Quantum Mechanics

4.   No randomness in the overall wavefunction
4.5   Summary

Summary of the chapter

The discussion of this chapter has led to a deep insight.

We have seen that in quantum theory, there is a randomness in the result of experiments. We cannot escape this randomness by building 'better' detectors; the randomness is an inescapable part of quantum theory.

But now we have understood the source of this randomness. The source is the process of measurement itself.

The meaning of measurement

The overall wavefunction - describing the measured object and measuring detector - evolves in a completely predictable way. Each time we repeat the experiment, this overall wavefunction will behave in exactly the same way, and its evolution can be predicted by the well-defined laws of wave mechanics.

But when we talk of a measurement we are not talking about the combined system of 'measured object' and 'measuring detector'. Instead, we are asking how the detector sees the object, since it is the detector which will record the state of the measured object.

Thus in our example of the electron measured by a muon detector, we will be looking at the height of the muon and asking what it says about the location of the electron.

The source of the problem

Now we see the problem.

The electron is a wave, and a wave can have any shape. We had taken this wave to have two well-separated bumps, so the electron wave did not have a well-localized position. We then asked the detector to tell us if the electron was on the left or the right.

But the muon doing the detection is also a wave. The muon wave can also split into two parts - one at the bottom of the detector and one at the top. But in that case the muon cannot give a unique answer for the location of the electron.

Let us emphasize this point again, since it is crucial. The muon was supposed to display the result of its detection by its height in the detector tube. If the muon wave splits between two different heights, then the muon has not chosen a unique response to the question it was supposed to answer: "where is the electron?"

The inescapable nature of the problem

A moments thought tells us that such a split of the muon wave is natural and inescapable. The muon is supposed to detect the electron, so the muon wave will have to respond to the location of the electron. If the electron wave is split into two parts (left and right) then the muon wave will have to correspondingly spilt into two parts: one saying that the electron is on the left and one saying that the electron is on the right.

To understand the way the muon wavefunction will split, we have to note that the combined electron-muon system is described by a wavefunction defined over a 2-dimensional plane: one direction $$x$$ corresponds to the position of the electron and the other direction $$z$$ corresponds to the height of the muon. We depict the evolution of this overall wavefunction again in fig.1. The muon detector is doing its job correctly: the height of the muon is gets correctly correlated with the position of the electron.

But after the correlation is correctly established, the overall wavefunction still has two bumps: one saying that the electron is on the left and the muon is correspondingly in the 'up' position, and one saying that the electron is on the right and the muon is correspondingly in the 'down' position.

This is from the viewpoint of the overall wavefunction. But we can still ask the question: where does the muon think the electron is? And then there is no unique answer, since we have two bumps in the overall wavefunction. In our example these bumps were of equal strength, so all we can say is that $$50\%$$ of the time the detector thinks the electron is on the left and $$50 \%$$ of the time the detector thinks the electron is on the right. This is the source of randomness.

As we will see in more detail later, building a more complicated detector will not resolve the problem of randomness. If the electron wave is split into two parts, the detector wavefunction will have to also split into two corresponding parts, no matter what the detector is.

We will revisit the issue of detection again in the context of Schrodinger's cat. For now, our next step will be to learn the third (and last) basic lesson of quantum mechanics: the superposition principle.