If we now place one detector in the transmitted path and another in the reflected path then only one will fire. There is only one particle and it can only be detected at one place. However, instead of placing detectors in these two paths, we can bring the two paths together at a second beam splitter, placing detectors in the rightward and upward paths of this second beam splitter. It is a consequence of quantum theory, and demonstrated many times in laboratories, that:

 

1. If we ensure that the two paths between the two beam splitters are equal in length, then the particle will always be detected in the rightward path of the second beam splitter.

    

    

2. If one path is longer than the other by a certain amount (equal to half the wavelength associated with the particle) then the particle will always be detected in the upward path of the second beam splitter.

    

    

Now imagine trying to explain this behaviour as if the particle went along only one of the two paths, though without us knowing which one. Imagine that, in actual fact, the particle went along the lower path while nothing went along the upper path. When the particle arrives at the second beam splitter, it has to choose whether to go up or to the right. This particle has only transversed the lower path whilst nothing has transversed the upper path and hence, the particle has no information as to whether the two paths are equal in which case it should go right, or differ by half a wavelength in which case it should go up, and hence there is no way the particle can behave in the way we actually observe in experiments. The same dilemma would apply if the particle took the upper path between the beam splitters. This means that we simply cannot think in this way. We cannot think in terms of one path being empty. There is some sense in which the particle is in two places at once.

This device, called a quantum interferometer, illustrates the central property of quantum theory that distinguishes it from classical theories. We have two possibilities—"the particle is in path A" and "the particle is in path B". In a classical theory only one of these two statements could be true at any given time, while in quantum theory, both statements can be partially true at once. This is called the superposition principle. The state of reality is given by superposing two states which are, from our usual classical way of thinking, mutually exclusive. Mathematically, we represent this by taking the two mutually exclusive statements and adding them together:

(Actually, the formalism is a little more complicated than this since we multiply each statement by a number which represents "how much" the atom is in one or the other box.) The superposition principle can be applied in many situations. For example, we can have an atom in two boxes at once, a photon over here and, at the same time, over there, a particle having clockwise spin, and at the same time having anticlockwise spin. We are forced to think in this way because all quantum objects are subject to interference experiments like the one we just described.

Another feature of quantum theory is that it is inherently probabilistic. Before we look to see whether the atom is in one box or the other, there is no way of knowing which box it will be found in. This fact is problematic for people who believe that every effect must be fully determined by some cause. In fact, one can attempt to remove this inherent indeterminism by supposing that quantum theory is a statistical theory derivable from a deeper deterministic theory in which the true state of affairs is given by a more complete description of the world than that provided by the quantum state. Such theories are often called "hidden variable theories". These hidden variables determine, in the above example, exactly which box the atom will actually be found in. The variables are called "hidden" since, before making a measurement, we do not know what values they take.

A big problem in quantum theory arises because we do not know when to stop using the superposition principle. Imagine that rather than an atom in box A or box B we have a molecule. This is a bit bigger than an atom but still very small and so definitely quantum. What about a slightly bigger object like a single cell? What about a really big object like a baseball or a cat or even a whole galaxy? In principle, there is nothing that tells us when the superposition principle does not apply. Thus, in principle we really could imagine a cat, for example, being in a superposition of being alive and dead at the same time. In practise it would be very difficult to actually see quantum interference with very big objects like baseballs or cats (though interference has been seen with large molecules) and hence we can, for all practical purposes, ignore the possibility that there is a superposition. However, if we are interested in the "in principle" question, we cannot ignore this possibility. This is a problem because we have to explain why we do not actually see superpositions of large objects in two places at once—our everyday experience is of a macroscopic world with well localized objects. There are various approaches to solving this problem. One is to say that, for sufficiently big systems, the state "collapses" onto one or the other of the two possibilities. Thus, it goes from being in the superposed state "A" + "B" to being in one of the states "A" or "B". Such theories of quantum theory are called "collapse theories".

One of the strangest features of quantum theory is quantum entanglement. This arises when we have two physical systems which have interacted with each other. After such an interaction, the total system becomes describable only as a single entity. There is no way of describing it by completely describing its components. Entanglement is a fascinating subject from an interpretational point of view, from a mathematical point of view and from the point of view of applications. The most striking effect that arises from quantum entanglement is quantum nonlocality—first discovered by John S Bell in 1964. Bell's theorem is one of the most important results in physics and has played a large role in motivating much recent work in quantum theory. Imagine that two quantum systems interact and then separate to a great distance where measurements are made. These measurements will reveal certain correlations between the two systems. So far, this is what we expect since the systems have interacted in the past. Bell showed, however, that we cannot explain these correlations purely in terms of the previous interaction between the systems. It appears that, even though the two systems are at a great distance, they continue to talk with one another. However, this effect is not so strong that it can actually be used to send signals, so the principle that information cannot travel faster than the speed of light is safe for the time being.