Here’s my five year old thread on the topic. The consensus among people who hired for technical positions was that a content-free thank you doesn’t help - no one is waiting for a “thank you” email and giving you a downcheck if you don’t send one - and it might possibly hurt. Bone, markn_1, and Pantastic all recommended against sending one.
If you have actual relevant information to send, then sure send it ASAP. This isn’t dating, they’re not going to think you’re too eager because you called back the next day.
And no, don’t send anything via USPS, unless specifically instructed to do so.