Glossary · History
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, entities that fit neither the classical idea of particles nor the classical idea of waves. The act of measurement causes the set of probabilities to immediately and randomly assume only one of the possible values. This feature of the mathematics is known as wavefunction collapse. The essential concepts of the interpretation were devised by Niels Bohr, Werner Heisenberg and others in the years 1924–27.
Classical physics draws a distinction between particles and energy, holding that only the latter exhibit waveform characteristics, whereas quantum mechanics is based on the observation that matter has both wave and particle aspects and postulates that the state of every subatomic particle can be described by a wavefunction—a mathematical expression used to calculate the probability that the particle, if measured, will be in a given location or state of motion.
In the early work of Max Planck, Albert Einstein, and Niels Bohr, the existence of energy in discrete quantities had been postulated, in order to explain phenomena, such as the spectrum of black-body radiation, the photoelectric effect, and the stability and spectrum of atoms such as hydrogen, that had eluded explanation by, and even appeared to be in contradiction with, classical physics. Also, while elementary particles showed predictable properties in many experiments, they became highly unpredictable in certain contexts, for example, if one attempted to measure their individual trajectories through a simple physical apparatus.
The Copenhagen interpretation is an attempt to explain the mathematical formulations of quantum mechanics and the corresponding experimental results. Early twentieth-century experiments on the physics of very small-scale phenomena led to the discovery of phenomena which could not be predicted on the basis of classical physics, and to the development of new models (theories) that described and predicted very accurately these micro-scale phenomena. These models 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 and caused much consternation among the physicists—often including their discoverers.
Origin of the term
Werner Heisenberg had been an assistant to Niels Bohr at his institute in Copenhagen during part of the 1920s, when they helped originate quantum mechanical theory. In 1929, Heisenberg gave a series of invited lectures at the University of Chicago explaining the new field of quantum mechanics. The lectures then served as the basis for his textbook, The Physical Principles of the Quantum Theory, published in 1930. In the book's preface, Heisenberg wrote:
On the whole the book contains nothing that is not to be found in previous publications, particularly in the investigations of Bohr. The purpose of the book seems to me to be fulfilled if it contributes somewhat to the diffusion of that 'Kopenhagener Geist der Quantentheorie' [i.e., Copenhagen spirit of quantum theory] if I may so express myself, which has directed the entire development of modern atomic physics.
The term 'Copenhagen interpretation' suggests something more than just a spirit, such as some definite set of rules for interpreting the mathematical formalism of quantum mechanics, presumably dating back to the 1920s. However, no such text exists, apart from some informal popular lectures by Bohr and Heisenberg, which contradict each other on several important issues. It appears that the particular term, with its more definite sense, was coined by Heisenberg in the 1950s, while criticizing alternate "interpretations" (e.g., David Bohm's) that had been developed. Lectures with the titles 'The Copenhagen Interpretation of Quantum Theory' and 'Criticisms and Counterproposals to the Copenhagen Interpretation', that Heisenberg delivered in 1955, are reprinted in the collection Physics and Philosophy. Before the book was released for sale, Heisenberg privately expressed regret for having used the term, due to its suggestion of the existence of other interpretations, that he considered to be "nonsense".
Because it consists of the views developed by a number of scientists and philosophers during the second quarter of the 20th Century, there is no definitive statement of the Copenhagen interpretation. Thus, various ideas have been associated with it; Asher Peres remarked that very different, sometimes opposite, views are presented as "the Copenhagen interpretation" by different authors. Nonetheless, there are several basic principles that are generally accepted as being part of the interpretation:
- A system is completely described by a wave function , representing the state of the system, which evolves smoothly in time, except when a measurement is made, at which point it instantaneously collapses to an eigenstate of the observable that is measured.
- The description of nature is essentially probabilistic, with the probability of a given outcome of a measurement given by the square of the modulus of the amplitude of the wave function. (The Born rule, after Max Born)
- It is not possible to know the value of all the properties of the system at the same time; those properties that are not known exactly must be described by probabilities. (Heisenberg's uncertainty principle)
- Matter exhibits a wave–particle duality. An experiment can show the particle-like properties of matter, or the wave-like properties; in some experiments both of these complementary viewpoints must be invoked to explain the results, according to the complementarity principle of Niels Bohr.
- Measuring devices are essentially classical devices, and measure only classical properties such as position and momentum.
- The quantum mechanical description of large systems will closely approximate the classical description. (This is the correspondence principle of Bohr and Heisenberg.)
Meaning of the wave function
The Copenhagen Interpretation denies that the wave function is anything more than a theoretical concept, or is at least non-committal about its being a discrete entity or a discernible component of some discrete entity.
The subjective view, that the wave function is merely a mathematical tool for calculating the probabilities in a specific experiment, has some similarities to the Ensemble interpretation in that it takes probabilities to be the essence of the quantum state, but unlike the ensemble interpretation, it takes these probabilities to be perfectly applicable to single experimental outcomes, as it interprets them in terms of subjective probability.
There are some[who?] 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 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 meta-physical. Bohr was heavily influenced by positivism. On the other hand, Bohr and Heisenberg were not in complete agreement, and they held different views at different times. Heisenberg in particular was prompted to move towards realism.
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."
Nature of collapse
All versions of the Copenhagen interpretation include at least a formal or methodological version of wave function collapse, in which unobserved eigenvalues are removed from further consideration. The Copenhagen interpretation has always treated wave function collapse as a fundamental, a priori principle. In 1952 David Bohm developed decoherence, an explanatory mechanism for the appearance of wave function collapse. Bohm applied decoherence to Louis DeBroglie's pilot wave theory, producing Bohmian mechanics, the first successful hidden variables interpretation of quantum mechanics. Decoherence was then used by Hugh Everett in 1957 to form the core of his many-worlds interpretation. However decoherence was largely ignored until the 1980s. Those who hold to the Copenhagen interpretation are willing to say that a wave function involves the various probabilities that a given event will proceed to certain different outcomes. But when an observer obtains one of those outcomes, no probabilities or superposition of the others linger.
Some argue that the concept of the collapse of a "real" wave function was introduced by Heisenberg and later developed by John von Neumann in 1932. However, Heisenberg spoke of the wavefunction as representing our knowledge of a system, and did not use the term "collapse" per se, but instead termed it "reduction" of the wavefunction to a new state representing the change in our knowledge which occurs once a particular phenomenon is registered by the experimenter (i.e. when an measurement takes place).
Acceptance among physicists
Throughout much of the twentieth century the Copenhagen interpretation had overwhelming acceptance among physicists. Although astrophysicist and science writer John Gribbin described it as having fallen from primacy after the 1980s, according to a poll conducted at a quantum mechanics conference in 1997, the Copenhagen interpretation remained the most widely accepted specific interpretation of quantum mechanics among physicists. In more recent polls conducted at various quantum mechanics conferences, varying results have been found.
The nature of the Copenhagen Interpretation is exposed by considering a number of experiments and paradoxes.
- This thought experiment highlights the implications that accepting uncertainty at the microscopic level has on macroscopic objects. A cat is put in a sealed box, with its life or death made dependent on the state of a subatomic particle. Thus a description of the cat during the course of the experiment—having been entangled with the state of a subatomic particle—becomes a "blur" of "living and dead cat." But this can't be accurate because it implies the cat is actually both dead and alive until the box is opened to check on it. But the cat, if he survives, will only remember being alive. Schrödinger resists "so naively accepting as valid a 'blurred model' for representing reality." How can the cat be both alive and dead?
- The Copenhagen Interpretation: The wave function reflects our knowledge of the system. The wave function means that, once the cat is observed, there is a 50% chance it will be dead, and 50% chance it will be alive.
- Wigner puts his friend in with the cat. The external observer believes the system is in the state . His friend, however, is convinced that the cat is alive, i.e. for him, the cat is in the state . How can Wigner and his friend see different wave functions?
- The Copenhagen Interpretation: The answer depends on the positioning of Heisenberg cut, which can be placed arbitrarily. If Wigner's friend is positioned on the same side of the cut as the external observer, his measurements collapse the wave function for both observers. If he is positioned on the cat's side, his interaction with the cat is not considered a measurement.
- 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, not exact.
- 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 and the Transactional interpretation (TI) maintain that Copenhagen interpretation is fatally non-local.
- The claim that EPR effects violate the principle that information cannot travel faster than the speed of light have been countered 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. However, this is a somewhat spurious argument, in that speed of light limitations applies to all information, not to what can or can not be subsequently done with the information. On the other hand, the special theory of relativity contains no notion of information at all. The fact that no classical body can exceed the speed of light (no matter how much acceleration applied) is a consequence of classical relativistic mechanics. As the correlation between the two particles in an EPR experiment is most probably not established by classical bodies or light signals, the displayed non-locality is not at odds with special relativity.
- A further argument is that relativistic difficulties about establishing which measurement occurred first also undermine the idea that one observer is causing what the other is measuring. This is totally spurious, since no matter who measured first the other will measure the opposite spin despite the fact that (in theory) the other has a 50% 'probability' (50:50 chance) of measuring the same spin, unless data about the first spin measurement has somehow passed faster than light (of course TI gets around the light speed limit by having information travel backwards in time instead).
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.
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.
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." and "Do you really think the moon isn't there if you aren't looking at it?" exemplify this. Bohr, in response, said, "Einstein, don't tell God what to do."
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.
E. T. Jaynes, from a Bayesian point of view, argued that probability is a measure of a human's information about the physical world. Quantum mechanics under the Copenhagen Interpretation interpreted probability as a physical phenomenon, which is what Jaynes called a Mind Projection Fallacy. A similar view is adopted in quantum information theories.
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". Although the Copenhagen interpretation is often confused with the idea that consciousness causes collapse, it defines an "observer" merely as that which collapses the wave function. Quantum information theories are more recent, and have attracted growing support.
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. For an atemporal interpretation that “makes no attempt to give a ‘local’ account on the level of determinate particles”, the conjugate wavefunction, ("advanced" or time-reversed) of the relativistic version of the wavefunction, and the so-called "retarded" or time-forward version are both regarded as real and the transactional interpretation results. 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 or Richard Feynman, it is in fact due to David Mermin.
- Bohr–Einstein debates
- Consciousness causes collapse
- Fifth Solvay Conference
- Interpretations of quantum mechanics
- Philosophical interpretation of classical physics
- Physical ontology
- Popper's experiment
- De Broglie–Bohm theory
Notes and references
- Hermann Wimmel (1992). Quantum physics & observed reality: a critical interpretation of quantum mechanics. World Scientific. p. 2. ISBN 978-981-02-1010-6. Retrieved 9 May 2011.
- J. Mehra and H. Rechenberg, The historical development of quantum theory, Springer-Verlag, 2001, p. 271.
- Howard, Don (2004). "Who invented the Copenhagen Interpretation? A study in mythology". Philosophy of Science: 669–682. JSTOR 10.1086/425941.
- Bohm, David (1952). "A Suggested Interpretation of the Quantum Theory in Terms of "Hidden" Variables. I & II". Physical Review 85 (2): 166–193. Bibcode:1952PhRv...85..166B. doi:10.1103/PhysRev.85.166.
- H. Kragh, Quantum generations: A History of Physics in the Twentieth Century, Princeton University Press, 1999, p. 210. ("the term 'Copenhagen interpretation' was not used in the 1930s but first entered the physicist’s vocabulary in 1955 when Heisenberg used it in criticizing certain unorthodox interpretations of quantum mechanics.")
- Werner Heisenberg, Physics and Philosophy, Harper, 1958
- Olival Freire Jr., "Science and exile: David Bohm, the hot times of the Cold War, and his struggle for a new interpretation of quantum mechanics", Historical Studies on the Physical and Biological Sciences, Volume 36, Number 1, 2005, pp. 31–35. ("I avow that the term ‘Copenhagen interpretation’ is not happy since it could suggest that there are other interpretations, like Bohm assumes. We agree, of course, that the other interpretations are nonsense, and I believe that this is clear in my book, and in previous papers. Anyway, I cannot now, unfortunately, change the book since the printing began enough time ago.")
- In fact Bohr and Heisenberg never totally agreed on how to understand the mathematical formalism of quantum mechanics. Bohr once distanced himself from what he considered to be Heisenberg's more subjective interpretation Stanford Encyclopedia of Philosophy
- "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.'", Asher Peres (2002). "Popper's experiment and the Copenhagen interpretation". Stud. History Philos. Modern Physics 33 (23): 10078. arXiv:quant-ph/9910078. Bibcode:1999quant.ph.10078P.
- "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. This attitude changed drastically with his paper 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 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.", from Claus Kiefer (2002). "On the interpretation of quantum theory - from Copenhagen to the present day". arXiv:quant-ph/0210152 [quant-ph].
- John Cramer on the Copenhagen Interpretation
- "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", Claus Kiefer (2002). "On the interpretation of quantum theory - from Copenhagen to the present day". arXiv:quant-ph/0210152 [quant-ph].
- David Bohm, A Suggested Interpretation of the Quantum Theory in Terms of "Hidden Variables", I, Physical Review, (1952), 85, pp 166–179
- David Bohm, A Suggested Interpretation of the Quantum Theory in Terms of "Hidden Variables", II, Physical Review, (1952), 85, pp 180–193
- Hugh Everett, Relative State Formulation of Quantum Mechanics, Reviews of Modern Physics vol 29, (1957) pp 454–462.
- H. Dieter Zeh, On the Interpretation of Measurement in Quantum Theory, Foundation of Physics, vol. 1, pp. 69-76, (1970).
- Wojciech H. Zurek, Pointer Basis of Quantum Apparatus: Into what Mixture does the Wave Packet Collapse?, Physical Review D, 24, pp. 1516–1525 (1981)
- Wojciech H. Zurek, Environment-Induced Superselection Rules, Physical Review D, 26, pp.1862–1880, (1982)
- "the “collapse” or “reduction” of the wave function. This was introduced by Heisenberg in his uncertainty paper  and later postulated by von Neumann as a dynamical process independent of the Schrodinger equation", Claus Kiefer (2002). "On the interpretation of quantum theory - from Copenhagen to the present day". arXiv:quant-ph/0210152 [quant-ph].
- W. Heisenberg "Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik," Zeitschrift für Physik, Volume 43, 172-98 (1927), as translated by John Wheeler and Wojciech Zurek, in Quantum Theory and Measurement (1983), p. 74. ("[The] determination of the position selects a definite "q" from the totality of possibilities and limits the options for all subsequent measurements. ... [T]he results of later measurements can only be calculated when one again ascribes to the electron a "smaller" wavepacket of extension λ (wavelength of the light used in the observation). Thus, every position determination reduces the wavepacket back to its original extension λ.")
- Gribbin, J. Q for Quantum
- Max Tegmark (1998). "The Interpretation of Quantum Mechanics: Many Worlds or Many Words?". Fortsch.Phys. 46 (6–8): 855–862. arXiv:quant-ph/9709032. Bibcode:1998ForPh..46..855T. doi:10.1002/(SICI)1521-3978(199811)46:6/8<855::AID-PROP855>3.0.CO;2-Q.
- M. Schlosshauer; J. Koer; A. Zeilinger (2013). "A Snapshot of Foundational Attitudes Toward Quantum Mechanics". Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 44 (3): 222–230. arXiv:arXiv:1301.1069.
- C. Sommer, "Another Survey of Foundational Attitudes Towards Quantum Mechanics", arXiv:1303.2719
- T. Norsen, S. Nelson, "Yet Another Snapshot of Foundational Attitudes Toward Quantum Mechanics", arXiv:1306.4646
- Erwin Schrödinger, in an article in the Proceedings of the American Philosophical Society, 124, 323-38.
- Nairz, Olaf; Brezger, Björn; Arndt, Markus; Zeilinger, Anton (2001). "Diffraction of Complex Molecules by Structures Made of Light". Physical Review Letters 87 (16). arXiv:quant-ph/0110012. Bibcode:2001PhRvL..87p0401N. doi:10.1103/PhysRevLett.87.160401.
- Brezger, Björn; Hackermüller, Lucia; Uttenthaler, Stefan; Petschinka, Julia; Arndt, Markus; Zeilinger, Anton (2002). "Matter-Wave Interferometer for Large Molecules". Physical Review Letters 88 (10): 100404. arXiv:quant-ph/0202158. Bibcode:2002PhRvL..88j0404B. doi:10.1103/PhysRevLett.88.100404. PMID 11909334.
- Michael price on nonlocality in Many Worlds
- Relativity and Causality in the Transactional Interpretation
- Collapse and Nonlocality in the Transactional Interpretation
- Werner Heisenberg, Physics and Philosophy, Harper, 1958, p. 137.
- "God does not throw dice" quote
- A. Pais, Einstein and the quantum theory, Reviews of Modern Physics 51, 863-914 (1979), p. 907.
- Bohr recollected his reply to Einstein at the 1927 Solvay Congress in his essay "Discussion with Einstein on Epistemological Problems in Atomic Physics", in Albert Einstein, Philosopher-Scientist, ed. Paul Arthur Shilpp, Harper, 1949, p. 211: "...in spite of all divergencies of approach and opinion, a most humorous spirit animated the discussions. On his side, Einstein mockingly asked us whether we could really believe that the providential authorities took recourse to dice-playing ("ob der liebe Gott würfelt"), to which I replied by pointing at the great caution, already called for by ancient thinkers, in ascribing attributes to Providence in everyday language." Werner Heisenberg, who also attended the congress, recalled the exchange in Encounters with Einstein, Princeton University Press, 1983, p. 117,: "But he [Einstein] still stood by his watchword, which he clothed in the words: 'God does not play at dice.' To which Bohr could only answer: 'But still, it cannot be for us to tell God, how he is to run the world.'"
- '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.', Claus Kiefer (2002). "On the interpretation of quantum theory - from Copenhagen to the present day". arXiv:quant-ph/0210152 [quant-ph].
- Jaynes, E. T. (1989). "Clearing up Mysteries--The Original Goal". Maximum Entropy and Bayesian Methods: 7.
- Kate Becker (2013-01-25). "Quantum physics has been rankling scientists for decades". Boulder Daily Camera. Retrieved 2013-01-25.
- "A Snapshot of Foundational Attitudes Toward Quantum Mechanics". 2013-01-06. Retrieved 2013-01-25.
- The Quantum Liar Experiment, RE Kastner, Studies in History and Philosophy of Modern Physics, Vol41, Iss.2,May2010
- The non-relativistic Schrödinger equation does not admit advanced solutions.
- N. David Mermin. "Could Feynman Have Said This?". Physics Today 57 (5).
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- H. Margeneau, The Nature of Physical Reality, McGraw-Hill 1950
- M. Chown, Forever Quantum, New Scientist No. 2595 (2007) 37.
- T. Schürmann, A Single Particle Uncertainty Relation, Acta Physica Polonica B39 (2008) 587.