Schrödinger's equation problem

[fluidistic]

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

$$-\frac{\hbar ^2}{2m} \cdot \frac{\partial ^2 \Psi}{\partial x^2}+V(x) \Psi =E\Psi$$.

The Attempt at a Solution

My idea: replace $$V(x)$$ by $$V(-x)$$ in the equation I just gave and then find $$\Psi$$. Then show that $$\Psi$$ is either odd or even.
I have no idea about the non degenerated eigenvalues of $$\Psi$$...
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 $$c_1y''+gy=c_2y$$ where $$y=\Psi$$ and $$g=V(-x)$$? 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.

 

[diazona]

Is it an equation of the form $$c_1y''+gy=c_2y$$ where $$y=\Psi$$ and $$g=V(-x)$$?

Yep, that's correct. A partial differential equation when there is only one independent variable is no different from an ordinary differential equation.

As for a hint: I think you can use the fact that any function $f(x)$ can be expressed as the sum of an odd part and an even part,
$$f_\text{even}(x) = \frac{1}{2}[f(x) + f(-x)]$$
and
$$f_\text{odd}(x) = \frac{1}{2}[f(x) - f(-x)]$$
Use this along with the principle of superposition (any linear combination of two solutions to the Schrödinger equation with the same energy is also a solution).

 

[fluidistic]

Thank you diazona.
Oh now I remember from calculus I the property you mention in the hint.

If I'm not wrong, the ODE I must solve is $$\Psi ''+\Psi \left ( \frac{g-c_2}{c_1} \right ) =0$$.
All my knowledge on ODE comes from a self study of Boyce-Di Prima's book. I don't really remember how to solve such an equation since g depends on x and isn't constant. If I still remember well, I should propose a solution of a particular form. Problem is, I don't know how to "guess" the particular form of the solution.

And yes, I do know that if I get 2 linearly independent solutions of the ODE, then any linear combination is also a solution.

I'll check out if I can encounter Boyce-Di Prima's book.

Edit: Ok I found the book but I'm still stuck. It seems I can get a second solution and therefore the general form of the solutions if I already know one solution. But since I don't know it, I'm totally stuck.
I realize the solution would be a simple harmonic motion kind of function if g would be constant. But since it's not, I have no idea.