I am referring to the section The Harmonic Oscillator in Griffiths's introductino to quantum mechanics (the older edition with the black cover). I understand how it all works, however there is a part that I am not sure about. How do we know when we apply a- or a+ (the ladder operators) to a state (to get the immediately lower or higher stationary states, as Griffiths stated) we are not skipping some other possible stationary states? how do we know this method finds all stationary states? Thanks!