I'm reading Griffiths's Introduction to Quantum Mechanics 3rd ed textbook . On p.43, the author explains:
What if I apply the lowering operator repeatedly? Eventually I’m going to reach a state with energy less than zero, which (according to the general theorem in Problem 2.3) does not exist! At some point the machine must fail. How can that happen?
We know that $a_-ψ$ is a new solution to the Schrödinger equation, but there is no guarantee that it will be normalizable—it might be zero, or its square-integral might be infinite. In practice it is the former: There occurs a “lowest rung” (call it $ψ_0$) such that $$a_−ψ_0 = 0 $$
I understood why $a_−ψ_0$ should not be normalized. But why should it be non-normalized like $a_−ψ_0 = 0$? As the author mentioned in the book, the possibility of its square-integral value being infinite may also exist (satisfying with the non-normalizable condition). The author went over this point, and I wonder what happens to the case I mentioned.
Griffiths, D. J.; Schroeter, D. F. Introduction to Quantum Mechanics 3rd ed; Cambridge University Press, 2018. ISBN 978-1107189638.