Twelve years ago, when I interviewed at teh prosecutor's office, i did not send a tank you note, but we scheduled a second interview that day. And I was offered the job at the second interview. The new job (magistrate), i was offered the position at the interview.
The whole idea of the thank you note is to favorably remind the interviewer of who you were, and try to bump yourself up on the list through a kind act. I would advise doing so these days, especially as Charby said, if you want the job. Couldn't hurt.