If you already have a 'start date' secured, just show up for work and carry on with your duties with the "I dont use drugs, so my drug test is a non-issue" mindset. In the world of drug testing, no news means good news. There wont be any ticker-tape parades with banners shouting "you passed", because a person is expected to be clean.
Dont bring up the subject of 'drug test' to anyone. You'll be notified one way or another if there were any problems with the drug test.