What is the point of absolute value squared in wave function's probability density?

Question

The wave function is defined as $$ \Psi(x,t) $$ To get the probability, they squared it with a modulus bracket $$ |\Psi(x,t)|^2 $$ Because amplitude can also be -ve but the probability cannot be. My question is, What is the actual point of both mod and squaring together?

Answer 1

This is physics. $\Psi$ can have complex values. So its square need not be positive.

In quantum mechanics the wave function must be complex. We (so far) have not found how to do it with only real values.

Answer 2

Keep in mind $$ |\Psi(x,t)|^2 = \Psi(x,t)^*\Psi(x,t)$$

Where $*$ indicates complex conjugacy. So the modulus is guaranteed to be real by this definition of the inner product.

The modulus indicates the probability density, the chance of finding the particle between $x$ and $x+dx$.