The particle's characteristics ARE the wavefunction - or to say it another way, the wavefunction is the only real characteristic a particle has.
When I run a quantum mechanics simulation at work, I might afterwards calculate the particle positions or some other classical characteristic of the system. But I need those characteristics only because I'm trying to get a classical model of the system. The system itself, when it's running, doesn't do any of that. It interacts wavefunction-to-wavefunction. As far as we know, that's how the universe works.
When I run a quantum mechanics simulation at work, I might afterwards calculate the particle positions or some other classical characteristic of the system. But I need those characteristics only because I'm trying to get a classical model of the system. The system itself, when it's running, doesn't do any of that. It interacts wavefunction-to-wavefunction. As far as we know, that's how the universe works.