Quantum mechanics describes particles on the smallest length and energy scales, at the level of atoms and subatomic particles. At these small scales, it is not possible to speak of a “particle”; it is more like the particle is spread out over space. Instead of being described as a point particle it is described by a wave packet, formed by interfering waves. Think, for example, of waves on a lake or river; if they are reflected from the banks they can interfere with the other waves, enhancing the up and down movement of the water. Such a region is quite similar to a wave packet, where the waves interfere, increasing their amplitude. However, now it is not water that is oscillating, but a “particle field”; the particle is spread out over all space, but the oscillations determine where the particle can be found.
These oscillations are described by a mathematical object called a function; this assigns a value to the field at any point in space and time. Due to its resemblance to waves it is called a wavefunction. The particle is more likely to be in places where the wavefunction becomes very large; the wavefunction describes the probability of the particle to be there. And if this was not confusing enough, when you measure the position of the particle, it will suddenly be in one place only! This is called the collapse of the wavefunction; instead of being spread out over all space, the wavefunction collapses into one point, and the probability is now 100% to find the particle there.
Wavefunctions do not just describe the position of the particle, but all kinds of properties; its energy, momentum, angular momentum, etc. Further, the outcome of a measurements can only take a certain, discrete, set of values instead of a continuous range; the properties are quantized. When measuring a property the wavefunction collapses to a state that has one specific value only of that property.
(In mathematical language, measuring a property is described by acting with an operator on the wavefunction. The operators can be represented by matrices and the wavefunctions by vectors. When a property is measured, the vector chooses to collapse onto one of the eigenvectors; the possible outcomes are the eigenvalues of the matrix.)
One other confusing observation is that different properties cannot be measured precisely at the same time; it is possible to measure one property very accurately but this will increase the uncertainty in measuring another property. For example position and momentum; you can only know how fast the particle is moving or where it is at some moment in time, but not both!
These observations are all part of the most common interpretation of quantum mechanics, the Copenhagen interpretation. However, there are also other interpretations possible. These will be mentioned below in the context of the double-slit experiment.
In this experiment a beam of light is shone upon two parallel slits and will produce an interference pattern on a screen, as if the two slits act as light sources themselves. However, it can be measured that the photons go through one of the two slits only. So light acts as a particle and a wave at the same time! This experiment can also be performed with electrons, that seem to act as waves as well. However, measuring through which slit the light or electrons went destroys the interference pattern on the screen.
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The Copenhagen interpretation states that the particles are not localised in space until they are detected. If there is no detector on the slits, there is no information about which slit the particle has passed through, they travel through both slits and create the interference pattern. If one slit has a detector on it, then the wavefunction collapses due to that detection, and the particles can only travel through one slit.
Another interpretation of quantum mechanics is the de Broglie-Bohm theory. There is only one wavefunction in this interpretation, that describes the whole universe. Its amplitude determines the regions where the particle can go, ensuring that each particle follows a specific trajectory and passes through one slit only. Where it ends up on the screen is determined by its initial position. This cannot be controlled, which makes the way the pattern appears on the screen seem random. When the wavefunction “collapses”, it only happens to the part of the wavefunction that describes this experiment. The rest of the wavefunction, describing everything else in of the universe, does not collapse.
There is another interpretation of quantum mechanics that deals with the collapse of the wavefunction: the many-worlds interpretation. In this interpretation the state of the observer and the observed object are correlated to each other; a measurement makes the two interact. After an observation is made the state of the observed object is collapsed and the observer is in a state that has observed the corresponding outcome. The two states have now combined in one new state, that evolves in such a way that future outcomes are always consistent with earlier outcomes. However, this new state exist next to the states that correspond to measuring different outcomes.
This can be illustrated by a tree that describes the evolution of the system; every time a measurement takes place a branch will split into two or more outcomes. In this picture, a “world” is the complete measurement history of the observer. The wavefunction has not truly collapsed, but only appears to be that way. It is actually still in superposition with the other possible outcomes. The only difference before and after an observation is the coupling between the state of the observer and the object. However, the terminology is misleading! This theory does not imply that there are multiple universes present at one moment.
Karl Popper devised an experiment to determine which interpretation of quantum mechanics is correct. In the Copenhagen interpretation the wavefunction collapses when a measurement is performed. This requires some kind of “action-at-a-distance”; the observer affects the object although there is no contact between the two. Popper designed an experiment that should decide on whether this action-at-a-distance takes place.
Imagine two particles that are emitted at the same time, travelling in opposite directions. They both travel through a slit and are scattered due to the uncertainty principle. Their position in the direction parallel to the slits is well known (as they pass through the slits) hence their momentum in this direction is uncertain, and they can scatter in any direction. This means that a narrow slit will cause more scattering. If only one slit is narrowed, however, the scattering at the other slit will be affected as well since the two particles states are correlated to each other. However, this increase in scatter is caused by our knowledge about the width of the slits. Hence the measurement depends on what the observer knows, and there is an action-at-a-distance. To summarize: if the scattering increases, the Copenhagen interpretation is correct.
This experiment was performed in 1999, with a photon source. However, there was no increased scattering observed. Rather, it was decreased. The authors concluded: “Popper was right in predicting the outcome of the experiment. However, Popper made the error by applying the results of two-particle physics to the explanation of the behavior of an individual particle. The two-particle entangled state is not the state of two individual particles. Our experimental result is emphatically NOT a violation of the uncertainty principle which governs the behavior of an individual quantum.“