How to use a graph of a wavefunction (psi) to create an approximate graph for potential (U)


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I don't understand how the top graph was used to approximate the bottom graph, could someone please explain to me what was going on?

I know that in regions 1, 3, 5, 7, there is no oscillation and therefore E<U, but I don't know what other information I can extract from the top graph.

Best Answer

I figured it out. From the Schrodinger wave function, psi, we know the following:


Where \$ E > U, ~~~ {\psi(x)}=Ae^{jkx}+Be^{-jkx}\$

Which is sinusoidal/oscillating.

Where \$ E < U, ~~~ {\psi(x)}=Ae^{kx}+Be^{-kx}\$

Which is not sinusoidal, it's exponential.

Now note in the first graph, regions 2, 4, and 6 have an oscillating wave and therefore E > U. On the contrary, regions 1, 3, 5, and 7 aren't oscillating and look exponential. So for each of these regions, they have a corresponding estimate whether U should be above or below \$E\$.

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