. Therefore, more precise energy means longer lifetime.

Can't run it that way, though. There's a certain lifetime and you have to take the hit on the precision. You could call it the "natural linewidth" for that transition.