We have seen that the overall wavefunction describing the system and the detector has a predictable behavior: each time we repeat the experiment we will get the same evolution for this total system. We can ask: where does the randomness seen in our experiment come from?
The answer is that the randomness arises when we look not at the whole system - the electron and the muon - but only at one part of the system - the electron. More precisely, in the process of detection, we try to see the state of the electron from the viewpoint of the detector particle, i.e., the muon.
From fig. 3 we see that the muon does not see a definite position for the electron. There is a \( 50\%\) chance that the muon sees the electron on the left - this is the bump in the top left corner. There is also a \( 50\%\) chance that the muon sees the electron as being on the right - this is the bump in the bottom right corner.
Thus the state of the muon is itself split among two possibilities, and so the muon cannot 'say' with definiteness what the electron is doing. The detector therefore just sees a probability \( 50\%\) that the electron is on the left and a probability \( 50 \%\) that the electron is on the right.
At this point one might wonder if there was any point is using the muon to detect the electron. If the state of the muon itself splits into two possibilities, then perhaps this detection process was meaningless?
Such is not the case. The important point is that even though the muon does not have a definite answer for where the electron is, it does have a correct correlation with the position of the electron.
In detail, this means the following. When the muon thinks the electron is on the left, then it rises up. When the muon thinks the electron is on the right, it does not rise up. These are the two bumps in fig. 2. But there are no bumps in the 'incorrect' locations. Thus there is no bump in the bottom left corner, which would describe an electron on the left and a muon that failed to be repelled to the top. Similarly, there is no bump in the top right corner, which would describe an electron on the right and the muon in our left detector rising up. Thus after interacting with the electron, the position of the muon always corresponds to 'correctly correlated' situations.
Let us summarize the overall situation with the help of an example.
After the detection process, the overall wavefunction of the electron-muon system has two bumps: one corresponding to the electron on the left and the muon in the 'up' position and one corresponding to the electron on the right and the muon in the 'down' position.
2(a): We start with this electron wavefunction
2(b): Squaring gives the probability graph; the area is \( 2/3\) on the left and \( 1/3\) on the right.
2(c): The evolution of the overall wavefunction is the same, each time the experiment is repeated.
2(d): Measurement however sees the result from the viewpoint of the muon; \( 2/3\) of the time the muon will rise (signaling an electron on the left) and \( 1/3\) of the time it will not.