A while back I had a thought about discrete energy levels. I emailed Dr. Garoff about it, and he explained the error in my argument.
The problem was that the energy levels are not infinitely narrow. They have a spread of at least the uncertainty principle. In addition, Dr. Garoff stated that the idea that the photon has to match the energy exactly is only correct to first order. Higher order perturbation theory allows for a widening of the acceptable photon energies.
There are similar sort of transitions in the nucleus (I think Schrodinger's equation can be applied to any sort of motion - just plug in the appropriate potential and Hamiltonian) that is slightly different. While I know that all particles obey the uncertainty principle, it is awkward to think about the uncertainty principle being applied to neutrons and protons since they are 1800 times heavier than an electron. However though, neutrons are the electrons of the nucleus; they move about in discrete transitions in the nucleus.
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