I studied this from Griffith Chapter 2, with the algebraic (raising and lowering operator) method, we reached the ground state by setting a_Ψ0 = 0 , then we got what the ground state is, and then plugged it in the Schrodinger equation to know the energy, and it turned out to be 0.5 ħω. My question is, if the lowering operator decreases the energy every time with ħω, how come the E0 isn't ħω ? so that when the lowering operator acts on it, it gives E=0 . I know I am getting something wrong here but I can't put my hand on it. Note: I am not familiar with the Dirac notations. If you planned on explaining using them.