To explain our main puzzle, let us collect here the rules we have learnt for quantum theory:
Putting these two observations together leads to a very deep puzzle.
Now we ask: what does this wave describe? Let us consider different possibilities:
We are thus forced to the following rather bizzare conclusion:
The wave of fig.1(c) describes one electron; further, this electron does not split into two halves. But there is a 50% probability that the whole electron is located at the left bump (i..e on Earth) and a 50% probability that the whole electron is located at the right bump (i.e. on Mars).The above conclusion is correct but very puzzling, and summarizes the entire mystique of quantum mechanics.
The conclusion is puzzling because in all earlier theories of physics, everything had been definite; we know whether an electron is on Earth or on Mars. But in quantum theory, for waveform of the kind in fig.1(c), all we have is a probability for the electron to be at different positions; we cannot say with definiteness where the electron is.
This situation was so uncomfortable to Einstein that at first he refused to accept it, saying "God does not play dice"; i.e., the description of nature must be definite and not in terms of probabilities.
Once quantum theory was better understood, the worries of the kind raised by Einstein disappeared. It turns out that two things are true:
In the rest of this chapter we will study the notion of probabilities in more detail; i.e., we focus on (1). In the next chapter we will understand the measurement process, thus clarifying (2).
 
 
 
 
 
 
 
 
 
1(a): A simple waveform for the electron, consisting of one 'bump'.
1(b): We can deform this waveform to get another one, consisting of two bumps.
1(c): We can deform the waveform further, getting two well-separated bumps.