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$\psi$ does carry a great amount of physical meaning. The annoying part is, that it doesn't represent anything directly observable. The way (non-relativistic) quantum mechanics is constructed, makes $\psi(x_i, s_i, t)$ ($x_i$ - physical coordinates of $i$-th electron, $s_i$ - spin projection on $z$ of said electron) - a direct analogue to trajectory, or, in ...

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