Qualitative plots of harmonic oscillator wave function

[eprparadox]

For the harmonic oscillator, I'm trying to study qualitative plots of the wave function from the one-dimensional time independent Schrödinger equation:

$$\frac{d^2 \psi(x)}{dx^2} = [V(x) - E] \psi(x)$$

If you look at the attached image, you'll find a plot of the first energy eigenfunction for the HO. In the image, for x > 0, the the wave function is greater than 0 and the graph is concave down.

My question is: why can't the wave function be less than 0 and have the graph be concave up for this first eigenfunction? Put another way, why can't we simply mirror the attached image wave function over the x-axis and have that be a valid first energy eigenfunction of the HO?

Is it just a choice of initial conditions of psi and the derivative of psi and that we chose them to be positive?

 

[phyzguy]

Remember that the wave function is complex and a function of time. You have only plotted one part of it (the real part perhaps) at one time. If you wait half a cycle, or if you multiply it by an arbitrary phase $e^{i \pi}$ then it will look like what you suggested.

 

[eprparadox]

Hey great points, thanks so much.

Just to confirm:

If I were presented with the wavefunction in the attached image above as well as the one I described and was asked to pick which of the two was a possible wave function for the first energy eigenfunction of the harmonic oscillator, then both are reasonable, correct?

 

[phyzguy]

Hey great points, thanks so much.

Just to confirm:

If I were presented with the wavefunction in the attached image above as well as the one I described and was asked to pick which of the two was a possible wave function for the first energy eigenfunction of the harmonic oscillator, then both are reasonable, correct?

I think so, yes.