What does this relation signify?

Question

I recently came across this in social media-

What doe the last equation mean?Is it somehow connected to probability or something?

Thanks a lot for any help or response!!

Answer 1

It's a quantum mechanical view ($\psi$ being the wavefunction) of the glass being both empty and half full simultaneously. The $1/\sqrt2$ being included as some kind of "normalisation" of the wavefunction. This is, in real examples, the link with probability where the normalisation ensures the integral of $|\psi|^2$ over all space is $1$.

Here is another relatively famous superposition of states, the Schrodinger's cat thought experiment/paradox where the cat is in a state of being alive and dead simultaneously.