I was in a similar situation few year ago. I started a new job and 2 weeks into orientation realized I do not like anything about it. I called superviser at my old job and she said that she wants me back, asap. I was hesitant to leave without giving an advance notice, but a friend (also an RN ) told me that since I was still on orientation/probation, I probably did not have to give a notice any way. That the probation goes both ways, they are checking me out, I am checking them out. And it is not like I have vacation time to lose.
Anyway, I approched ADN the next day, and told her that I do not feel this job is working out for me for a number of reasons, and since I made up my mind about leaving, I do not see why they should waste their time orienting me. I also told her that I want to return to my old job. I said everything nicely, told her how great everyone was to me ( that was true), and she was fine with letting me go.
I do not think that after being on the job for only 1 month you are obligated to give any lengthy advance notice. While you would definately not want to burn bridges you have to look out for yourself.
BTW, I also always gave notice otherwise.