Wave Functions and Probability

One of the more difficult concepts is the exact nature of the wave function (Ψ) itself. We can easily represent wave functions with mathematical formulas, but what exactly are we representing? The standard Copenhagen interpretation doesn’t take a position on whether the wave function is a physical reality or not. Wave functions can include imaginary components that can’t even be observed.

But the Copenhagen interpretation does take a stand on the square of the wave function |Ψ|²; namely, that it represents the probability density associated with the subatomic entity in question. While the wave function itself may be just an abstract idea, it provides the tool we need to make statistical predictions of how measurements are likely to turn out. The Copenhagen viewpoint does not say that a particle is accompanied by its wave function; they do not have separate existences. The wave function is all that there is.

Another challenging aspect of this interpretation is the fact that we are inherently limited to probabilities. In science we have a longstanding tradition of believing in causal and predictive determinism. Our sense is that the state of any physical system should flow smoothly and logically from its previous state. We also want the laws of physics to make definite predictions of how quantum systems change in time under the influence of forces and such. Otherwise, what good are those laws?

For a given initial state, quantum physics sometimes tells us that multiple outcomes are equally likely, and gives us no way to determine ahead of time which one will actually happen. Does that mean that something is missing from the theory? We’ll revisit this important question.