# Is the wave function of a particle re-created after a measurement stops?

Assuming wave-function collapse is correct (which can be a relatively hefty philosophical claim in some circles), then think of measurement this way:

When you measure an observable on a system, you collapse the wave-function of the system into a Dirac delta function in the eigenbasis for that observable.

If you measure position, you get a delta function in position-space. If you measure momentum, you get a delta function in momentum-space (or a sine wave in position space). If you measure energy, you get an energy eigenfunction.

Then - after the collapse - the system begins evolving according to Schroedinger's Equation once again, but this time your initial conditions for the system are whatever shape you collapsed the wave-function into with your measurement.

Remember, particles obey Schroedinger's Equation. It tells you what they do in Quantum Mechanics - just like Newton's 2nd Law tells you what they do in Classical Mechanics. Give me the Hamiltonian and the initial conditions of a system, and I will tell you how it evolves in time. That is the name of the game for much of Quantum Mechanics.

(As an interesting side-note: if you make another measurement of the same observable very quickly after making the first measurement (and I mean VERY quickly), you will get back the same result because the wave-function has not had time to evolve away from that state yet.)

The electron doesn't get destroyed when you measure it (though photons usually do), but its wavefunction doesn't go back to how it was before. Instead it gets a new wavefunction, different from the old one. If you measured the position of the electron, this new wavefunction will be a delta function (a single infinitely sharp spike) centred at the position you measured. This change in the wavefunction is what's meant by "collapse".

If this didn't happen then you would be able to measure both the position and the momentum simultaneously, by making multiple measurements: first measure the position, and then measure the momentum. But in reality you can't do this, because the first measurement changes the wavefunction. The delta function it turns into doesn't have a well-defined momentum (i.e. its momentum could be anything), and this is essentially how the Heisenberg uncertainty principle works.

Two cents from an experimentalist.

It is always good to keep in mind that a wavefunction for a real particle in the lab is a solution of Schroedinger's equation with specific boundary conditions given by the experimental set-up that makes the measurement. Every measurement changes the boundary conditions for the solution that describes the particle's $(x,y,z,t)\;\;\&\;\; (p_x,p_y,p_z,E)$ which are the vectors we can normally measure.

It is also good to keep in mind that the solution of S's equation that describes the specific particle in the lab is a function whose square gives the probability of finding the specific measurement one finds with the experiment. That is the reason one does not try to devise experiments chasing after the "same" electron, because a single measurement in space and time ( or momentum and energy) cannot give any information on probability distributions and whether one has the correct potentials in the S equation ( or the more advanced formalisms of quantum mechanics). We do scattering experiments with beams with an enormous number of particles for that reason. Same boundary conditions and a plethora of particles will give us the probability function, and thus help us discriminate between theories, which is the reason for experiments.

After the measurement each particle is described by a new probability function given by the new boundary conditions , because each measurement changes the boundary conditions.

And finally it should also be stressed that the probability distribution describing a particle is just that , a distribution in space (or energy momentum space) of the probability of finding the particle whole when you measure it at that specific coordinate. It is not a solution with the particle's mass spread out like a splash in coordinate space. Thus the concept of "collapse" is a misleading concept. The "collapse" happens in probability space not in real space, in the same way that when one throws the dice, each of the 6 numbers is spread in probability space equally and the throw "collapses it " to a specific number. Nothing material gets collapsed. It is not a balloon being punctured.