Why does a photon emit discrete frequencies of light?

[enwa]

Why does an atom emit discrete frequencies of light?

Solving the Schrodinger wave function for the hydrogen atom (that is a single particle representing an electron bound by a spherical potential) we find that it has discrete energy levels. Plotting every possible value of $f$ in $E'-E = \hbar f$ where $E'$ and $E$ are the different energy levels of eigenstates we recover the emission spectra.

What I don't understand is why we only see these discrete energy levels. According to the superposition principle, the wave function could be in a superposition with expected energy $\tfrac{1}{2}(E'+E)$ but differences from these levels don't show up on the emission spectra. The measurement postulate seems relevant, "measurement" collapses a wave function into an eigenstate, this seems to be happening before and after the photon emission, can anyone explain why?

 

[Drakkith]

I don't think there is an answer as to WHY electrons can only move between orbitals and energies in values of h, just that they do. Plank discovered this constant without having any answer to why it was this way and that particular value, only that it was.

 

[LearningDG]

The reason is that a photon in a superposition of (|E1>+|E2>)/sqrt(2) is not the same as a photon with energy (E1+E2)/2.

 

[PhilDSP]

If you're looking for a physical rationale one of the clearest is in David Bohm's "Quantum Theory" chapter 6 or thereabouts (don't have the book here)

 

[KOSS]

To elaborate on LearningDG's remark, while superpositions are allowed and inevitable, when photon emissions are measured the wavefunction is collapsed according to the Born principle, which picks out a particular discrete pair of energy levels.

All the superposition tells you is the probability of particular transitions. When you observe a transition (photon emission) it will almost always correspond to the energy difference of two levels. Why so? I'm not sure it is any more than a fundamental empirical fact---this is how nature is, so it is a postulate of QM. That's why I have to say "almost always", simply because no one has ever observed otherwise, but there is no known necessity for nature to be this way.

You could ask the further question "why then are the energy levels discrete?": the answer is again deeply one of the empirical facts about nature, but we understand it in terms of quantum theory as a result of the wave nature of particle states, i.e., since the electrons are wave-like they exist in stationary orbitals only by virtue of being held in 3D standing wave formation. If they were not standing wave solutions they would not be stationary, they'd instead be time-dependent. Hence the mathematics for time independent solutions dictates spherical harmonics (standing waves in 3D), these are labeled by integer solutions, hence discrete energies. Most students understand this first from the simpler mathematical analysis of an idealized quantum particle in a box (infinite square well potential).

If you were to relax the requirement of stationary (i.e., time independent) states then presumably the wave function solutions would no longer necessarily correspond to spherical harmonics, and the energy levels could be all over the place. This might be set up, say for example, if you bombard an atom with an intense laser, or impose some other kind of drastic nonlinear dynamical perturbation of the atomic electrons. Good luck to you if you want to calculate the energy levels and transition amplitudes in such situations!

 

[Ken G]

I think one can give a kind of reason as to "why this is." It has to do with the importance of interference and resonance in wave mechanics, and the presence of a quantized action. An atom emitting light is usually only perturbed a very tiny amount, such that the perturbation wouldn't amount to squat over a single cycle of the atom's periodic oscillation (periodicities in the wave function are pretty much the main crux of orbital quantum mechanics). So to get the atom to emit light, one has to "tickle" it over and over, in synch with the natural frequencies that can yield constructive interference within the atom. That's called a resonance, it's pretty much why a guitar string emits notes rather than white noise. Indeed, your question is quite intimately related to "why do musical instruments emit sound at special frequencies." It comes from the combination of having a resonant frequency in the system, and obeying the interference-regulated rules of waves. Now, it should be noted that guitar strings can be any length, so can make any note, but on the atomic scale we encounter the quantization of the action-- the scale is small enough to probe the minimum scale on which action can appear. That's what creates "guitar strings" in atoms that cannot be just any length.