QT3 - Teachers

The Correspondence Principle

When a successful physical theory is replaced with an improved theory, there are still areas where the original theory is good enough. Under these conditions, the new theory must predict the same results as the original theory. When Einstein’s theory of General Relativity supplanted Newton’s theory of gravity, it had to replicate Newton’s results for slow speeds and modest densities of matter. In the limit, it has to correspond with the earlier theory. Indeed, NASA only uses Newton’s theory of gravity for spacecraft trajectories; Newton’s theory is mathematically simpler and accurate enough that other sources of error completely swamp the difference.

Since quantum mechanics is a theory that supercedes classical physics, in the limit of large objects, it too must predict the same results as classical physics. This is called the correspondence principle. Stars, rocks, bumble bees, even bacteria, follow classical physics. It isn’t until we reach the scale of molecules, atoms, and subatomic particles that classical physics fails and must be supplanted by quantum mechanics.

Because Tic-Tac-Toe has only 9 squares, the correspondence principle is not dramatic, but it is still there. Consider a game where O decides on the strategy of immediately collapsing X’s previous move. She places both her spooky marks in the same squares as X did. After a few moves, the board has many classical marks on it, but at most only one pair of spooky marks. It looks very much like a game of Classical Tic-Tac-Toe. That’s the correspondence principle. Even when collapses are rare, any game that goes through all nine moves, ends up with nine collapsed squares, i.e., nine classical marks, and not a spooky mark to be found.

The appearance of the board is not the only place where the correspondence principle applies; it also applies to strategy. In general, the strategy for Quantum Tic-Tac-Toe is significantly different than for Classical Tic-Tac-Toe. However, the smaller the entanglements and the more rapidly they are collapsed, the more the strategy resembles that in Classical Tic-Tac-Toe. So the correspondence principle applies to both states and the evolution from state to state.