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fluidistic
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Homework Statement
Show that for a unidimensional potential of the form V(x)=v(-x), the solutions to the time independent Schrödinger's equation have a defined parity as long as these solutions does not correspond to eigenvalues not degenerated.
Homework Equations
[tex]-\frac{\hbar ^2}{2m} \cdot \frac{\partial ^2 \Psi}{\partial x^2}+V(x) \Psi =E\Psi[/tex].
The Attempt at a Solution
My idea: replace [tex]V(x)[/tex] by [tex]V(-x)[/tex] in the equation I just gave and then find [tex]\Psi[/tex]. Then show that [tex]\Psi[/tex] is either odd or even.
I have no idea about the non degenerated eigenvalues of [tex]\Psi[/tex]...
I'm stuck on starting to solve the equation. I'm a bit familiar with differential equations but not with partial ones.
Is it an equation of the form [tex]c_1y''+gy=c_2y[/tex] where [tex]y=\Psi[/tex] and [tex]g=V(-x)[/tex]? I realize I've no idea why I even bothered changing V(x) for V(-x).
I don't really know what to do. I'd like a little push.