This has already been adressed here: https://www.physicsforums.com/showthread.php?t=173896 , but I still didn't get the answer. The Harmonic Oscillator is fully described (according to my favourite QM book) by the HO Hamiltonian, and the commutation relations between the position and momentum operators. So starting there, we define the ladder operators and show that the energy eigenvalues are: [itex] E_n = (n + 1/2) \hbar ω [/itex] where n is a non-negative integer I know how to do this. But how do we know there are no other eigenvalues? Or equivalently - that the corresponding eigenkets form a complete set? Can we show it without any extra assumptions? I know it can be shown from solving the pde, that all other solutions blow up, but I'd like to do it purely by ladder operators.