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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

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

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 question, I plugged the separable solution into (1) and found after some rearrangement:

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

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.

StephenG - Help Ukraine
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user13514
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1 Answers1

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Yes. Remember what role mathematics plays in models of the physical world. Once we have modeled the physical scenario in terms of a mathematical model, we "forget" the physical world and simply solve the mathematical problem presented by the model. Once we have a solution, we can test it against the physical scenario with the aid of experiments to see if the solution is valid.

In this particular case, the solution that you would get as a separable product of functions would be a valid solution of the mathematical problem. However, you would get a whole set of such solutions. The physical scenario that the model describes may be obtained as a superposition of these solutions.

As for solving this particular set of equations, there are various approaches. Much of it would depend on the details of $m^*(x)$ and $V(x)$. So I don't think I can give a generic answer to this part of the question, without more information.

flippiefanus
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