Turning in your two weeks notice can be terrifying__because no matter how well you think you know your supervisor, you can never be 100% sure of his or her reaction. Will he be furious and insist on walking you out the door immediately? Will she beg you to stay, enticing you with an enormous raise? Or will he be completely civil and wish you the best of luck?
However, at most companies, there's no formal requirement that you do so. Most employment in the United States is called "at will", which means that your current employer can fire you at any time and for any reason (except those that are prohibited under law).