Hi, I've been reading about the use of the lowering and raising operators to solve quantum mechanical problems and I have this question in mind. In the book "Introduction to Quantum Mechanics", by David Griffiths, page 35, he notes out that there should be only one energy ladder, so what we obtain as energies is the only possible solutions. This pulls me to the following questions. 1. How did we make sure the ladder we obtained is the only energy ladder? 2. If this is true, this means that using the raising operator will get me the subsequent higher energy level. Is this true? How could I make sure there are no solutions between the points I obtained on the ladder?