Mott problem
Encyclopedia
In quantum mechanics
, the Mott problem is a paradox
that illustrates some of the difficulties of understanding the nature of wave function collapse and measurement in quantum mechanics. The problem was first formulated in 1929 by Sir Nevill Francis Mott
and Werner Heisenberg
, illustrating the paradox of the collapse of a spherically symmetric wave function into the linear tracks seen in a cloud chamber
.
In practice, virtually all high energy physics experiments, such as those conducted at particle colliders, involve wave functions which are inherently spherical. Yet, when the results of a particle collision are detected, they are invariably in the form of linear tracks (see, for example, the illustrations accompanying the article on bubble chamber
s). It is somewhat strange to think that a spherically symmetric wave function should be observed as a straight track, and yet, this occurs on a daily basis in all particle collider experiments.
A related variant formulation was given in 1953 by Mauritius Renninger
, and is now known as Renninger's negative-result gedanken experiment. In this formulation, it is noted that the absence of a particle detection can also constitute a quantum measurement; namely, that a measurement can be performed even if no particle whatsoever is detected.
was considered. It was noted that the result of such a decay is always observed as linear tracks seen in Wilson's cloud chamber
. Intuitively, one might think that such a wave function should randomly ionize atoms throughout the cloud chamber, but this is not the case. Mott demonstrated that by considering the interaction in configuration space, where all of the atoms of the cloud chamber play a role, it is overwhelmingly probable that all of the condensed droplets in the cloud chamber will lie close to the same straight line. What is uncertain is which straight line the wave packet will reduce to; the probability distribution of straight tracks is spherically symmetric.
Mott's analysis is reminiscent of the Feynman path integral approach, but predates it by twenty years. By considering every possible combination of the location of the atoms ionized by the alpha particle, Mott showed that the joint probability distribution is overwhelmingly dominated by the case where the trail of ionization is close to a classical path.
or other astrophysical phenomena is considered.
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
, the Mott problem is a paradox
Paradox
Similar to Circular reasoning, A paradox is a seemingly true statement or group of statements that lead to a contradiction or a situation which seems to defy logic or intuition...
that illustrates some of the difficulties of understanding the nature of wave function collapse and measurement in quantum mechanics. The problem was first formulated in 1929 by Sir Nevill Francis Mott
Nevill Francis Mott
Sir Nevill Francis Mott, CH, FRS was an English physicist. He won the Nobel Prize for Physics in 1977 for his work on the electronic structure of magnetic and disordered systems, especially amorphous semiconductors. The award was shared with Philip W. Anderson and J. H...
and Werner Heisenberg
Werner Heisenberg
Werner Karl Heisenberg was a German theoretical physicist who made foundational contributions to quantum mechanics and is best known for asserting the uncertainty principle of quantum theory...
, illustrating the paradox of the collapse of a spherically symmetric wave function into the linear tracks seen in a cloud chamber
Cloud chamber
The cloud chamber, also known as the Wilson chamber, is a particle detector used for detecting ionizing radiation. In its most basic form, a cloud chamber is a sealed environment containing a supersaturated vapor of water or alcohol. When a charged particle interacts with the mixture, it ionizes it...
.
In practice, virtually all high energy physics experiments, such as those conducted at particle colliders, involve wave functions which are inherently spherical. Yet, when the results of a particle collision are detected, they are invariably in the form of linear tracks (see, for example, the illustrations accompanying the article on bubble chamber
Bubble chamber
A bubble chamber is a vessel filled with a superheated transparent liquid used to detect electrically charged particles moving through it. It was invented in 1952 by Donald A. Glaser, for which he was awarded the 1960 Nobel Prize in Physics...
s). It is somewhat strange to think that a spherically symmetric wave function should be observed as a straight track, and yet, this occurs on a daily basis in all particle collider experiments.
A related variant formulation was given in 1953 by Mauritius Renninger
Mauritius Renninger
Mauritius Renninger was a theoretical physicist noted for his work on crystallography and x-ray optics. He's known for the Renninger effect and for the Renninger negative-result experiment.-External links:*...
, and is now known as Renninger's negative-result gedanken experiment. In this formulation, it is noted that the absence of a particle detection can also constitute a quantum measurement; namely, that a measurement can be performed even if no particle whatsoever is detected.
Mott's analysis
In the original 1929 formulation by Mott and Heisenberg, the spherical wave function of an alpha ray emitted from the decay of a radioactive atomic nucleusAtomic nucleus
The nucleus is the very dense region consisting of protons and neutrons at the center of an atom. It was discovered in 1911, as a result of Ernest Rutherford's interpretation of the famous 1909 Rutherford experiment performed by Hans Geiger and Ernest Marsden, under the direction of Rutherford. The...
was considered. It was noted that the result of such a decay is always observed as linear tracks seen in Wilson's cloud chamber
Cloud chamber
The cloud chamber, also known as the Wilson chamber, is a particle detector used for detecting ionizing radiation. In its most basic form, a cloud chamber is a sealed environment containing a supersaturated vapor of water or alcohol. When a charged particle interacts with the mixture, it ionizes it...
. Intuitively, one might think that such a wave function should randomly ionize atoms throughout the cloud chamber, but this is not the case. Mott demonstrated that by considering the interaction in configuration space, where all of the atoms of the cloud chamber play a role, it is overwhelmingly probable that all of the condensed droplets in the cloud chamber will lie close to the same straight line. What is uncertain is which straight line the wave packet will reduce to; the probability distribution of straight tracks is spherically symmetric.
Mott's analysis is reminiscent of the Feynman path integral approach, but predates it by twenty years. By considering every possible combination of the location of the atoms ionized by the alpha particle, Mott showed that the joint probability distribution is overwhelmingly dominated by the case where the trail of ionization is close to a classical path.
Modern applications
In modern times, the Mott problem is occasionally considered theoretically in the context of astrophysics and cosmology, where the evolution of the wave function from the Big BangBig Bang
The Big Bang theory is the prevailing cosmological model that explains the early development of the Universe. According to the Big Bang theory, the Universe was once in an extremely hot and dense state which expanded rapidly. This rapid expansion caused the young Universe to cool and resulted in...
or other astrophysical phenomena is considered.