- Ensemble Interpretation
The Ensemble Interpretation, or Statistical Interpretation of
quantum mechanics , is an interpretation that can be viewed as a minimalist interpretation; it is a quantum mechanical interpretation that claims to make the fewest assumptions associated with the standard mathematical formalization. At its heart, it takes the statistical interpretation ofMax Born to the fullest extent. The interpretation states that the wave function does not apply to an individual system – or for example, a single particle – but is an abstract mathematical, statistical quantity that only applies to an ensemble of similar prepared systems or particles. Probably the most notable supporter of such an interpretation wasAlbert Einstein :To date, probably the most prominent advocate of the Ensemble Interpretation is
Leslie E. Ballentine , Professor atSimon Fraser University , and writer of the graduate-level textbook "Quantum Mechanics, A Modern Development".cite book |title= Quantum Mechanics: A Modern Development |page=Chapter 9 |url=http://books.google.com/books?id=sHJRFHz1rYsC&printsec=frontcover&dq=intitle:Quantum+intitle:Mechanics+intitle:A+intitle:Modern+intitle:Development&lr=&as_brr=0#PPA230,M1
isbn=9810241054 |year=1998 |publisher=World Scientific |author=Leslie E. Ballentine]The ensemble interpretation, unlike other interpretations to the Copenhagen Interpretation Fact|date=April 2007, does not attempt to justify, or otherwise derive, or explain quantum mechanics from any deterministic process, or make any other statement about the real nature of quantum phenomena; it is simply a statement as to the manner of wave function interpretation. It is identical in all of its predictions as is the standard interpretations.
Measurement and Collapse
The attraction of the ensemble interpretation is that it immediately dispenses with the metaphysical issues associated with reduction of the state vector, Schrödinger cat states, and other issues related to the concepts of multiple simultaneous states. As the ensemble interpretation postulates that the wave function only applies to an ensemble of systems, there is no requirement for any single system to exist in more than one state at a time, hence, the wave function is never physically required to be "reduced". This can be illustrated by an example:
Consider a classical die. If this is expressed in Dirac notation, the "state" of the die can be represented by a "wave" function describing the probability of an outcome given by:
:
It is clear that on each throw, only one of the states will be observed, but it is also clear that there is no requirement for any notion of collapse of the wave function/reduction of the state vector, or for the die to physically exist in the summed state. In the ensemble interpretation, wave function collapse would make as much sense as saying that the number of children a couple produced, collapsed to 3 from its average value of 2.4.
The state function is not taken to be physically real, or be a literal summation of states. The wave function, is taken to be an abstract statistical function, only applicable to the statistics of repeated preparation procedures, similar to classical
statistical mechanics . It does not directly apply to a single experiment, only the statistical results of many.Criticism
David Mermin sees the Ensemble interpretation as being motivated by an adherence ("not always acknowledged") to classical principles.
"For the notion that probabilistic theories mustbe about ensembles implicitly assumes that probability is about ignorance. (The “hiddenvariables” are whatever it is that we are ignorant of.) But in a non-determinstic worldprobability has nothing to do with incomplete knowledge, and ought not to require anensemble of systems for its interpretation".
He also emphasises the importance of "describing" single systems, rather than ensembles.
"The second motivation for an ensemble interpretation is the intuition that becausequantum mechanics is inherently probabilistic, it only needs to make sense as a theory ofensembles. Whether or not probabilities can be given a sensible meaning for individualsystems, this motivation is not compelling. For a theory ought to be able to describe aswell as predict the behavior of the world. The fact that physics cannot make deterministicpredictions about individual systems does not excuse us from pursuing the goal of beingable to describe them as they currently are." [ [http://arxiv.org/abs/quant-ph/9609013v1 Mermin, N.D. "The Ithaca interpretation"] ]
ingle particles
According to proponents of this interpretation, no single system is ever required to be postulated to exist in a physical mixed state so the state vector does not need to collapse.
It can also be argued that this notion is consistent with the standard interpretation in that, in the CI, statements about the exact system state prior to measurement can not be made. That is, if it were possible to absolutely, physically measure say, a particle in two positions at once, then QM would be falsified as QM explicitly postulates that the result of any measurement must be a single
eigenvalue of a single eigenstate.Criticism
Alfred Nuemaier find fault with the applicability of the Ensemble Interpretation to small systems.
"Among the traditional interpretations, the statistical interpretationdiscussed by Ballentine in Rev. Mod. Phys. 42, 358-381 (1970) is theleast demanding (assumes less than the Copenhagen interpretationand the Many Worlds interpretation) and the most consistent one.It explains almost everything, and only has the disadvantage thatit explicitly excludes the applicability of QM to single systems or very small ensembles (such as the few solar neutrinos or top quarksactually detected so far), and does not bridge the gulf betweenthe classical domain (for the description of detectors) and thequantum domain (for the description of the microscopic system)".(spelling emended) [ [http://www.mat.univie.ac.at/~neum/physics-faq.txt Alfred Neumaier's FAQ] ]
Schrödinger's cat
The Ensemble Interpretation states that superpositions are nothing but subensembles of a larger statistical ensemble. That being the case, the state vector would not apply to individual cat experiments, but only to the statistics of many similar prepared cat experiments. Proponents of this interpretation state that this makes the
Schrödinger's cat paradox a trivial non issue. However, the application of state vectors to individual systems, rather than ensembles, has explanatory benefits, in areas like single-particle twin-slit experiments and quantum computing. (See here). As an avowedly minimalist approach, the Ensemble Interpretation does not offer any specific alternative explanation for these phenomena.The frequentist probability variation
The claim that the wave functional approach fails "to apply" to single particle experiments cannot be taken as a claim that quantum mechanics fails in describing single-particle phenomena. In fact, it gives correct results within the limits of a probablistic or
stochastic theory.Probability always require a set of multiple data, and thus single-particle experiments arereally part of an ensemble — an ensemble of individual experiments that are performed one after the other over time. In particular, the interference fringes seen in the
double slit experiment require repeated trials to be observed.Criticism
One possible criticism is that the procedure described is essentially
frequentist . The frequentist approach suffices for "classical" probability, but quantum mechanics is a theory of quantum probability, which is more general. [ [http://plato.stanford.edu/entries/qt-quantlog/ Stanford Encyclopedia of Philosophy] ] [ [http://math.ucr.edu/home/baez/bayes.html Baez, J. "Bayesian probability Theory and Quantum Mechanics"] ]The quantum Zeno effect
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