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Main Question or Discussion Point
For the harmonic oscillator, I'm trying to study qualitative plots of the wave function from the onedimensional time independent schrodinger equation:
[tex] \frac{d^2 \psi(x)}{dx^2} = [V(x)  E] \psi(x) [/tex]
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?
[tex] \frac{d^2 \psi(x)}{dx^2} = [V(x)  E] \psi(x) [/tex]
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?
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