Interacting Systems

When two systems, I and II, with state vectors and interact, we start the interaction by assigning to the coupled system (i.e., the "joint" system I+II) a state which we can write as a product . Then, as the interaction proceeds, evolves according to the Schrödinger equation . Once the interaction is going, in general, each individual system (I or II) will no longer have its own state vector. Rather the probabilities for measurement outcomes on each system will be obtained from the evolved state vector for the joint system.

For example let

                  (1)

where and are "up" and "down" eigenstates of an observable A on system I. Suppose we represent Schrödinger evolution for the coupled system by an arrow (" ") and suppose that the evolution looks like this

                    (2)

and

                    (3)

where and are "up" and "down" eigenstates of an observable B on system II. What this means, for example in (2), is that if system I starts out in eigenstate of an observable A and system II starts out in the state then after they interact if we make an A-measurement on system I and a B-measurement on system II we will find (with probability 1) that A and B both have the "up" value. That is, we can regard system I as finishing where it began, in the A-eigenstate , with system II finishing in the B-eigenstate .

But something different happens if the interaction starts with system I in state and system II in state . Then, since Schrödinger evolution is linear,

The evolved state (4) of the coupled system

                (4)

is not a simple product of system I and system II states (as in eqs. (2) and (3)) but is a superposition over such products. Here we cannot assign separate state vectors to system I and system II. We can say what happens if we make a measurement, though. If, for instance, we measure B on system II then collapses either to the product

with probability .

Thus the result of the B-measurement will be either "up" or "down" with a 50:50 chance of each. The same is true for a measurement of A on system I. But if we jointly measured A on I and B on II then, although each result would have a probability of 1/2, the results would be linked. Whenever one system came out "up" so would the other, and the same for the "down" result. That linkage (or "entanglement", to use a term introduced by Schrödinger) is a consequence of the evolution described in equations (2) and (3), and a characteristic feature of quantum interactions.