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If the following list of functions defined on $x \in [0,L] \,$ $$\phi_n(x) \equiv \sqrt{\frac{2}{L}}\sin\left(\frac{n\pi}{L}x\right)$$ where $n$ is positive integer and the functions are zero at the boundary points.

The question is to show that the following identity $$-\frac{h^2}{2m}\frac{\partial^2\phi_n(x)}{\partial x^2} = E_n\phi_n(x)$$

So here can we assume that $V$ is zero to be able to reach the desired goal ? Because in the question there is no information about $V$.

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  • $\begingroup$ The infinite well problem usually states that $V(x)=0$ for $x \in [0,L] \,$. But to prove the identity you'll need to know the value of $E_n$ $\endgroup$
    – Gert
    Nov 4, 2020 at 20:08
  • $\begingroup$ I don't understand. Is the question explicitly telling you to show the differential equation is valid? Or did they say to show the TISE is valid but they didn't tell you what $V$ is? $\endgroup$ Nov 4, 2020 at 20:15
  • $\begingroup$ Is this your question -- Given a wave function can you determine the potential for the system? $\endgroup$ Nov 4, 2020 at 20:30
  • $\begingroup$ Actually the question is derive the second equality by only knowing the given list of functions. $\endgroup$
    – malachi
    Nov 5, 2020 at 8:37
  • $\begingroup$ Yes! @BioPhysicist. Interestingly, it is wanted the show the second equality by only using the given list of functions. $\endgroup$
    – malachi
    Nov 5, 2020 at 9:39

1 Answer 1

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You could quite easily reformulate the Schrodinger equation for a finite potential within an infinite potential well in one dimension (Note: I do no not mean finite well height, I mean a finite potential within an infinite potential well). Here, I am using $\psi(x)$ as a trial solution.

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

The spatial dependence of $V$ is then:

$V(x) = V_0$, for $x \in [0,L]$

$V(x) = \infty$, for $x < 0, x>L$

$V_0$ is a constant as the well has a constant potential at the base. Restricting the domain of the differential equation to $[0,L]$ as well as specifying the boundary conditions of $\psi(x)$ to vanish at the boundaries, the equation can then be simplified.

$\frac{-\hbar^2}{2m} \frac{\partial^2}{\partial x^2}\psi(x) = (E-V_0)\psi(x)$, $x \in [0,L]$

This simply transforms $E_n$ to $E-V_0$. From here you can insert your solution, the actual value of the potential inside the well becomes irrelevent as the eigenvalue energy just gives the energy with respect to the base of the well.

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  • $\begingroup$ Not sure what was wrong with my answer, I am quite new to this. Feedback would be appreciated. Cheers. $\endgroup$ Nov 4, 2020 at 21:54
  • $\begingroup$ Actually, in the question there is no knowledge about a well. I am just guessing it because I know the infinite well potential solution. The only given thing in the question is the list functions. $\endgroup$
    – malachi
    Nov 5, 2020 at 9:12
  • $\begingroup$ Okay then, I think the other answers comments cover well enough. $\endgroup$ Nov 5, 2020 at 15:42
  • $\begingroup$ Thank you all :) $\endgroup$
    – malachi
    Nov 6, 2020 at 17:11

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