Question

how does uncertainty relation account for natural width of spectral lines

Answer 1

Lets take an electron which is "in orbit" around the nucleus. As far as I know, its (ANY conceivable) wavefunction can be represented as a superposition of energy eigenfunctions, which correspond to the discrete eigenvalues of the electron's enrergy. What I do not understand is where the natural line width of spectral lines comes from. I've heard that it's a consequence of the energy time uncertainty relation. I think I understand the position-momentum uncertainty relation, as both x and p take on a continuous spectrum of values. The energy levels in an atom are, on the other hand, DISCRETE. My understanding of this is that an electron can be in one energy level, or another, or in a superposition of several. Natural line width seems to imply that energy can have a CONTINUOS range of values. As I have mentioned previously, I do not quite get how the energy-time uncertainty relation applies to the case of DISCRETE energy values. Could someone explain this? Now, it is usually said that an electron drops from one enegy level to another and emits a photon. But, due to the time-energy relation, doesn't the electron have to be in a superposition of at least a couple energy levels unless we wait for an infinite amount of time? So, the difference in energy eingenvalues of the electron has some EXACT value, and the electron is in a superposition of these eigenstates, although the energy of the emitted photon does not exactly correspond to this energy difference? After the emission, the atom (electron) is only APPROXIMATELY in its ground state, right (i.e. the COEFFICIENT of the higher energy level eigenfunction is not exactly 0).