Renninger negative-result experiment
Encyclopedia
In quantum mechanics
, the Renninger negative-result experiment is a thought experiment
that illustrates some of the difficulties of understanding the nature of wave function collapse and measurement in quantum mechanics. The statement is that a particle need not be detected in order for a quantum measurement to occur, and that the lack of a particle detection can also constitute a measurement. The thought experiment was first posed in 1953 by Mauritius Renninger
. It can be understood to be a refinement of the paradox presented in the Mott problem
.
. Formulated in 1929 by Sir Nevill Francis Mott
and Werner Heisenberg
, it was resolved by a calculation done by Mott that showed that the correct quantum mechanical system must include the wave functions for the atoms in the cloud chamber as well as that for the alpha ray. The calculation showed that the resulting probability is non-zero only on straight lines raying out from the decayed atom; that is, once the measurement is performed, the wave-function becomes non-vanishing only near the classical trajectory of a particle.
s, completely surrounding a radioactive atom at the center that is about to decay by emitting an alpha ray. For the purposes of the thought experiment, the detectors are assumed to be 100% efficient, so that the emitted alpha ray is always detected.
By consideration of the normal process of quantum measurement, it is clear that if one detector registers the decay, then the other will not: a single particle cannot be detected by both detectors. The core observation is that the non-observation of a particle on one of the shells is just as good a measurement as detecting it on the other.
The strength of the paradox can be heightened by considering the two hemispheres to be of different diameters; with the outer shell a good distance farther away. In this case, after the non-observation of the alpha ray on the inner shell, one is led to conclude that the (originally spherical) wave function has "collapsed" to a hemisphere shape, and (because the outer shell is distant) is still in the process of propagating to the outer shell, where it is guaranteed to eventually be detected.
In the standard quantum-mechanical formulation, the statement is that the wave-function has partially collapsed, and has taken on a hemispherical shape. The full collapse of the wave function, down to a single point, does not occur until it interacts with the outer hemisphere. The conundrum of this thought experiment lies in the idea that the wave function interacted with the inner shell, causing a partial collapse of the wave function, without actually triggering any of the detectors on the inner shell. This illustrates that wave function collapse can occur even in the absence of particle detection.
invalidates the result. This objection can be dispelled by sizing the hemispheres appropriately with regards to the half-life of the nucleus. The radii are chosen so that the more distant hemisphere is much farther away than the half-life of the decaying nucleus, times the flight-time of the alpha ray.
To lend concreteness to the example, assume that the half-life of the decaying nucleus is 0.01 microsecond (most elementary particle
decay half-lives are much shorter; most nuclear decay half-lives are much longer; some atomic electromagnetic excitations have a half-life about this long). If one were to wait 0.4 microseconds, then the probability that the particle will have decayed will be
; that is, the probability will be very very close to one. The outer hemisphere is then placed at (speed of light) times (0.4 microseconds) away: that is, at about 120 meters away. The inner hemisphere is taken to be much closer, say at 1 meter.
If, after (for example) 0.3 microseconds, one has not seen the decay product on the inner, closer, hemisphere, one can conclude that the particle has decayed with almost absolute certainty, but is still in-flight to the outer hemisphere. The paradox then concerns the correct description of the wave function in such a scenario.
, and is false. The wave function was truly spherical, and is not the incoherent superposition (mixed state) of a large number of plane waves. The distinction between mixed and pure states is illustrated more clearly in a different context, in the debate comparing the ideas behind hidden variables
and their refutation by means of the Bell inequalities.
pattern to be observed on the outer hemisphere. This is not really an objection, but rather an affirmation that a partial collapse of the wave function has occurred. If a diffraction pattern were not observed, one would be forced to conclude that the particle had collapsed down to a ray, and stayed that way, as it passed the inner hemisphere; this is clearly at odds with standard quantum mechanics. Diffraction from the inner hemisphere is expected.
) or a photon
(spin-1). This is taken to mean that the decay is not truly sphere symmetric, but rather has some other distribution, such as a p-wave. However, on closer examination, one sees this has no bearing on the spherical symmetry of the wave-function. Even if the initial state could be polarized; for example, by placing it in a magnetic field, the non-spherical decay pattern is still properly described by quantum mechanics.
is not in conflict with quantum mechanics.
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 Renninger negative-result experiment is a thought experiment
Thought experiment
A thought experiment or Gedankenexperiment considers some hypothesis, theory, or principle for the purpose of thinking through its consequences...
that illustrates some of the difficulties of understanding the nature of wave function collapse and measurement in quantum mechanics. The statement is that a particle need not be detected in order for a quantum measurement to occur, and that the lack of a particle detection can also constitute a measurement. The thought experiment was first posed 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:*...
. It can be understood to be a refinement of the paradox presented in the Mott problem
Mott problem
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 Mott problem
The Mott problem concerns the paradox of reconciling the spherical wave function describing the emission of an alpha ray by a radioactive nucleus, with the linear tracks seen in a cloud chamberCloud 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...
. 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...
, it was resolved by a calculation done by Mott that showed that the correct quantum mechanical system must include the wave functions for the atoms in the cloud chamber as well as that for the alpha ray. The calculation showed that the resulting probability is non-zero only on straight lines raying out from the decayed atom; that is, once the measurement is performed, the wave-function becomes non-vanishing only near the classical trajectory of a particle.
Renninger's negative-result experiment
In the Renninger formulation, the cloud chamber is replaced by a pair of hemispherical particle detectorParticle detector
In experimental and applied particle physics, nuclear physics, and nuclear engineering, a particle detector, also known as a radiation detector, is a device used to detect, track, and/or identify high-energy particles, such as those produced by nuclear decay, cosmic radiation, or reactions in a...
s, completely surrounding a radioactive atom at the center that is about to decay by emitting an alpha ray. For the purposes of the thought experiment, the detectors are assumed to be 100% efficient, so that the emitted alpha ray is always detected.
By consideration of the normal process of quantum measurement, it is clear that if one detector registers the decay, then the other will not: a single particle cannot be detected by both detectors. The core observation is that the non-observation of a particle on one of the shells is just as good a measurement as detecting it on the other.
The strength of the paradox can be heightened by considering the two hemispheres to be of different diameters; with the outer shell a good distance farther away. In this case, after the non-observation of the alpha ray on the inner shell, one is led to conclude that the (originally spherical) wave function has "collapsed" to a hemisphere shape, and (because the outer shell is distant) is still in the process of propagating to the outer shell, where it is guaranteed to eventually be detected.
In the standard quantum-mechanical formulation, the statement is that the wave-function has partially collapsed, and has taken on a hemispherical shape. The full collapse of the wave function, down to a single point, does not occur until it interacts with the outer hemisphere. The conundrum of this thought experiment lies in the idea that the wave function interacted with the inner shell, causing a partial collapse of the wave function, without actually triggering any of the detectors on the inner shell. This illustrates that wave function collapse can occur even in the absence of particle detection.
Common objections
There are a number of common objections to the standard interpretation of the experiment; all of the objections demonstrate a lack of understanding of quantum mechanics. Some of these objections, and standard rebuttals, are listed below.Finite radioactive lifetime
It is sometimes noted that the time of the decay of the nucleus cannot be controlled, and that the finite half-lifeHalf-life
Half-life, abbreviated t½, is the period of time it takes for the amount of a substance undergoing decay to decrease by half. The name was originally used to describe a characteristic of unstable atoms , but it may apply to any quantity which follows a set-rate decay.The original term, dating to...
invalidates the result. This objection can be dispelled by sizing the hemispheres appropriately with regards to the half-life of the nucleus. The radii are chosen so that the more distant hemisphere is much farther away than the half-life of the decaying nucleus, times the flight-time of the alpha ray.
To lend concreteness to the example, assume that the half-life of the decaying nucleus is 0.01 microsecond (most elementary particle
Elementary particle
In particle physics, an elementary particle or fundamental particle is a particle not known to have substructure; that is, it is not known to be made up of smaller particles. If an elementary particle truly has no substructure, then it is one of the basic building blocks of the universe from which...
decay half-lives are much shorter; most nuclear decay half-lives are much longer; some atomic electromagnetic excitations have a half-life about this long). If one were to wait 0.4 microseconds, then the probability that the particle will have decayed will be
; that is, the probability will be very very close to one. The outer hemisphere is then placed at (speed of light) times (0.4 microseconds) away: that is, at about 120 meters away. The inner hemisphere is taken to be much closer, say at 1 meter.If, after (for example) 0.3 microseconds, one has not seen the decay product on the inner, closer, hemisphere, one can conclude that the particle has decayed with almost absolute certainty, but is still in-flight to the outer hemisphere. The paradox then concerns the correct description of the wave function in such a scenario.
Classical trajectories
Another common objection states that the decay particle was always travelling in a straight line, and that only the probability of the distribution is spherical. This, however, is a mis-interpretation of the Mott problemMott problem
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...
, and is false. The wave function was truly spherical, and is not the incoherent superposition (mixed state) of a large number of plane waves. The distinction between mixed and pure states is illustrated more clearly in a different context, in the debate comparing the ideas behind hidden variables
Hidden variables
Hidden variables may refer to:* Hidden variable theories, in physics a class of theories trying to explain away the statistical nature of quantum mechanics* Latent variables, in statistics, variables that are inferred from other observed variables...
and their refutation by means of the Bell inequalities.
Diffraction
A true quantum-mechanical wave would diffract from the inner hemisphere, leaving a diffractionDiffraction
Diffraction refers to various phenomena which occur when a wave encounters an obstacle. Italian scientist Francesco Maria Grimaldi coined the word "diffraction" and was the first to record accurate observations of the phenomenon in 1665...
pattern to be observed on the outer hemisphere. This is not really an objection, but rather an affirmation that a partial collapse of the wave function has occurred. If a diffraction pattern were not observed, one would be forced to conclude that the particle had collapsed down to a ray, and stayed that way, as it passed the inner hemisphere; this is clearly at odds with standard quantum mechanics. Diffraction from the inner hemisphere is expected.
Complex decay products
In this objection, it is noted that in real life, a decay product is either spin-1/2 (a fermionFermion
In particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....
) or a photon
Photon
In physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...
(spin-1). This is taken to mean that the decay is not truly sphere symmetric, but rather has some other distribution, such as a p-wave. However, on closer examination, one sees this has no bearing on the spherical symmetry of the wave-function. Even if the initial state could be polarized; for example, by placing it in a magnetic field, the non-spherical decay pattern is still properly described by quantum mechanics.
Non-relativistic language
The above formulation is inherently phrased in a non-relativistic language; and it is noted that elementary particles have relativistic decay products. This objection only serves to confuse the issue. The experiment can be reformulated so that the decay product is slow-moving. At any rate, special relativitySpecial relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
is not in conflict with quantum mechanics.