We don't like the idea of determinism; it conflicts with our dearly held notion of free will. Quantum Mechanics, by allowing only the calculation of probabilities of various outcomes but refusing to predict a unique future, seems to provide an escape from determinism. However, upon closer inspection this route comes to a dead end too.
First of all, the math behind Quantum Mechanics is totally deterministic. We have a wave function and a differential equation—the Schroedinger equation—to predict the evolution of the wave function with absolute certainty. The fuzziness happens at the level of physical interpretation. One of the interpretations called the Copenhagen interpretation was agreed upon by a committee of famous physicists, including Niels Bohr and Werner Heisenberg, in the 1920s. According to this agreement, the wave function itself has no physical meaning. It's only the square of the magnitude of the wave function that yields the probabilities of classical outcomes (the value of the wave function is a complex number which has magnitude and phase). The word classical is central here, because what we observe every day using our senses is not some weird quantum effects, but a reasonably classical world, where objects have well defined velocities, positions and energies.
By squaring the wave function we lose some information about it—specifically its phase. And, to a large extent, it's the phase that makes quantum effects so different from classical. In Quantum Mechanics a particle may produce an interference pattern with itself, like in the famous two-slot experiment (see sidebar), because the phase of its wave function can add or cancel depending on its path. But in our classical world we cannot observe these phases. Physicists think the phases are there because they can see the interference pattern. So is wave function only a mathematical device?
By far the weakest and most controversial part of the Copenhagen interpretation is the ad hoc introduction of the so-called quantum collapse. In order to stitch Quantum Mechanics to our classical world, we have to assume that each experiment ends with a collapse of the wave function. The phases are suddenly discarded and we see a classical event take place with a certain probability. In the two-slot experiment, what we calculate is the wave function, but what we see is a particle hitting a screen at a specific point. A moment ago the particle behaved like a wave smeared all over space, but then it collapsed, and we quickly measured its exact position on the screen.
In this experiment there is a source of particles, let's say electrons, which have to pass through a screen with two slots separated by very small distance. The electrons hit the scintillating screen, which measures their positions. Quantum-mechanically, each individual electron behaves like a wave (it has a wave function), so after passing through a double slot, it interferes with itself. Whenever the two crests or two troughs arrive at the screen at the same time, we see constructive interference. When a crest from one slot meets the trough from the other slot, they cancel each other. After a while, this produces the usual interference pattern on the screen (see Fig 4.).
This experiment can be performed with a very weak source, say a source that emits one electron per second. The result, after a long run, is still the same, even though the electron must clearly interfere with itself, not with other electrons. (Without interference, you'd just see two smeared images of the two slots.)
Notice that each detection event produces a classical point-like scintillation. At the moment of the observation, the wave function turns into a particle. What the troughs and crests of the magnitude of the wave function communicate to us is the probability of detecting an electron at each point.
To an outside observer the actual experiments look similar to Fig 5. A single electron is detected at some specific position each time. Only after the observer accumulates enough events, can he see that the consecutive electrons struck certain regions more times than others. These places correspond to the peaks of the interference pattern.
Even if we believe that the world jumps from one classical state to another classical state, and the outcome is only known with some probability, this still doesn't allow for free will. Whether our actions are predicted with certainty or with quantum probabilities, the bottom line is that the choices are not ours.
There is another interpretation of Quantum Mechanics that is steadily gaining popularity among physicists. The many-universe interpretation assumes that all possible outcomes happen. At every moment, our world branches into infinitely many parallel universes. The observer also branches into infinitely many future observers, each seeing a different outcome of the experiment. Of course, next moment all those observers branch into another multitude, and so on.
Even though at first sight this idea might seem pretty far-fetched, it doesn't conflict with anything we know from experience. We, as observers, are always in a single universe—"our" universe. It's our doubles who branch into other universes. Of course, each of these doubles believes deeply that he's the one, but that's okay. As long as we cannot communicate with parallel universes, there is nothing that can disprove their existence.
Notice that I use the word "existence" very loosely. It's not necessary for something to be directly observable in order to exist. It's enough that we can observe the indirect results of their existence or that some accepted physical theory requires them in order to explain our view of the world. We have the same situation with quarks—no physicist doubts their existence but, as far as we know, it might never be possible to observe a single quark directly. They always come in pairs, threes, etc.—never alone!
One problem with the many-universe interpretation of Quantum Mechanics is that it's not clear what the meaning of probability is. If all outcomes, no matter how small their probabilities are, are realized, why do we observe the most likely ones most of the time? In the many-universe theory, the square of the magnitude of the wave function is interpreted as the "density" of universes, but as long as the "rare" universes exist on the same footing as the "dense" ones, we can't really talk about their probabilities. You can't say, for instance, that it's more probable that you will branch into one particular future, rather than another, because you will branch into all of them.
There is another problem with Quantum Mechanics—it isn't consistent with Special Relativity. Which brings us to the next section.