Quantum Mechanics

3.   The randomness of measurement results
         3.4   Summary
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Summary of the chapter

We have come to a bizarre conclusion: in quantum theory, the result of an experiment cannot be predicted with certainty; all we can give is the probability for different outcomes. If we perform the same experiment many times, then these probabilities will tell us the fraction of times we can expect one outcome or another, but there is no way to know what outcome we will get in any one instance of the experiment.

The reasoning behind this conclusion was simple. The first lesson of quantum theory tells us that the electron, like everything else, is a wave. This wave can have any shape. In particular we can take a shape that has two well separated bumps.

The second lesson of quantum theory says that we can have zero electrons or one electron, but not a fraction of an electron. But in that case what should we see at the location of any one bump in the above waveform?

Only a fraction of the wave is near any one bump. Thus we conclude that this fraction must represent the probability that we find a whole electron at the location of this bump.

But once we introduce the notion of probability, we are trapped into a situation where we cannot predict with definiteness where the electron will be found.

Finding probabilities

Since probabilities are central to quantum mechanics, we should have a clear prescription of how to obtain the probability to find the electron at different places. There is indeed a natural prescription, which is borne out by experiments.

We first square the wavefunction. This gives a function, called the \( P \) function, which is not negative anywhere. Further, since the wave should have the strength representing one electron, the area under this \( P \) function should be unity.

There is now a natural prescription for finding probabilities. Suppose we want to find that the probability for the electron to be between positions \( x_1\) and \( x_2 \). Then we find the area under the \( P \) graph, over the line segment from \( x_1 \) to \( x_2 \). Suppose this area is \( 0.3\). Then the probability is \( 0.3\) that if we look for the electron using a detector, we will find it between the positions \( x_1\) and \( x_2 \).

Since the probability cannot be a negative number, it is important that the \( P \) function was obtained by squaring the wavefunction, so that it is not negative anywhere. Further, the probability that the electron is somewhere should be \( 1 \), which agrees with the fact that the total area under the \( P \) graph is \( 1 \).

The actual measurement

We then came to the most crucial step: actually setting up an experiment which would detect the location of the electron. To detect the electron we made a detector, as follows. We took a negatively charged muon, which is repelled by the negatively charged electron, and placed this in a vertical tube. If an electron is present at a given location, then the muon will be repelled to a certain height; if a fraction of an electron was present, the muon would be repelled to a correspondingly smaller height.

We took an electron wavefunction having the shape of two equal and well-separated bumps, and placed the detector near the left bump. The results confirmed the picture of probabilities that we had arrived at by abstract reasoning. The detector never found a fraction of an electron; sometimes it found no electron and sometimes it found the whole electron. These two possibilities occurred at random, with a \( 50 \% \) probability for each.

The next question is: does this inherent randomness mean that there are no definite laws of physics that can tell us how things should behave?


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