Quantum non-equilibrium
Encyclopedia
Quantum non-equilibrium is a concept within stochastic formulations of the De Broglie–Bohm theory of quantum physics.
In the Copenhagen interpretation
, that is, the most widely used interpretation of quantum mechanics, the Born rule
defines that 
, the probability density function
of a particle (that is, the probability to find the particle in the differential volume
at time t) equals the absolute square of the wave function 
, and it constitutes one of the basic assumptions of the theory.
This is not the case for the De Broglie–Bohm theory, where the Born rule is not a basic law. Rather, in this theory the link between the probability density and the wave function is considered a hypothesis, called the quantum equilibrium hypothesis, which is additional to the basic principles governing the wave function, the dynamics of the quantum particles and the Schrödinger equation
. (For mathematical details, refer to the derivation by Peter R. Holland.)
That is, a quantum non-equilibrium is characterized in that the Born rule is not fulfilled; that is, the probability to find the particle in in the differential volume
at time t is unequal to 
.
Recent advances in investigations into properties of quantum non-equilibrium states has been performed mainly by theoretical physicist Antony Valentini
, and before him steps in this direction were undertaken by David Bohm
, Jean-Pierre Vigier
, Basil Hiley
and Peter R. Holland
. The existence of quantum non-equilibrium states has not been verified experimentally; quantum non-equilibrium is so far a theoretical construct. The relevance of quantum non-equilibrium states to physics lies in the fact that they can lead to different predictions for results of experiments, depending on whether the De Broglie–Bohm theory in its stochastic form or the Copenhagen interpretation
is assumed to describe reality. (The Copenhagen interpretation, which stipulates the Born rule a priori, does not foresee the existence of quantum non-equilibrium states at all.) That is, properties of quantum non-equilibrium can make certain classes of Bohmian theories falsifiable
according to the criterion of Karl Popper
.
In practice, when performing Bohmian mechanics computations in quantum chemistry
, the quantum equilibrium hypothesis is simply considered to be fulfilled, in order to predict system behaviour and the outcome of measurements.
Bohm and other physicists, including Valentini, view the Born rule
linking
to the probability density function
as representing not a basic law, but rather as constituting a result of a system having reached quantum equilibrium during the course of the time development under the Schrödinger equation
. It can be shown that, once an equilibrium has been reached, the system remains in such equilibrium over the course of its further evolution: this follows from the continuity equation associated with the Schrödinger evolution of
. However, it is less straightforward to demostrate whether and how such an equilibrium is reached in the first place.
In 1991, Valentini provided indications for deriving the quantum equilibrium hypothesis which states that
in the framework of the pilot wave theory. (Here, 
stands for the collective coordinates of the system in configuration space
). Valentini showed that the relaxation
→ 
may be accounted for by a H-theorem constructed in analogy to the Boltzmann H-theorem of statistical mechanics.
Valentini's derivation of the quantum equilibrium hypothesis was criticized by Detlef Dürr and co-workers in 1992, and the derivation of the quantum equilibrium hypothesis has remained a topic of active investigation.
” for non-equilibrium cases in which
≠
, thereby violating the assumption that signals cannot travel faster than the speed of light
.
Valentini furthermore showed that an ensemble of particles with known wave function and known nonequilibrium distribution could be used to perform, on another system, measurements that violate the uncertainty principle
.
These predictions differ from predictions that would result from approaching the same physical situation by means of the Copenhagen interpretation
and therefore would in principle make the predictions of this theory accessible to experimental study. As it is unknown whether or how quantum non-equilibrium states can be produced, it is difficult or impossible to perform such experiments.
However, also the hypothesis of quantum non-equilibrium big bang
gives rise to quantitative predictions for nonequilibrium deviations from quantum theory which appear to be more easily accessible to observation.
| Quantum non-equilibrium: |
  |
| Relaxation to quantum equilibrium: |
→  |
| Quantum equilibrium hypothesis: |
with  representing the probability density functionProbability density function In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the... |
and  representing the wave function. |
In the Copenhagen interpretation
Copenhagen interpretation
The Copenhagen interpretation is one of the earliest and most commonly taught interpretations of quantum mechanics. It holds that quantum mechanics does not yield a description of an objective reality but deals only with probabilities of observing, or measuring, various aspects of energy quanta,...
, that is, the most widely used interpretation of quantum mechanics, the Born rule
Born rule
The Born rule is a law of quantum mechanics which gives the probability that a measurement on a quantum system will yield a given result. It is named after its originator, the physicist Max Born. The Born rule is one of the key principles of quantum mechanics...
defines that , the probability density functionProbability density function
In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...
of a particle (that is, the probability to find the particle in the differential volume
at time t) equals the absolute square of the wave function , and it constitutes one of the basic assumptions of the theory.This is not the case for the De Broglie–Bohm theory, where the Born rule is not a basic law. Rather, in this theory the link between the probability density and the wave function is considered a hypothesis, called the quantum equilibrium hypothesis, which is additional to the basic principles governing the wave function, the dynamics of the quantum particles and the Schrödinger equation
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
. (For mathematical details, refer to the derivation by Peter R. Holland.)
That is, a quantum non-equilibrium is characterized in that the Born rule is not fulfilled; that is, the probability to find the particle in in the differential volume
at time t is unequal to .Recent advances in investigations into properties of quantum non-equilibrium states has been performed mainly by theoretical physicist Antony Valentini
Antony Valentini
Antony Valentini is a theoretical physicist and a professor at Clemson University. He is known for his work on the foundations of quantum physics.- Education and career :...
, and before him steps in this direction were undertaken by David Bohm
David Bohm
David Joseph Bohm FRS was an American-born British quantum physicist who contributed to theoretical physics, philosophy, neuropsychology, and the Manhattan Project.-Youth and college:...
, Jean-Pierre Vigier
Jean-Pierre Vigier
Jean-Pierre Vigier earned his Ph.D. in Mathematics from University of Geneva in 1946 and in 1948 was appointed assistant to Louis de Broglie, a position he held until the latter's retirement in 1962. Vigier was professor emeritus at in the Department of Gravitational Physics at Pierre et Marie...
, Basil Hiley
Basil Hiley
Basil Hiley, born 1935, is a British quantum physicist and professor emeritus of the University of London.- Work :Hiley published a paper in 1961 on the random walk of a macromolecule, which was followed by further papers on the Ising model, and lattice constant systems defined in graph theoretical...
and Peter R. Holland
Peter R. Holland
Peter R. Holland is a theoretical physicist, known for his book on the pilot wave theory and the de Broglie-Bohm causal interpretation of quantum mechanics and his work on foundational problems in quantum physics....
. The existence of quantum non-equilibrium states has not been verified experimentally; quantum non-equilibrium is so far a theoretical construct. The relevance of quantum non-equilibrium states to physics lies in the fact that they can lead to different predictions for results of experiments, depending on whether the De Broglie–Bohm theory in its stochastic form or the Copenhagen interpretation
Copenhagen interpretation
The Copenhagen interpretation is one of the earliest and most commonly taught interpretations of quantum mechanics. It holds that quantum mechanics does not yield a description of an objective reality but deals only with probabilities of observing, or measuring, various aspects of energy quanta,...
is assumed to describe reality. (The Copenhagen interpretation, which stipulates the Born rule a priori, does not foresee the existence of quantum non-equilibrium states at all.) That is, properties of quantum non-equilibrium can make certain classes of Bohmian theories falsifiable
Falsifiability
Falsifiability or refutability of an assertion, hypothesis or theory is the logical possibility that it can be contradicted by an observation or the outcome of a physical experiment...
according to the criterion of Karl Popper
Karl Popper
Sir Karl Raimund Popper, CH FRS FBA was an Austro-British philosopher and a professor at the London School of Economics...
.
In practice, when performing Bohmian mechanics computations in quantum chemistry
Quantum chemistry
Quantum chemistry is a branch of chemistry whose primary focus is the application of quantum mechanics in physical models and experiments of chemical systems...
, the quantum equilibrium hypothesis is simply considered to be fulfilled, in order to predict system behaviour and the outcome of measurements.
Relaxation to equilibrium
The causal interpretation of quantum mechanics has been set up by de Broglie and Bohm as a causal, deterministic model, and it was extended later by Bohm, Vigier, Hiley, Valentini and others to include stochastic properties.Bohm and other physicists, including Valentini, view the Born rule
Born rule
The Born rule is a law of quantum mechanics which gives the probability that a measurement on a quantum system will yield a given result. It is named after its originator, the physicist Max Born. The Born rule is one of the key principles of quantum mechanics...
linking
to the probability density functionProbability density function
In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...
as representing not a basic law, but rather as constituting a result of a system having reached quantum equilibrium during the course of the time development under the Schrödinger equationSchrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
. It can be shown that, once an equilibrium has been reached, the system remains in such equilibrium over the course of its further evolution: this follows from the continuity equation associated with the Schrödinger evolution of
. However, it is less straightforward to demostrate whether and how such an equilibrium is reached in the first place.In 1991, Valentini provided indications for deriving the quantum equilibrium hypothesis which states that
in the framework of the pilot wave theory. (Here, stands for the collective coordinates of the system in configuration spaceConfiguration space
- Configuration space in physics :In classical mechanics, the configuration space is the space of possible positions that a physical system may attain, possibly subject to external constraints...
). Valentini showed that the relaxation
→ may be accounted for by a H-theorem constructed in analogy to the Boltzmann H-theorem of statistical mechanics.Valentini's derivation of the quantum equilibrium hypothesis was criticized by Detlef Dürr and co-workers in 1992, and the derivation of the quantum equilibrium hypothesis has remained a topic of active investigation.
Predicted properties of quantum non-equilibrium
Valentini showed that his expansion of the De Broglie–Bohm theory would allow “signal nonlocalityNonlocality
In Classical physics, nonlocality is the direct influence of one object on another, distant object. In Quantum mechanics, nonlocality refers to the absence of a local, realist model in agreement with quantum mechanical predictions.Nonlocality may refer to:...
” for non-equilibrium cases in which
≠, thereby violating the assumption that signals cannot travel faster than the speed of lightSpeed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
.
Valentini furthermore showed that an ensemble of particles with known wave function and known nonequilibrium distribution could be used to perform, on another system, measurements that violate the uncertainty principle
Uncertainty principle
In quantum mechanics, the Heisenberg uncertainty principle states a fundamental limit on the accuracy with which certain pairs of physical properties of a particle, such as position and momentum, can be simultaneously known...
.
These predictions differ from predictions that would result from approaching the same physical situation by means of the Copenhagen interpretation
Copenhagen interpretation
The Copenhagen interpretation is one of the earliest and most commonly taught interpretations of quantum mechanics. It holds that quantum mechanics does not yield a description of an objective reality but deals only with probabilities of observing, or measuring, various aspects of energy quanta,...
and therefore would in principle make the predictions of this theory accessible to experimental study. As it is unknown whether or how quantum non-equilibrium states can be produced, it is difficult or impossible to perform such experiments.
However, also the hypothesis of quantum non-equilibrium big bang
Big 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...
gives rise to quantitative predictions for nonequilibrium deviations from quantum theory which appear to be more easily accessible to observation.