Hi, In one of my advanced quantum mechanics classes, the instructor posed a problem, namely to show that the ground state of a one dimensional quantum harmonic oscillator is unique, without getting into differential equations. I know that the equation [tex]a\left|0\right\rangle = 0[/tex] when written in the position space representation gives a simple differential equation, the solution to which is the familiar ground state Gaussian wavefunction. So, any such state which is a solution to the differential equation must be the same (up to a phase, which we can fix by a normalization choice). But how do you reason without using the differential equation approach? I was thinking about arriving at a proof by contradiction somehow, but it does not seem to me that the ground state being unique is a theorem (unless I am missing something more fundamental here), but rather a postulate. I digged into PF archives and found vanesch's statement that the ground state isn't in fact unique. The references are https://www.physicsforums.com/showthread.php?t=173896 and https://www.physicsforums.com/showpost.php?p=1356434&postcount=2 So, is the question wrong?