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I am solving the Schrödinger equation for a particle in a hybrid system. Specifically I have to solve the following differential equation

$$ - \frac{\hbar^2}{2}\frac{d}{dx}(\frac{1}{m^{*}(x)}\frac{d\psi}{dx}) -\frac{\hbar^2}{2m^{*}(x)} \frac{d^2\psi}{dy^2} + V(x)\psi = E\psi \tag{1} $$

I would like to know: Is this problem separable? I.e. can the solutions be written as $\psi(x,y)=\Phi(x)\chi(y)$? To investigate this we plug it in into the differential equation above and find after some rearrangement:

$$ \frac{1}{\chi(y)}\frac{d^2\chi(y)}{dy^2} + \frac{m^{*}(x)}{\Phi(x)} \cdot \frac{d}{dx}(\frac{1}{m^{*}(x)}\frac{d\Phi}{dx}) + 2m^{*}(x) \cdot \frac{V(x)-E}{\hbar^2} = 0 $$

Now, the first term depends only on y while the two other terms depend only on x. Therefore one can separate the above into two differential equations, a simple one for $\chi(y)$ and a nasty one for $\Phi(x)$. Does this prove that the solutions of (1) are separable? And say they are, how would I go about calculating the solutions to the differential equation for $\Phi(x)$. The spatial dependence of the mass $m^{*}(x)$ is rather annoying.

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The solutions can't all be written in that form, nor would one expect them to be written in that form. For example, since this is a linear equation, you could have a linear combination of different solutions of that form. The most you can hope for is that solutions of the form $\Phi(x) \chi(y)$ are in some sense a basis of the space of solutions.

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