Origins
Excitement was not always welcome. When excite entered English in the 14th century, it meant 'to stir up' or 'to provoke' — closer to agitate than to thrill. An excited crowd was a dangerous one. The purely positive sense — eager anticipation — only became dominant in the 18th century.
The word comes from Latin excitāre, a frequentative of exciēre: ex- ('out') + ciēre ('to set in motion, to call, to summon'). To excite was to call something out from rest into action.
The Latin ciēre, from Proto-Indo-European *ḱey- ('to set in motion'), generated a large English family. To cite is to call a source into evidence. To recite is to call words back. To incite is to set someone in hostile motion. Resuscitate means 'to stir up from below again' — re- + sub- + citāre.
Latin Roots
The physics sense of excite preserves the original Latin meaning precisely. An excited electron has been moved to a higher energy state — called out of its ground state into motion. An excited atom is one that has been roused.
Solicit also belongs to this family. Latin sollicitāre meant 'to shake the whole of something' — sollus ('whole') + citāre ('to set moving'). To solicit is to thoroughly stir someone into responding.