# Orbital and energy shell transitions

1. Nov 26, 2012

### DiracPool

I've always learned that putting energy-photons into an atom can bump an electron up to a "higher energy state", and that when the electron "falls" back down into a lower energy state, it then emits a photon, and so forth. What I never seem to find, however, in these descriptions are any specifics.

My question, say for a hydrogen atom, is 1) Is this higher energy state an electron gets bumped up to simply a generally higher energy shell? Say n=1 to n=2 or n=4? Or is there some specific orbital within that higher energy shell the electron prefers, like say the Px orbital in n=2 versus the Pz orbital...or, perhaps one of the D orbitals versus a P orbital if the electron is bumped to even a higher energy level. I guess, more specifically, does bumping up an electron in a hydrogen atom actually CREATE one of the p,d,f,etc. orbitals we know are found in larger atoms? Or, alternatively, is it just bumped into some temporary amorphous higher energy "state?"

The second related question is, correspondingly, does the electron have any preferred "path" down to lower energy levels via some orbital hierarchy-cascade? Thanks.

2. Nov 27, 2012

### tom.stoer

I think the answer to your question is the understanding of so-called selection rules.

Think about two eigenstates of a system $|\psi^i\rangle$ and $|\psi^f\rangle$ with i="initial" and f="final" and some additional interaction represented by an operator $\hat{T}$. In non-rel. QM this could e.g. be an electromagnetic field (instead of a photon b/c we do not want to deal with field quantization which is beyond QM and requires QFT).

The transition between the two states induced by the interaction is described by the matrix elements

$$\langle\psi^f|\hat{T}|\psi^f\rangle$$

Now looking at the hydrogen atom states $|nlm\rangle$ and using an electromagnetic wave you get the usual selection rules for different multipoles. Neglecting spin, spin-orbit coupling and other tiny effects one can derive (via symmetry considerations), that an electromagnetic wave representing a dipole results in transitions for which the following rules must hold:

$$|n^i\,l^i\,m^i\rangle\;\to\;|n^f\,l^f\,m^f\rangle$$
is allowed for

$$|l^f - l^i| = 1$$
$$|m^f - m^i| = 0,1$$

The probabilities for these transitions can be calculated from the matrix elements

$$\langle n^f\;l\pm 1\;m\pm 1,0|\hat{T}_\text{dipole}|n^i\,l\,m\rangle$$

For all other combinations, i.e. for transitions violating the dipole rules the matrix element is exactly zero and the transition is forbidden.

http://en.wikipedia.org/wiki/Selection_rule

Last edited: Nov 27, 2012
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