Copenhagen interpretation

Quantum mechanics
\Delta x\, \Delta p \ge \frac{\hbar}{2}
Uncertainty principle
Introduction · Mathematical formulations
Interpretations
de Broglie–Bohm · CCC · Consistent histories · Copenhagen · Ensemble · Hidden variables · Many-worlds · Pondicherry · Quantum logic · Relational · Stochastic · Transactional · Objective collapse

The Copenhagen interpretation is an interpretation of quantum mechanics. A key feature of quantum mechanics is that the state of every particle can be described by a wavefunction, which is a mathematical representation used to calculate the probability for it to be found in a location or a state of motion. According to this interpretation, the act of measurement causes the calculated set of probabilities to "collapse" to the value defined by the measurement. This feature of the mathematical representations is known as wavefunction collapse.

Early twentieth-century experiments on the physics of very small-scale phenomena led to the discovery of phenomena that could not be predicted on the basis of classical physics, and to new models (theories) that described and predicted very accurately those micro-scale phenomena so recently discovered. These models of the real world being observed at this micro scale, could not easily be reconciled with the way objects are observed to behave on the macro scale of everyday life. The predictions they offered often appeared counter-intuitive to observers. Indeed, they touched off much consternation—even in the minds of their discoverers. The Copenhagen interpretation consists of attempts to explain the experiments and their mathematical formulations.

The work of relating the experiments and the abstract mathematical and theoretical formulations that constitute quantum physics to the experience that all of us share in the world of everyday life fell first to Niels Bohr and Werner Heisenberg in the course of their collaboration in Copenhagen around 1927. Bohr and Heisenberg stepped beyond the world of empirical experiments and pragmatic predictions of such phenomena as the frequencies of light emitted under various conditions. In the earlier work of Planck, Einstein and Bohr himself, discrete quantities of energy had been postulated in order to avoid paradoxes of classical physics when pushed to extremes. Bohr and Heisenberg now found a new world of energy quanta, entities that fit neither the classical ideas of particles nor the classical ideas of waves. Elementary particles showed predictable properties in many experiments. But they became highly unpredictable in certain contexts, for example if one attempted to measure their individual trajectories through a simple physical apparatus.

The new theories were inspired by laboratory experiments and based on the idea that matter has both wave and particle aspects. One of the consequences, derived by Heisenberg, was that knowledge of the position of a particle limits how precisely its momentum can be known – and vice-versa. The results of their own burgeoning understanding disoriented Bohr and Heisenberg, and some physicists concluded that human observation of a microscopic event changes the reality of the event.

The Copenhagen interpretation was a composite statement about what could and could not be legitimately stated in common language to complement the statements and predictions that could be made in the language of instrument readings and mathematical operations. In other words, it attempted to answer the question, "What do these amazing experimental results really mean?" The insight that quantum mechanics does not yield an objective description of microscopic reality but that measurement plays an ineradicable role is probably the most telling characteristic of the Copenhagen interpretation.

Contents

Overview

There is no definitive statement of the Copenhagen Interpretation[1] since it consists of the views developed by a number of scientists and philosophers during the second quarter of the 20th century. Thus, there are a number of ideas that have been associated with the Copenhagen interpretation. Asher Peres remarked that very different, sometimes opposite, views are presented as "the Copenhagen interpretation" by different authors.[2]

Principles

  1. A system is completely described by a wave function \psi, which represents an observer's knowledge of the system. (Heisenberg)  
  2. The description of nature is essentially probabilistic. The probability of an event is related to the square of the amplitude of the wave function related to it. (Born rule, due to Max Born)
  3. Heisenberg's uncertainty principle states the fact that it is not possible to know the values of all of the properties of the system at the same time; those properties that are not known with precision must be described by probabilities.
  4. Complementarity principle: matter exhibits a wave-particle duality. An experiment can show the particle-like properties of matter, or wave-like properties, but not both at the same time.(Niels Bohr)
  5. Measuring devices are essentially classical devices, and measure classical properties such as position and momentum.
  6. The correspondence principle of Bohr and Heisenberg: the quantum mechanical description of large systems should closely approximate the classical description.

Meaning of the wave function

The Copenhagen Interpretation denies that any wave function is anything more than an abstraction, or is at least non-committal about its being a discrete entity or a discernible component of some discrete entity.

There are some who say that there are objective variants of the Copenhagen Interpretation that allow for a "real" wave function, but it is questionable whether that view is really consistent with logical positivism and/or with some of Bohr's statements. Bohr emphasized that science is concerned with predictions of the outcomes of experiments, and that any additional propositions offered are not scientific but rather meta-physical. Bohr was heavily influenced by positivism. On the other hand, Bohr and Heisenberg were not in complete agreement, and held different views at different times. Heisenberg in particular was prompted to move towards realism.[3]

Even if the wave function is not regarded as real, there is still a divide between those who treat it as definitely and entirely subjective, and those who are non-committal or agnostic about the subject. An example of the agnostic view is given by Carl Friedrich von Weizsäcker, who, while participating in a colloquium at Cambridge, denied that the Copenhagen interpretation asserted: "What cannot be observed does not exist". He suggested instead that the Copenhagen interpretation follows the principle: "What is observed certainly exists; about what is not observed we are still free to make suitable assumptions. We use that freedom to avoid paradoxes."[4]

The subjective view, that the wave function is merely a mathematical tool for calculating probabilities of specific experiment, is a similar approach to the Ensemble interpretation.

Nature of collapse

All versions of the Copenhagen interpretation include at least a formal or methodological version of wave function collapse,[5] in which unobserved eigenvalues are removed from further consideration. (In other words, Copenhagenists have always made the assumption of collapse, even in the early days of quantum physics, in the way that adherents of the Many-worlds interpretation have not.) In more prosaic terms, those who hold to the Copenhagen understanding are willing to say that a wave function involves the various probabilities that a given event will proceed to certain different outcomes. But when one or another of those more- or less-likely outcomes becomes manifest the other probabilities cease to have any function in the real world. So if an electron passes through a double slit apparatus there are various probabilities for where on the detection screen that individual electron will hit. But once it has hit, there is no longer any probability whatsoever that it will hit somewhere else. Many-worlds interpretations say that an electron hits wherever there is a possibility that it might hit, and that each of these hits occurs in a separate universe.

An adherent of the subjective view, that the wave function represents nothing but knowledge, would take an equally subjective view of "collapse".

Some argue that the concept of collapse of a "real" wave function was introduced by John Von Neumann in 1932 and was not part of the original formulation of the Copenhagen Interpretation.[6]

Acceptance among physicists

According to a poll at a Quantum Mechanics workshop in 1997[7], the Copenhagen interpretation is the most widely-accepted specific interpretation of quantum mechanics, followed by the many-worlds interpretation.[8] Although current trends show substantial competition from alternative interpretations, throughout much of the twentieth century the Copenhagen interpretation had strong acceptance among physicists. Astrophysicist and science writer John Gribbin describes it as having fallen from primacy after the 1980s.[9]

Consequences

The nature of the Copenhagen Interpretation is exposed by considering a number of experiments and paradoxes.

1. Schrödinger's Cat

A cat is put in a box with a radioactive substance and a radiation detector (such as a Geiger counter). The half-life of the substance is the period of time in which there is a 50% chance that a particle will be emitted (and detected). The detector is activated for that period of time. If a particle is detected, a poisonous gas will be released and the cat killed. Schrödinger set this up as what he called a "ridiculous case" in which "The psi-function of the entire system would express this by having in it the living and dead cat (pardon the expression) mixed or smeared out in equal parts." He resisted an interpretation "so naively accepting as valid a 'blurred model' for representing reality."[10] How can the cat be both alive and dead?
The Copenhagen Interpretation: The wave function reflects our knowledge of the system. The wave function (|\text{dead}\rangle + |\text{alive}\rangle)/\sqrt 2 means that, once the cat is observed, there is a 50% chance it will be dead, and 50% chance it will be alive.

2. Wigner's Friend

Wigner puts his friend in with the cat. The external observer believes the system is in the state (|\text{dead}\rangle + |\text{alive}\rangle)/\sqrt 2. His friend however is convinced that cat is alive, i.e. for him, the cat is in the state |\text{alive}\rangle. How can Wigner and his friend see different wave functions?
The Copenhagen Interpretation: Wigner's friend highlights the subjective nature of probability. Each observer (Wigner and his friend) has different information and therefore different wave functions. The distinction between the "objective" nature of reality and the subjective nature of probability has led to a great deal of controversy. Cf. Bayesian versus Frequentist interpretations of probability.

3. Double Slit Diffraction

Light passes through double slits and onto a screen resulting in a diffraction pattern. Is light a particle or a wave?
The Copenhagen Interpretation: Light is neither. A particular experiment can demonstrate particle (photon) or wave properties, but not both at the same time (Bohr's Complementarity Principle).
The same experiment can in theory be performed with any physical system: electrons, protons, atoms, molecules, viruses, bacteria, cats, humans, elephants, planets, etc. In practice it has been performed for light, electrons, buckminsterfullerene, and some atoms. Due to the smallness of Planck's constant it is practically impossible to realize experiments that directly reveal the wave nature of any system bigger than a few atoms but, in general, quantum mechanics considers all matter as possessing both particle and wave behaviors. The greater systems (like viruses, bacteria, cats, etc.) are considered as "classical" ones but only as an approximation.

4. EPR (Einstein–Podolsky–Rosen) paradox

Entangled "particles" are emitted in a single event. Conservation laws ensure that the measured spin of one particle must be the opposite of the measured spin of the other, so that if the spin of one particle is measured, the spin of the other particle is now instantaneously known. The most discomforting aspect of this paradox is that the effect is instantaneous so that something that happens in one galaxy could cause an instantaneous change in another galaxy. But, according to Einstein's theory of special relativity, no information-bearing signal or entity can travel at or faster than the speed of light, which is finite. Thus, it seems as if the Copenhagen interpretation is inconsistent with special relativity.
The Copenhagen Interpretation: Assuming wave functions are not real, wave function collapse is interpreted subjectively. The moment one observer measures the spin of one particle, he knows the spin of the other. However another observer cannot benefit until the results of that measurement have been relayed to him, at less than or equal to the speed of light.
Copenhagenists claim that interpretations of quantum mechanics where the wave function is regarded as real have problems with EPR-type effects, since they imply that the laws of physics allow for influences to propagate at speeds greater than the speed of light. However, proponents of Many worlds[11] and the Transactional interpretation[12][13] maintain that their theories are fatally non-local.
The claim that EPR effects violate the principle that information cannot travel faster than the speed of light can be avoided by noting that they cannot be used for signaling because neither observer can control, or predetermine, what he observes, and therefore cannot manipulate what the other observer measures. Relativistic difficulties about establishing which measurement occurred first also undermine the idea that one observer is causing what the other is measuring.

Criticisms

The completeness of quantum mechanics (thesis 1) was attacked by the Einstein-Podolsky-Rosen thought experiment which was intended to show that quantum physics could not be a complete theory.

Experimental tests of Bell's inequality using particles have supported the quantum mechanical prediction of entanglement.

The Copenhagen Interpretation gives special status to measurement processes without clearly defining them or explaining their peculiar effects. In his article entitled "Criticism and Counterproposals to the Copenhagen Interpretation of Quantum Theory," countering the view of Alexandrov that (in Heisenberg's paraphrase) "the wave function in configuration space characterizes the objective state of the electron." Heisenberg says,

Of course the introduction of the observer must not be misunderstood to imply that some kind of subjective features are to be brought into the description of nature. The observer has, rather, only the function of registering decisions, i.e., processes in space and time, and it does not matter whether the observer is an apparatus or a human being; but the registration, i.e., the transition from the "possible" to the "actual," is absolutely necessary here and cannot be omitted from the interpretation of quantum theory.
—Heisenberg, Physics and Philosophy, p. 137

Many physicists and philosophers have objected to the Copenhagen interpretation, both on the grounds that it is non-deterministic and that it includes an undefined measurement process that converts probability functions into non-probabilistic measurements. Einstein's comments "I, at any rate, am convinced that He (God) does not throw dice."[14] and "Do you really think the moon isn't there if you aren't looking at it?"[15] exemplify this. Bohr, in response, said "Einstein, don't tell God what to do".

Steven Weinberg in "Einstein's Mistakes", Physics Today, November 2005, page 31, said:

All this familiar story is true, but it leaves out an irony. Bohr's version of quantum mechanics was deeply flawed, but not for the reason Einstein thought. The Copenhagen interpretation describes what happens when an observer makes a measurement, but the observer and the act of measurement are themselves treated classically. This is surely wrong: Physicists and their apparatus must be governed by the same quantum mechanical rules that govern everything else in the universe. But these rules are expressed in terms of a wave function (or, more precisely, a state vector) that evolves in a perfectly deterministic way. So where do the probabilistic rules of the Copenhagen interpretation come from?

Considerable progress has been made in recent years toward the resolution of the problem, which I cannot go into here. It is enough to say that neither Bohr nor Einstein had focused on the real problem with quantum mechanics. The Copenhagen rules clearly work, so they have to be accepted. But this leaves the task of explaining them by applying the deterministic equation for the evolution of the wave function, the Schrödinger equation, to observers and their apparatus.

The problem of thinking in terms of classical measurements of a quantum system becomes particularly acute in the field of quantum cosmology, where the quantum system is the universe.[16]

Alternatives

The Ensemble Interpretation is similar; it offers an interpretation of the wave function, but not for single particles. The consistent histories interpretation advertises itself as "Copenhagen done right". Consciousness causes collapse is often confused with the Copenhagen interpretation.

If the wave function is regarded as ontologically real, and collapse is entirely rejected, a many worlds theory results. If wave function collapse is regarded as ontologically real as well, an objective collapse theory is obtained. Dropping the principle that the wave function is a complete description results in a hidden variable theory.

Many physicists have subscribed to the instrumentalist interpretation of quantum mechanics, a position often equated with eschewing all interpretation. It is summarized by the sentence "Shut up and calculate!". While this slogan is sometimes attributed to Paul Dirac[17] or Richard Feynman, it is in fact due to the lesser known David Mermin.[18]

See also

Notes and references

  1. ↑ In fact Bohr and Heisenberg never totally agreed on how to understand the mathematical formalism of quantum mechanics, and none of them ever used the term "the Copenhagen interpretation" as a joint name for their ideas. Bohr once distanced himself from what he considered to be Heisenberg's more subjective interpretation Stanford Encyclopedia of Philosophy
  2. ↑ "There seems to be at least as many different Copenhagen interpretations as people who use that term, probably there are more. For example, in two classic articles on the foundations of quantum mechanics, Ballentine (1970) and Stapp(1972) give diametrically opposite definitions of 'Copenhagen.'", A. Peres, Popper's experiment and the Copenhagen interpretation, Stud. History Philos. Modern Physics 33 (2002) 23, preprint
  3. ↑ "Historically, Heisenberg wanted to base quantum theory solely on observable quantities such as the intensity of spectral lines, getting rid of all intuitive (anschauliche) concepts such as particle trajectories in space-time [2]. This attitude changed drastically with his paper [3] in which he introduced the uncertainty relations – there he put forward the point of view that it is the theory which decides what can be observed. His move from positivism to operationalism can be clearly understood as a reaction on the advent of Schrödinger’s wave mechanics [1] which, in particular due to its intuitiveness, became soon very popular among physicists. In fact, the word anschaulich (intuitive) is contained in the title of Heisenberg’s paper [3]."Kiefer, C. On the interpretation of quantum theory – from Copenhagen to the present day
  4. ↑ John Cramer on the Copenhagen Interpretation
  5. ↑ "To summarize, one can identify the following ingredients as being characteristic for the Copenhagen interpretation(s)[...]Reduction of the wave packet as a formal rule without dynamical significance"Kiefer, C. On the interpretation of quantum theory – from Copenhagen to the present day
  6. ↑ "the “collapse” or “reduction” of the wave function. This was introduced by Heisenberg in his uncertainty paper [3] and later postulated by von Neumann as a dynamical process independent of the Schrodinger equation"Kiefer, C. On the interpretation of quantum theory – from Copenhagen to the present day
  7. ↑ Tegmark, M. (1997), The Interpretation of Quantum Mechanics: Many Worlds or Many Words?.
  8. ↑ The Many Worlds Interpretation of Quantum Mechanics
  9. ↑ Gribbin, J. Q for Quantum
  10. ↑ Erwin Schrödinger, in an article in the Proceedings of the American Philosophical Society, 124, 323-38.
  11. ↑ Michael price on nonlocality in Many Worlds
  12. ↑ Relativity and Causality in the Transactional Interpretation
  13. ↑ Collapse and Nonlocality in the Transactional Interpretation
  14. ↑ "God does not throw dice" quote
  15. ↑ A. Pais, Einstein and the quantum theory, Reviews of Modern Physics 51, 863-914 (1979), p. 907.
  16. ↑ 'Since the Universe naturally contains all of its observers, the problem arises to come up with an interpretation of quantum theory that contains no classical realms on the fundamental level.'Kiefer, C. On the interpretation of quantum theory from Copenhagen to the present day
  17. ↑ http://home.fnal.gov/~skands/slides/A-Quantum-Journey.ppt
  18. ↑ "Shut up and calculate" quote.

Further reading

External links