I haven't taken a course on quantum mechanics yet, but I was asked to solve (numerically)(adsbygoogle = window.adsbygoogle || []).push({});

##[-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+V(x)]\phi(x)=E\phi(x) ##

##V(x)=2000(x-0.5)^2##

by supposing the solution is ##\sum_{0}^{\infty} a_n \phi_n(x)## and ##\phi_n(x)## is the typical solution to the a square potential ##\phi_n(x) = \sqrt{\frac{2}{L}} sin(\frac{n \pi x}{L})##.

Now, to solve this I've done the approximation that my sum is actually a finite sum. Doing some manipulations one can show that you can find coefficients via the matrix equation

##\mathbf{M}\mathbf{a}=E\mathbf{a}##.

where ##M_{mn} = E_m \delta_{mn} + \int_{0}^{L} \phi_m \phi_n V(x) dx##.

And ##E_m= \frac{\hbar^2 m^2 \pi^2}{2 M L^2}##

Now, I've implemented a matlab program to solve eigenvalues and eigenvectors for ##\mathbf{M}## and used those coefficients to construct the solution to this problem.

Now, my question ishow do I mathematically know ##\phi(x)## is normalized as well? Regarding the eigenvalues ##E##, does QM say that my system can have any of the eigenvalues as energies when I'm not observing? Is that the interpretation?

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# Matrix method to find coefficients of 1-d S.E.

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