Valid solutions to the Schrodinger equation give you the wave function amplitudes in multiple places; the particles in these places can interact with each other still, even if they are 'the same particle'.
However, the wave function at different places interacts with the environment and start to shift in phase, eventually becoming unable to interfere with itself - this is called decoherence, and is a valid explanation about why and how we can't observe wave-like behaviors at large scales or in hot systems.
On the other hand, we can only postulate, based on observations, that when a particle interacts with a measurement device, the measurement device will show a single value with a probability determined by the amplitude of the particle's wave function at that point. We can postulate that the wave function collapses, or we can postulate that the device branches out into different devices in different worlds (enough such devices&worlds to achieve the probability distribution through observer selection somehow), or many other ways of formulating the Born rule. But whichever way you put it, this rule must be added to your system to predict experimental results, it does not derive from the Schrodinger equation.
>Valid solutions to the Schrodinger equation give you the wave function amplitudes in multiple places; the particles in these places can interact with each other still, even if they are 'the same particle'.
I suppose it's destructive interference. It's qualitatively interesting, but its observation is complicated by orthogonal states: when you multiply orthogonal states you get zero. If you can thoroughly dismantle the state to observe it, you still can do it only on microscale, then you'll have a problem lifting it to macroscale evading destructive interference while orthogonal states are all over the place. Anyway, Schrodinger equation describes behavior of quantum states with mathematical precision and the math is quite conclusive that a linear equation behaves in a linear way. When you feel intuition doesn't get you much, you can resort to math, that's why math is seen as an indispensable part of science, because intuition isn't guaranteed to work, which is exactly your case.
>it does not derive from the Schrodinger equation
MWI derives it from the Schrodinger equation. Observation is experience of the observer and can be calculated. Unless you assume that the observer is supernatural and is thus unknowable.
> MWI derives it from the Schrodinger equation. Observation is experience of the observer and can be calculated. Unless you assume that the observer is supernatural and is thus unknowable.
This posits the notion of an observer that only observes one outcome, whereas the SE predicts that an observer will observe several different outcomes with different amplitudes. The MWI is postulating that we should only look at each outcome separately.
Furthermore, it is not possible to derive the actual probability value from the wave function amplitude without some additional postulate equivalent to the Born rule, for example that the number of observers that observe one outcome is proportional to the wave function amplitude of that outcome.
The result of calculation of the state of observer is linear evolution: the state of observer splits and entangles with the observed state and each part observes the respective outcome. Ironically Copenhagen gave the same result for Schrodinger's cat experiment: even before measurement it's known what states are in superposition and those states are "dead" and "alive", and it's still known without measurement too.
>that the number of observers that observe one outcome is proportional to the wave function amplitude of that outcome
If you mean the number, the norm of each state of observer that observes the respective outcome can be calculated. The statistics over the outcomes can be calculated too.
However, the wave function at different places interacts with the environment and start to shift in phase, eventually becoming unable to interfere with itself - this is called decoherence, and is a valid explanation about why and how we can't observe wave-like behaviors at large scales or in hot systems.
On the other hand, we can only postulate, based on observations, that when a particle interacts with a measurement device, the measurement device will show a single value with a probability determined by the amplitude of the particle's wave function at that point. We can postulate that the wave function collapses, or we can postulate that the device branches out into different devices in different worlds (enough such devices&worlds to achieve the probability distribution through observer selection somehow), or many other ways of formulating the Born rule. But whichever way you put it, this rule must be added to your system to predict experimental results, it does not derive from the Schrodinger equation.