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Illustration Squared magnitude of a wave function (example)

Absolute square of a wave function (example)
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Squared magnitude \( |\mathit{\Psi}|^2\) of the wave function of a particle in the infinite potential well (for \(n=3\)):$$ \mathit{\Psi}(x) ~=~ \sqrt{\frac{2}{L}}\sin\left(\frac{3\pi}{L}\,x\right) $$

The area under the \( |\mathit{\Psi}|^2\) is the probability. The probability between the points \(x=a\) and \(x=b\) is given by$$ P ~=~ \int_{a}^{b} |\mathit{\Psi}|^2 \, \text{d}x $$

The particle is most likely to find at the maxima of the function and most unlikely at the minima. The total area is 1 (normalization condition).