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Illustration Energies of an Electron with Antisymmetric Wave Function in the Finite Potential Box (Graph)

Energies of an Electron with Antisymmetric Wave Function in the Finite Potential Box (Graph)
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Graph of the following transcendental equation:$$ -\cot\left( \frac{L}{2}\sqrt{\frac{2m}{\hbar^2} \, W^-} \right) ~=~ \sqrt{\frac{V_0}{W^-} ~-~ 1} $$

Here, the right and left sides of the equation were plotted (a bit rescaled) separately as a function of \(W^-\). This equation was obtained by solving the Schrödinger equation for an electron in a finite potential well described by a antisymmetric wave function. And \(W^-\) corresponds to the energy of the electron. The intersection \(W^-_1 \) is the only allowed energy of the electron inside the potential box. In this case, the electron can only occupy one energy in the potential box if it has an antisymmetric state.