How can I fix a floor joist that has separated from the cross beam?
As stated in everybody's comments, this is not a place to go blindly.

For the joist to "slip out of" its notch, something else has probably moved. This can lead to a house-sized game of "Jenga®" where the players (being you and anybody else in the house) can be crushed by the falling timbers. 

Hire yourself a civil/structural consulting engineer, preferably one that really understands old wooden houses if yours is (as it sounds like it might be) a post and beam (P&amp;B) structure. Those working for new P&amp;B companies might be your best bet, and if they are not, they will usually know someone who does do work with old P&amp;B houses, in most cases.