Which of these interpretations of the modulus squared of wavefunction is right?

dEdt

Does [itex]|\psi(\mathbf{x},t)|^2d^3\mathbf{x}[/itex] or [itex]|\psi(\mathbf{x},t)|^2d^3\mathbf{x}dt[/itex] give the probability of a particle to collapse at the point [itex]\mathbf{x}[/itex] at time [itex]t[/itex]?

Griffiths sides with the former, but I'm having doubts.

cattlecattle

It's the former, it doesn't make sense to integrate over time, at any instant t, the integration over space gives you the overall probability at that time, which is 1.

dEdt

Here are my issues: 1) space and time seem to be treated on different footings, and 2) the probability that say a Geiger counter goes off at a given time t is zero, but the probability that it goes off over some time interval is non-zero.

The_Duck

Yes. This is nonrelativistic quantum mechanics, which treats space and time differently. To fix this we have relativistic quantum field theory.

If you do an ideal position measurement at time t, the probability of finding the particle *somwhere* is 1. Geiger counters don't do ideal position measurements; the quantum mechanical analysis of radioactive decay is somewhat more complicated.

Jano L.

This is true, but I think the probability given by [itex]|\psi^2|dV[/itex] according to Born's interpretation should not be used directly to predict the frequency of counts of the Geiger detector (unless one smuggles the source intensity to [itex]\psi[/itex], which can allow us to do just that; but then the above form of Born's rule is not applicable.)

Instead, if the wave function for a particle is normalized (the most clear approach), it gives us the probability that this particle is at some point of space (without the necessity to detect it there).

You are right that the number of counts(clicks) of detector set in some definite distance from the piece of matter scattering charged particles will be proportional to time interval of the measurement, but this is because larger interval allows more [itex]\textit{distinct particles}[/itex] to come at the detector; however, each one can be ascribed by normalized wave function that gives density of probability in space by the rule [itex]|\psi|^2dV[/itex].