http://en.wikipedia.org/wiki/Atomic_orbital
Think of a violin string as an analogy: the ends are constrained, so it can have only certain tones...certain vibrational patterns and associated energies. it's energy levels are constained to certain values...it's degrees of freedom are limited.
Another helpful analogy is to think of the electron as a wave...when it's in free space the wave is everywhere, it extends all over the place. But when attracted by a proton in a nucleus, for example, that wave is now localized...it's constrained and so its different from the free space case. And the constraint is also modified by the presence of other electrons and additional protons. Since the energy is contained in the wave, changing it's configuration via the presence of nearby particles changes the wave characteristic and likely energy levels. It's very unlikely for the electron to be found between allowed energy levels.
When a particle is part of an atom or a larger structure, it's constrained...it's degrees of freedom are determined and limited by the whole structure. So electron energy levels and degrees of freedom are determined by the numbers of protons in the nucleus as well as the particular structure of a lattice, as examples. The Schrodinger wave equation describes these.
tomstoer posted this related description:
In an atom it's not the [bound] electron that absorbs the energy of the incoming photon but the whole atom; the usual QM description of the hydrogen atom is a bit misleading here b/c it treats the proton classically, but it should be clear that a more realistic picture is a two-particle Schrödinger equation where the proton-electron system as whole can absorb the photon whereas a single, free electron can't due to energy-momentum conservation... for a particle to absorb a photon you need internal degrees of freedom which can be excited. A free electron can't absorb a photon due to the non-existence of these inner degrees of freedom. An electron bound in an atom can b/c the whole atom (proton-electron bound state) provides these inner degrees of freedom.
Think of a violin string as an analogy: the ends are constrained, so it can have only certain tones...certain vibrational patterns and associated energies. it's energy levels are constained to certain values...it's degrees of freedom are limited.
Another helpful analogy is to think of the electron as a wave...when it's in free space the wave is everywhere, it extends all over the place. But when attracted by a proton in a nucleus, for example, that wave is now localized...it's constrained and so its different from the free space case. And the constraint is also modified by the presence of other electrons and additional protons. Since the energy is contained in the wave, changing it's configuration via the presence of nearby particles changes the wave characteristic and likely energy levels. It's very unlikely for the electron to be found between allowed energy levels.
In contrast, a free electron can take on any energy level. But when it is part of an atom or a larger structure, it's constrained...it's degrees of freedom are determined and limited by the whole structure. So an electron's energy levels and degrees of freedom are determined by the numbers of protons in the nucleus as as well as the particular structure of a lattice, as examples. The Schrodinger wave equation describes these.
