In QM wavefunctions are written down as it’s square is the probability distribution where you find the particle in the space. Is it possible to flip the space axis to time axis and this way you describe the probability distribution in time that the particle was in exactly that point in space?
Yes, you can. As a matter of fact, time-dependent QM was first developed.
Since this is a well-documented subject (it is even in books), you may ask ChatGPT for a detailed explanation, which will also solve additonal inqueries you may have or will clarifications in terms you are more familiar with.
Best, Pablo
No, the wavefunction is just one way of obtaining the probability distribution. It's the latter that matters, not the former. And the probability distribution does depend on time, of course.
The probability distribution in quantum mechanics has the same meaning as in any other context: P(x,t) is the probability of finding the psrticle at time t at the state of position x and is given by the corresoonding integral of the probability density, ρ(x,t) that's the probability of finding the particle in a state of position between x and x+dx at time between t and t+dt. One way of calculating ρ(x,t) is by writing it as ρ(x,t)=|ψ(x,t)|^2 and solving the Schrödinger equation. But it's not the only way.
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