One of the first physicsts to be publically troubled by the philosophical
interpretations of quantum mechanics was Albert Einstein. In 1935, he
co-authored a paper which was intended to show that Quantum Mechanics could
not be a complete theory of nature. The arguments in the EPR paper are
very similar to ones which Einstein himself made in correspondences to
friends, but are not exactly the same.1
The first thing to notice
is that Einstein was not trying to disprove Quantum Mechanics in any way.
In fact, he was well aware of its power to predict the outcomes of various
experiments. What he was trying to show was that Quantum Mechanics could
not be a complete theory of nature and that some other theory would
have to be invoked in order to fully describe nature.
The argument begins by assuming that there are two systems, A and B (which might be two free particles), whose wavefunctions are known. Then, if A and B interact for a short period of time, one can determine the wavefunction which results after this interaction via the Schroedinger equation or some other Quantum Mechanical equation of state. Now, let us assume that A and B move far apart, so far apart that they can no longer interact in any fashion. In other words, A and B have moved outside of each others light cones and are therefore spacelike separated.
With this situation in mind, Einstein asked the question: what happens if one makes a measurement on system A? Say, for example, one measures the momentum value for system A. Then, using the conservation of momentum and our knowledge of the system before the interaction, one can infer the momentum of system B. Thus, by making a momentum measurement of A, one can also measure the momentum of B. Recall now that A and B are spacelike separated, and thus they can not communicate in any way. This separation means that B must have had the inferred value of momentum not only in the instant after one makes a measurement at A, but also in the few moments before the measurement was made. If,on the other hand, it were the case that the measurement at A had somehow caused B to enter into a particular momentum state, then there would need to be a way for A to signal B and tell it that a measurement took place. But, the two systems cannot communicate in any way!
If one examines the wavefunction at the moment just before the measurement at A is made, one finds that there is no certainty as to the momentum of B because the combined system is in a superposition of multiple momentum eigenstates of A and B. So, even though system B must be in a definite state before the measurement at A takes place, the wavefunction description of this system can not tell us what that momentum is! Therefore, since system B has a definite momentum and since Quantum Mechanics cannot predict this momentum, Quantum Mechanics must be incomplete.