Does quantum mechanics predict instantaneous action at a distance even without entanglement? The suggestion that quantum mechanics implies that instantaneous action at a distance occurs is normally based on the contention that this follows from the entanglement of particles that share a common origin. My question is: does quantum mechanics predict that this happens even when there is no entanglement? In particular, does this happen when two quantum systems interact in such a way that the wave function of at least one of them changes drastically?
Take, for example, an interaction between a hydrogen atom and a hydrogen-oxide radical: H + HO = H2O. While the hydrogen atom is free, the wave function of its electron allows it to be anywhere in the universe. Or rather, it is everywhere in the universe. Instantly after the interaction has completed, the amplitude of the quantum wave of the same electron is smaller everywhere, except in the very immediate neighbourhood of the water molecule of which it has become a part. In the rest of the universe, the wave function has taken on other values, if not instantaneously – the reaction could take some time – then at least far quicker than light.
One could object that the electron is never really at more than one point at any one time, only we don’t know which until we make a measurement. But that amounts to a refusal to accept quantum mechanics. The effect of a wave on its environment is determined by the totality of the wave, including even those places where its amplitude is partly or entirely imaginary. For example, the hydrogen atoms in a water molecule are positioned at an angle of 108 degrees from each other, as seen from the oxygen nucleus, due to some very particular properties of the wave functions which define the hydrogen-oxygen bonds. That results in water having some very peculiar properties, including the property that its solid form – ice – is less dense than the liquid form and therefore floats on it. Even the mere stability of water molecules is due to the wave function of the electrons which the hydrogen atoms share with the oxygen atom: in this wave function the electrons are as much as near the oxygen atom as the hydrogen atom.
 A: It seems to me that you have misunderstood the nature of the quantum mechanical wave. It is not a wave in the sense of an acoustic wave, nor a wave in the water. It is a probability wave: a mathematical function that tells us how probable it  is to find the electron, for example, at a specific (x,y,z,t) . Find means detection, detection means measurement.
There is a basic misunderstanding here:

The effect of a wave on its environment is determined by the totality of the wave, including even those places where its amplitude is partly or entirely imaginary. 

The probability wave has no effect. Probability  is just a number that allows one to guess at the outcome of a measurement.Heads or tails on a coin? the probability is 50% . Neither heads nor tails exist until the coin is thrown.( measurement).

Take, for example, an interaction between a hydrogen atom and a hydrogen-oxide radical: H + HO = H2O. While the hydrogen atom is free, the wave function of its electron allows it to be anywhere in the universe.Or rather, it is everywhere in the universe.

Basic misunderstanding. The wave function just tells you the probability to find this bound electron  anywhere in the universe, if you go there and try to measure its existence ( good luck). In any case, to go there for your measurement you would travel at most close to the velocity of light, so there is nothing instantaneous. If you do not go there, the wave function gives you just a number for what you would find if you could go there and measure/check if that electron is there. In the same way that 50% is your probability to get heads in a throw of a coin. The probability will be a very small number, and if you wanted to do the experiment you would need zillions of H atoms to get one real electron out there at that (x,y,z,t). 

Instantly after the interaction has completed, the amplitude of the quantum wave of the same electron is smaller everywhere, except in the very immediate neighbourhood of the water molecule of which it has become a part.

Yes, except again it is a probability amplitude . It is .99999999... probable that the electron under study is tied up with the water molecule within a nanometer or so, though it will now be in an indistinguishable 1/9 of the electron orbitals of the molecule. (there is no individuality in identical elementary particles).
The quantum mechanical wave, although it describes the behavior of matter, is not a  matter wave. Once again, the square of the wave function gives the probability of the particle under study to be at a particular ( x,y,z,t) in space time.
A: There is not any "instantaneous action at a distance" in quantum mechanics. Moreover the term is a misnomer. Yes, the wavefunction for an electron in a free hydrogen atom allows the electron "to be anywhere in the universe", but your:

the amplitude of the quantum wave of the same electron is smaller everywhere, except in the very immediate neighbourhood of the water molecule of which it has become a 
  part. In the rest of the universe, the wave function has taken on other values, if not instantaneously – the reaction could take some time – then at least far quicker than light.

is completely incorrect because when the electron is in the water molecule, no wavefunction can describe it due to electron-electron correlations. The quantum state of the electron in a water molecule is not given by any wavefunction.
