Description of the theory

There are a number of mathematically equivalent formulations of quantum mechanics. One of the oldest and most commonly-used formulations is the transformation theory invented by Paul Dirac, which unifies and generalizes the two earliest formulations of quantum mechanics, matrix mechanics (invented by Werner Heisenberg) and wave mechanics (invented by Erwin Schrodinger).

In this formulation, the instantaneous state of a quantum system encodes the probabilities of its measurable properties, or "observables". Examples of observables include energy, position, momentum, and angular momentum. Observables can be either continuous (e.g., the position of a particle) or discrete (e.g. the energy of an electron bound to a hydrogen atom.)

Generally, quantum mechanics does not assign definite values to observables. Instead, it makes predictions about probability distributions; that is, the probability of obtaining each of the possible outcomes from measuring an observable. Naturally, these probabilities will depend on the quantum state at the instant of the measurement. There are, however, certain states that are associated with a definite value of a particular observable. These are known as "eigenstates" of the observable ("eigen" meaning "own" in German).

A concrete example will be useful here. Let us consider a free particle. Its quantum state can be represented as a wave, of arbitrary shape and extending over all of space, called a wavefunction. The position and momentum of the particle are observables. An eigenstate of position is a wavefunction that is very large at a particular position x, and zero everywhere else. If we perform a position measurement on such a wavefunction, we will obtain the result x with 100% probability. An eigenstate of momentum, on the other hand, has the form of a plane wave. It turns out that the wavelength is equal to h/p, where h is Planck's constant and p is the momentum of the eigenstate.

Usually, a system will not be in an eigenstate of whatever observable we are interested in. However, if we measure the observable, the wavefunction will immediately become an eigenstate of that observable. This process is known as wavefunction collapse. If we know the wavefunction at the instant before the measurement, we will be able to compute the probability of collapsing into each of the possible eigenstates. For example, the free particle in our previous example will usually have a wavefunction that is a wave packet centered around some mean position x0, neither an eigenstate of position nor of momentum. When we measure the position of the particle, it is impossible for us to predict with certainty the result that we will obtain. It is probable, but not certain, that it will be near x0, where the amplitude of the wavefunction is large. After we perform the measurement, obtaining some result x, the wavefunction collapses into a position eigenstate centered at x.

Wave functions can change as time progresses. An equation known as the Schrodinger equation describes how wave functions change in time, a role similar to Newton's second law in classical mechanics. The Schrodinger equation, applied to our free particle, predicts that the center of a wave packet will move through space at a constant velocity, like a classical particle with no forces acting on it. However, the wave packet will also spread out as time progresses, which means that the position becomes more uncertain. This also has the effect of turning position eigenstates into broadened wave packets that are not position eigenstates.

Some wave functions produce probability distributions that are constant in time. Many systems that are treated dynamically in classical mechanics are described by such "static" wave functions. For example, an electron in an unexcited atom is pictured classically as a particle moving in a circular trajectory around the atomic nucleus, whereas in quantum mechanics it is described by a static, spherically symmetric probability cloud surrounding the nucleus (Fig. 1).

The time evolution of wave functions is deterministic in the sense that, given a wavefunction at an initial time, it makes a definite prediction of what the wavefunction will be at any later time. During a measurement, the change of the wavefunction into another one is not deterministic, but rather unpredictable, i.e. random.

The probabilistic nature of quantum mechanics thus stems from the act of measurement: the object interacts with an apparatus, and their respective wavefunctions become entangled. In effect, the object ceases to exist as an independent entity. This may introduce some uncertainty into the prediction of what the outcome of a measurement may become, at some stage in the future, insofar as this prediction is required to draw information only from the object wavefunction. However, it might be thought that, by preparing the apparatus beforehand, its influence during the measurement might be predictable, or at least afterwards detectable, so that this kind of uncertainty might be merely a matter of insufficient data. But, as it turns out, to actually detect such influence data, by handling the apparatus, is incompatible with its functioning as a measurement device. That is, for practical reasons, the apparatus cannot do both at the same time.

It is therefore a matter of principle, rather than practice, that one has the uncertainty which necessitates a probabilistic prediction. This is one of the most difficult ideas to understand about the nature of quantum systems. It was the central topic in the famous Bohr-Einstein debates, in which they sought to clarify these fundamental principles by way of thought experiments.

There are some interpretations of quantum mechanics that do away with the concept of "wavefunction collapse" by altering the concept of what constitutes a "measurement" in quantum mechanics. For example, see relative state interpretation.