Ensemble Interpretation
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In quantum mechanics, the ensemble interpretation or statistical interpretation is an interpretation of quantum mechanics 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. It maintains that quantum mechanics is not applicable to single particles.
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[edit] Introduction
At its heart, it takes the statistical interpretation of Max 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 was Albert Einstein:
| “ | The attempt to conceive the quantum-theoretical description as the complete description of the individual systems leads to unnatural theoretical interpretations, which become immediately unnecessary if one accepts the interpretation that the description refers to ensembles of systems and not to individual systems. | „ |
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—Albert Einstein[1] |
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To date, probably the most prominent advocate of the Ensemble Interpretation is Leslie E. Ballentine, Professor at Simon Fraser University, and writer of the graduate-level textbook "Quantum Mechanics, A Modern Development".
[edit] Variations
[edit] Underlying classicism
Early proponents of the statistical approach regarded quantum mechanics as an approximation to a classical theory. John Gribbin writes:
"The basic idea is that each quantum entity (such as an electron or a photon) has precise quantum properties (such as position or momentum) and the quantum wavefunction is related to the probability of getting a particular experimental result when one member (or many members) of the ensemble is selected by a experiment"
However, hopes for turning quantum mechanics back into a classical theory were dashed. Gribbin continues:
"There are many difficulties with the idea, but the killer blow was struck when individual quantum entities such as photons were observed behaving in experiments in line with the quantum wave function description. The Ensemble interpretation is now only of historical interest"
So why does the ensemble interpretation still have adherents? It has modified its claims in various ways.
[edit] The frequentist probability variation
If the original claim there is an underlying classical theory must be abandoned, how can the ensemble interpretation be modified? The claim that the wave functional approach fails to apply to single particle experiments cannot be taken as a claim that quantum fails in describing single-particle phenomena. In fact, it gives correct results within the limits of a probablistic or stochastic theory. However, classical physics is wrong as a description of how individual particles behave. Whatever the motivation is for denying that individual particles have wave functions, it is not incorrect results.
According to the ensemble interpretation, 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, in much the same way as classical statistical mechanics. It does not directly apply to a single experiment, only the statistical results of many.
Earlier it was stated that conventional quantum mechanics gives results within the limits of a probablistic theory. It can be claimed that no single single-particle experiment gives meaningful results. Probability always require a set of multiple data, and thus single-particle experiments are really 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.
This ingenious suggestions is open to some criticisms.
One 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.[3][4]
Another objection is that single experiments can be meaningful. If photons were tiny billiard-balls obeying classical physics, they would not be able to pass through the slits of the double slit experiment and so nothing would be observed on the screen. The fact that something is observed in a single experiment falsifies classical physics.
[edit] The non-comittal variation
Since it cannot be asserted that classical physics is a correct description of the microworld, or that quantum physics is false, Ensemblists are often left with no positive interpretation of microscopic behavior.
"The Ensemble Interpretation does not attempt any explanation as to why Quantum Mechanics is the way it is. It simply states a rational way of interpreting and calculating results without introducing the conceptual difficulties that are inherent in the Copenhagen Interpretation".
It is interesting to compare this apprach to the Copenhagen Interpretation. Since Ensemblists deny collapse (unlike Copenhagenists), they effectively assert that unmeasured entities are already in the state that they will eventually be meaasured in, unlike the Copenhagen Interpretation, which is non-commital on the point.
[edit] The particle-only variation
Ensemble proponent Kevin Aylward writes:
"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 eigen value of a single eigen state".
However, wave like behavior can be inferred. The photon in the double slit experiment must go through both slits — the interference pattern is destroyed if either of the slits is blocked. Moreover, the passage assumes that only position eigenstates matter, but a well-defined momentum eigentstate is automatically a poorly defined position eigenstate — that is the, position of the quantum entity is spread out or blurred.
The wave-particle duality is one of the most common misconception of Quantum Mechanics. Particles, apparently, are always particles and never waves. What they do do, is operate on the basis of Quantum Mechanics, not Newtonian Mechanics.
Aylward's motivation for denying the existence of wave behavior is his belief that only particles are observed.
"Whether the phenomena is electrons or light, what is always observed, are small localized impacts on screens that build up statistically in a pattern that is similar to that expected of continuous pure waves."
However, photons (for example) can be observed as waves using a simple aerial (that is, an aerial extract frequency information but no position information).[6]
[edit] Implications
[edit] Measurement and collapse
According to the Ensemble interpetation, 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. For example, it can be assumed that before the measurement, that the system was simply in the measured state, although this assumption is not strictly necessary."
However, in the single-particle double-slit experiment, the measured state is a particle state with a well-defined position. However, if the photon had been in a particle state throughout its transition through the apparatus, an interference pattern could not have been obtained. There are specific experimental reasons for thinking collapse occurs at the point of measurement.
[edit] Superposition and Schrödinger's cat
The attraction of the ensemble interpretation is that it appears to dispense 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.
However, the Dirac notation is not just a notation for classical probability; it can also express quantum probability structures (featuring complex numbers) that have no classical equivalent.
Moreover, the real existence of quantum superpositions is a useful idea. It explains the single-particle double slit experiment, and also the efficacy of quantum computing. (Thus, it is justifiable by abduction even if not directly observable). There is even some (fairly) direct observational evidence.[7]. The denial of ontologically real wave functions is perhaps motivated by the "particle only" variation of the Ensemble interpretation, but as has already been stated, that is based on the incorrect premise that wave-states cannot be directly observed.
If Ensemblists maintaint the view that superpositions never occur, they are left with no explanation of single-particle (e.g Young's slits) experiments and quantum computing. No framework but superposition has been put foreward to explain why these phenomena exist at all.[8]. On the other hand, if they admit that there is such a phenomenon as superposition, they are left without any response to the challenge of the Schrödinger's_catparadox: whether and how collapse occurs.
[edit] The quantum Zeno effect
Leslie Ballantine promoted the Ensemble Interpretation in his book "Quantum Mechanics, A Modern Development". In it [9], he described what he called the "Watched Pot Experiment". His argument was that, under certain circmstances, a repeatedly measured system, such as an unstable nucleus, would be prevented from decaying by the act of measurement itself. He initially presented this as a kind of reductio ad absurdum of wave function collapse. [10]
The effect has been shown to be real. (It is more widely known as the quantum Zeno effect). He later wrote papers claiming that it could be explained without wave function collapse.[11]
[edit] References
- ^ Einstein: Philosopher-Scientist, ed. P.A. Schilpp (Harper & Row, New York)
- ^ John Gribbin, Q for Quantum
- ^ Stanford Encyclopedia of Philosophy
- ^ Baez, J. Bayesian probability Theory and Quantum Mechanics
- ^ Kevin Aylward
- ^ ""Whereas some physicists have indeed taken the view that all measurements are ultimately measurements of position, I would myself regard such a perspective as being much too narrow. Indeed, the way the quantum formalism is normally presented does no require all measurements to be only of position". (R. Penrose, "Road to Reality", p517)"
- ^ PhysicsWeb New Life for Schrödinger's Cat
- ^ Can Schrodinger's Cat Factor large Numbers
- ^ Ballentine, L. Quantum Mechanics, A Modern Development(p 342)
- ^ "Like the old saying "A watched pot never boils", we have been led to the conclusion that a contineously observed system never changes its state! This conclusion is, of course false. The fallacy clearly results from the assertion that if an observation indicates no decay, then the state vector must be |y_u>. Each successive observation in the sequence would then "reduce" the state back to its initial value |y_u>, and in the limit of continuous observation there could be no change at all. Here we see that it is disproven by the simple empirical fact that [..] continuous observation does not prevent motion. It is sometimes claimed that the rival interpretations of quantum mechanics differ only in philosophy, and can not be experimentally distinguished. That claim is not always true. as this example proves" ".Ballentine, L. Quantum Mechanics, A Modern Development(p 342)
- ^ "The quantum Zeno effect is not a general characteristic of continuous measurements. In a recently reported experiment [Itano et al., Phys. Rev. A 41, 2295 (1990)], the inhibition of atomic excitation and deexcitation is not due to any ‘‘collapse of the wave function,’’ but instead is caused by a very strong perturbation due to the optical pulses and the coupling to the radiation field. The experiment should not be cited as providing empirical evidence in favor of the notion of ‘‘wave-function collapse.’’" Physical Review
