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Illustration Squared magnitude - Normalization condition - Integration over the entire space

Absolute square (example) - normalization condition results in probability 1
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For the statistical interpretation of quantum mechanics (Copenhagen interpretation) to make any sense at all, the normalization condition must be fullfilled:$$ \int_{-\infty}^{\infty} |\mathit{\Psi}(x,t)|^2 \, \text{d}x ~=~ 1 $$

The normalization condition states that the probability of finding the particle somewhere in space must be 1. The integral is the area under the \( |\mathit{\Psi}(x,t)|^2 \)-curve.