In quantum physics, the relation
$$ \int_{-\infty}^{\infty} (\psi[x,t]^*)(\psi[x,t]) dx=1 \tag{1} $$
is paramount. What would the consequence be of defining the normalization condition as
$$ \int_{-\infty}^{\infty} \sqrt{(\psi[x,t]^*)(\psi[x,t])}dx=1 \tag{2} $$
Of course, it goes without saying the mathematics will now be more complicated due to the square root.
However, (2) is simply the complex norm and thus I feel it is closer to my natural intuition on how complex probabilities ought to connect to real probabilities. So is there at least a trade-off, or perhaps an equivalence?
