In quantum mechanics, the results of measurement or observation can only be predicted probabilistically. To understand how that is done, let us talk some more about wave functions.

What exactly is this wave function? After all a particle, by its nature is located at a single point, whereas the wave function (as its name suggests) is spread out in space (it’s a function of x for any given time t). How can such an object represent the state of a particle? The answer is provided by Born’s statistical interpretation of the wave function, which says that \mid\psi (x,t)\mid^2 gives the probability of finding the function at a point x, at a time t. More precisely

\displaystyle\int_{b}^{a} |\psi (x,t)|^2 dx \begin{Bmatrix}\mbox{Probability of finding the} \\ \mbox{particle between a and b,} \\ \mbox{at time}\end{Bmatrix}

A typical wave function. The shaded area represents the probability of finding the particle between a and b. The particle would be relatively likely to be found near A, and relatively unlikely to find it near point B.

Probability is the area under the graph of \mid\psi (x,t)\mid^2 . For the wave function in the Figure, one would quite likely find the particle in the vicinity of point A, where \mid\psi\mid^2 is large, and relatively unlikely to find it near point B.

Just as the positions and momenta specify the state of an object in classical mechanics, the state of an object in quantum mechanics is specified by a wave function (as a consequence of de Broglie’s wave particle duality hypothesis—all matter can exhibit wave-like behavior). All properties of an isolated system, such as an atom, are encapsulated mathematically by the system’s wave function. For example, if you want to predict how likely it is that an electron of that atom will appear at a certain location, you can calculate that probability from its wave function in the following way. For every possible outcome of an observation, the wave function assigns a number, called the amplitude, which can be either be positive or negative. For any possible observation or measurement, the amplitude tells us what the probability could be of a certain result. The probability of a certain observational outcome X is expressed mathematically as square of the amplitude (see box):

\mbox{Probability of observing}\ X = \mbox{(Amplitude assigned to}\ X)^2

It is important to realize that a wave function describes a superposition of all possible observational outcomes before any measurement or observation is made. The wave function can be spread out over a large region, implying that the probability can also be broadly distributed, although it gives an accurate estimate of the probability that the electron will be found at any given place. Of course, when a measurement is actually made at some chosen point, the electron must either be detected or not be detected. But when it is actually detected, the wave function is never “smeared” out but always has a definite position. In technical terms, the wave function is said to collapse as soon as a quantum system is subjected to a measurement. The wave function converts instantaneously from describing a superposition of various possible observational outcomes to one that “assigns 100 percent probability to the outcome that was actually measured, and 0 percent to anything else”. In other words, the wave function has now collapsed into an “eigenstate.”

The collapse of a wave function is neither smooth, nor perfectly predictable. Furthermore, the information is not conserved and not reversible. The amplitude (squared) associated with any particular outcome is a measure of the probability that the wave function will collapse to a state that is concentrated entirely on that outcome.

The fact that the wave function of a quantum system collapses when it is subjected to a measurement is a bizarre concept and formed the basis of the Copenhagen interpretation of quantum mechanics, which I shall describe shortly. To the physicists it was not entirely clear what physical interpretation should be given to the wave function and the concept of wave function collapse was troubling. “…suspicion lingered that some equation ought to describe when and how this collapse occurred. Many physicists took this lack of an equation to mean that something was intrinsically wrong with quantum mechanics and that it would soon be replaced by a more fundamental theory that would provide such an equation” (Tegmark, M. & Wheeler, J. A., 2001). And because of ambiguities such as this, many physicists found it most sensible to regard quantum mechanics as merely a set of rules that prescribe the outcome of experiments.

About a dozen of “interpretations” of quantum mechanics have been proposed that try to explain why this wave function collapse happens, or whether it happens at all, or whether the wave function itself is physically real or merely a mathematical entity. These include the Copenhagen interpretation, the many-worlds interpretation, the guiding field interpretation, and the recent quantum Bayesian model. Each of these claim to eliminate some of the paradoxes of quantum mechanics—or at least put them in a less troubling form, although none has yet proven sufficiently compelling to achieve a scientific consensus. A comprehensive discussion of these interpretations is beyond the scope of this article, though we will focus on a few that exposes certain important features of quantum mechanics. But before diving into their subtleties, it is necessary to understand the Heisenberg uncertainty principle which is the cornerstone of quantum mechanics.

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