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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,anysuch 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?

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# Is the ground state of a harmonic oscillator unique?

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