was invented. But its
interpretation was ticklish. It was a probability
amplitude wave. But we thought we were doing mechanics and not
tossing dice. Unfortunately the wave picture persisted for
very long. And probabilities continued to bother the great
masters of the subject.
The real essence of the new science was grasped by Dirac right at the inception. In his classic textbook written in 1929 Dirac emphasised the abstract approach. He begins by observing that there should be no surprise that a new constant of nature h presents itself; otherwise there would be no real distinction between the micro and the macro worlds. No fundamental scales would distinguish the two, and in principle the same stories we know at large scale should boringly repeat however finely we probed.
He then goes on to emphasise the new principle Quantum Mechanics presents to us. This is the Principle of Superposition of States. Classically we cannot imagine superposing two potential states of a system. Either the ball is inside the boundary or outside. It does not exist in a superposition of these states. States in QM are, like in Classical Mechanics, labelled by the possible values of observables. But with the difference that some of the classically allowed states may not occur and vice versa. We are to list all the possible observables of a system and their possible values, these are called eigenstates. And the general state of the system would be a superposition of the various eigenstates. The mathematical structure is therefore of a linear vector space.
The persistence of the wave picture had to do with this Principle. But this is the new principle revealed by QM. There is nothing uncertain about it. Nor is there any uncertainty in how the states evolve. Time evolution of the state is completely deterministic, by a first order differential equation. Identically prepared systems could however result in different answers in individual experiments because the system when observed can only be found in one its eigenstates. This is the result of a radically distinct way that states are listed in QM. An uneasy feeling of uncertainty results from this very concrete and crisp principle. The problem clearly lies in the eyes of the beholder.
There is no such thing as ``wave''function ``collapse.'' By the latter is usually meant how the system is ``forced'' into one of its eigenstates upon ``observation''. By ``observation'' is meant a macroscopic apparatus that interacts with a particular observable associated with the quantal system. Actually we may turn the definition around. Whatever interaction always leaves behind the system in a specific eigenstate is an act of observation. But there are other interactions allowed which do not ``collapse'' the ``wave''function. They alter how the state is made of its eigenstates without reducing it to a pure eigenstate. But you say, anything macroscopic always leads to collapse so how can we think of it in the same class as a quantal interaction? Well, there are macroscopic entities that do not collapse the wavefunction. One of them is Gravity. This is a classical field for all practical purposes. It can never be switched off, i.e., no system can be isolated from it. If Gravity were an observer it would have reduced the entire Universe to a single eigenstate by now. We can also easily construct classical, macroscopic arrangements of electromagnetic fields which will not collapse a wavefunction.