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| | In [[theoretical physics]], the '''pilot wave theory''' was the first known example of a [[hidden variable theory]], presented by [[Louis de Broglie]] in 1927. Its more modern version, the [[Bohm interpretation]], |
| | remains a controversial attempt to interpret [[quantum mechanics]] as a [[deterministic]] theory, avoiding |
| | troublesome notions such as instantaneous [[wavefunction collapse]] and the paradox of [[Schrödinger's cat]]. |
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| | ==The pilot wave theory== |
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| | The pilot wave theory is one of several [[interpretations of quantum mechanics]]. It uses the same mathematics as other interpretations of quantum mechanics; consequently, it is |
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| | also supported by the current experimental evidence to the same extent as the other interpretations. |
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| | The pilot wave theory is a [[hidden variable theory]]. Consequently: |
| | * the theory has realism (meaning that its concepts exist independently of the observer); |
| | * the theory has [[determinism]]. |
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| | The positions and momenta of the particles are considered to be the hidden variables. However, the observer not only doesn't know the precise value of these variables, but more importantly, cannot know them precisely because any measurement disturbs them – as stipulated by the Heisenberg [[uncertainty principle]]. |
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| == crimeboss Nike Tn Sale Australia == | | A collection of particles has an associated matter wave, which evolves according to the [[Schrödinger Equation]]. Each particle follows a deterministic trajectory, which is guided by the wave function; collectively, the density of the particles conforms to the magnitude of the wave function. The wave function is not influenced by the particle and can exist also as an [[#Empty wave function|empty wave function]].<ref name="bell-1992">J. S. Bell: [http://www.springerlink.com/content/l671u1wkl6212670/ ''Six possible worlds of quantum mechanics''], Foundations of Physics, vol. 22, no. 10, Part I. Invited Papers Dedicated To Louis De Broglie, 1992, pp. 1201-1215, {{doi|10.1007/BF01889711}}, p. 1212</ref> |
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| | The theory brings to light [[Action at a distance|nonlocality]] that is implicit in the non-relativistic formulation of quantum mechanics and uses it to satisfy [[Bell's theorem]]. Interestingly, these nonlocal effects are compatible with [[no-communication theorem]], which prevents us from using them for faster-than-light communication. |
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| == the UK. Ugg Boots For Sale Adelaide == | | ===Consequences=== |
| | The pilot wave theory shows that it is possible to have a realistic and deterministic hidden variable theory, which reproduces the experimental results of ordinary quantum mechanics. The price which has to be paid for this is manifest nonlocality. |
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| == Pandora Online Australia Shop == | | By plugging this into the Schrödinger-equation, one can derive two new equations for the real variables. |
| | The first is the [[Probability current#Continuity equation for quantum mechanics|continuity equation for the probability density]]<math>\rho</math> :<ref>{{Cite journal| first = David | last = Bohm | title = A Suggested Interpretation of the Quantum Theory in Terms of "Hidden Variables" I | journal = Physical Review | year = 1952 | volume = 85 | pages = 166–179 | doi = 10.1103/PhysRev.85.166|bibcode = 1952PhRv...85..166B }} ([https://www.nd.edu/~dhoward1/Bohm%20HV-I%20Phys%20Rev%201952.pdf full text])</ref> |
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| | Ordinary quantum mechanics and pilot wave theory are based on the same partial differential equation. |
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| | The main difference is that in ordinary quantum mechanics, the Schrödinger-equation is connected to reality by the Born postulate, which states that the probability density of the particle's position is given by <math> \rho = |\psi|^2 </math>. Pilot wave theory considers the guidance equation to be the fundamental law, and sees the Born rule as a derived concept. |
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| | - \frac{\partial S}{\partial t} = \frac{\left(\nabla S\right)^2}{2m} + V +Q \; , |
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| | Q = - \frac{\hbar^2}{2m} \frac{\nabla^2 \sqrt{\rho}}{\sqrt{\rho}} |
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| | \frac{d}{dt} = \frac{ \partial }{ \partial t } + \vec{v} \cdot \nabla \; . |
| | </math> |
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| | ===Mathematical formulation for multiple particles=== |
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| == jump on their back Longchamp Leather Bags Australia ==
| | The Schrödinger equation for the many-body wavefunction <math> \psi(\vec{r}_1, \vec{r}_2, \cdots, t) </math> is given by |
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| | <math> |
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| | i \hbar \frac{\partial \psi}{\partial t} =\left( -\frac{\hbar^2}{2} \sum_{i=1}^{N} \frac{\nabla_i^2}{m_i} + V(\bold{r}_1,\bold{r}_2,\cdots\bold{r}_N) \right) \psi |
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| | </math> |
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| <li>[http://www.harrietoneilllaw.com/node/573243 http://www.harrietoneilllaw.com/node/573243]</li>
| | The complex wavefunction can be represented as: |
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| | <math>\psi = \sqrt{\rho} \; \exp \left( \frac{i \, S}{\hbar} \right) </math> |
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| | The pilot wave guides the motion of the particles. The guidance equation for the jth particle is: |
|
| | |
| </ul>
| | <math> |
| | \vec{v}_j = \frac{\nabla_j S}{m}\; . |
| | </math> |
| | |
| | The velocity of the jth particle explicitly depends on the positions of the other particles. |
| | This means that the theory is nonlocal. |
| | |
| | ===Empty wave function=== |
| | [[Lucien Hardy]]<ref>Lucien Hardy: ''On the existence of empty waves in quantum theory'', Physics Letters A, vol. 167, no. 1, 6 July 1992, pp. 11-16, {{doi|10.1016/0375-9601(92)90618-V}} ([http://www.sciencedirect.com/science/article/pii/037596019290618V abstract])</ref> and [[John Stewart Bell|J. S. Bell]]<ref name="bell-1992"/> have emphasized that in the de Broglie–Bohm picture of quantum mechanics there can exist '''empty waves''', represented by wave functions propagating in space and time but not carrying energy or momentum,<ref>[[Franco Selleri]], [[Alwyn Van der Merwe]]: ''Quantum paradoxes and physical reality'', [http://books.google.com/books?id=qUgX3B02ofAC&pg=PA86 p. 86]</ref> and not associated with a particle. The same concept was called ''ghost waves'' (or "Gespensterfelder", ''ghost fields'') by [[Albert Einstein]].<ref>[[Franco Selleri]], [[Alwyn Van der Merwe]]: ''Quantum paradoxes and physical reality'', Fundamental Theories of Physics, Kluwer Academic, 1990, ISBN 0-7923-0253-2, [http://books.google.com/books?id=qUgX3B02ofAC&pg=PA85 p. 85–86]</ref> |
| | |
| | The empty wave function notion has been discussed controversially.<ref>Marek Zukowski: ''"On the existence of empty waves in quantum theory": a comment'', Physics Letters A, vol. 175, no. 3-4, 12 April 1993, pp. 257-258, {{doi|10.1016/0375-9601(93)90837-P}} ([http://www.sciencedirect.com/science/article/pii/037596019390837P abstract])</ref><ref>H. D. Zeh: ''Why Bohm's Quantum Theory?'', Found. Phys. Lett. 12 (1999) pp. 197-200, [http://arxiv.org/abs/quant-ph/9812059v2 quant-ph/9812059v2]</ref><ref>L. Vaidman: ''The Reality in Bohmian Quantum Mechanics or Can You Kill with an Empty Wave Bullet?'', [http://arxiv.org/abs/quant-ph/0312227 quant-ph/0312227] (submitted on 31 Dec 2003)</ref> In contrast, the [[many-worlds interpretation of quantum mechanics]] does not call for empty wave functions.<ref name="bell-1992"/> |
| | |
| | ==History== |
| | In his 1926 paper,<ref>Born M. 1926, Z Phys. 38;803. Wave Mechanics</ref> [[Max Born]] suggested that the wave function of Schrödinger's wave equation represents the probability density of finding a particle. |
| | |
| | From this idea, de Broglie developed the pilot wave theory, and worked out a function for the guiding wave. Initially, de Broglie proposed a ''double solution'' approach, in which the quantum object consists of a physical wave (''u''-wave) in real space which has a spherical singular region that gives rise to particle-like behaviour; in this initial form of his theory he did not have to postulate the existence of a quantum particle.<ref name="dewdney-et-al-1992">C. Dewdney, G. Horton, M. M. Lam, Z. Malik, M. Schmidt: ''Wave-particle dualism and the interpretation of quantum mechanics'', Foundations of Physics, vol. 22, no. 10, pp. 1217-1265, 1992, {{doi|10.1007/BF01889712}}, therein: page 1221.</ref> He later formulated it as a theory in which a particle is accompanied by a pilot wave. He presented the pilot wave theory at the 1927 Solvay Conference.<ref>Solvay Conference, 1928, Electrons et Photons: Rapports et Discussions du Cinquieme Conseil de Physique tenu a Bruxelles du 24 au 29 Octobre 1927 sous les auspices de l'Institut International Physique Solvay.</ref> However, [[Wolfgang Pauli]] raised an objection to it at the conference, saying that it did not deal properly with the case of [[inelastic scattering]]. De Broglie was not able to find a response to this objection, and he and Born abandoned the pilot-wave approach. Unlike [[David Bohm]], de Broglie did not complete his theory to encompass the many-particle case.<ref name="dewdney-et-al-1992"/> |
| | |
| | Later, in 1932, [[John von Neumann]] published a paper claiming to prove that all hidden variable theories were impossible.<ref>von Neumann J. 1932, Mathematische Grundlagen der Quantenmechanik.</ref> (A result found to be flawed by [[Grete Hermann]] three years later, though this went unnoticed by the physics community for over fifty years). However, in 1952, [[David Bohm]], dissatisfied with the prevailing orthodoxy, rediscovered de Broglie's pilot wave theory. Bohm developed pilot wave theory into what is now called the [[De Broglie–Bohm theory|De Broglie-Bohm theory]].<ref>Bohm, David (1952). "A suggested Interpretation of the Quantum Theory in Terms of Hidden Variables, I and II, Physical Review 85.</ref> |
| | |
| | The de Broglie-Bohm theory itself might have gone unnoticed by most physicists, if it had not been championed by [[John Stewart Bell|John Bell]], who also countered the objections to it. In 1987, John Bell<ref>Bell, John S, Speakable and Unspeakable in Quantum Mechanics, Cambridge University Press</ref> rediscovered Grete Hermann's work, and thus showed the physics community that Pauli's and von Neumann's objections really only showed that the pilot wave theory did not have locality. |
| | |
| | The de Broglie-Bohm theory is now considered by some to be a valid challenge to the prevailing orthodoxy of the [[Copenhagen Interpretation]], but it remains controversial. |
| | |
| | Yves Couder and co-workers recently discovered a macroscopic pilot wave system in the form of ''walking droplets''. This system exhibits behaviour of a pilot wave, heretofore considered to be reserved to microscopic phenomena.<ref>Y. Couder, A. Boudaoud, S. Protière, Julien Moukhtar, E. Fort: ''Walking droplets: a form of wave-particle duality at macroscopic level?'' , {{doi|10.1051/epn/2010101}}, ([http://www.df.uba.ar/users/dasso/fis4_2do_cuat_2010/walker.pdf PDF])</ref> |
| | |
| | ==References== |
| | {{reflist}} |
| | |
| | ==External links== |
| | *[http://www.tcm.phy.cam.ac.uk/~mdt26/pilot_waves.html "Pilot waves, Bohmian metaphysics, and the foundations of quantum mechanics"], lecture course on pilot wave theory by [[Mike Towler]], Cambridge University (2009). |
| | |
| | *[http://www-math.mit.edu/~bush/PNAS-2010-Bush.pdf "Quantum mechanics writ large"], article about walking droplets by John W. M. Bush, MIT (2010). |
| | *[http://math.mit.edu/~bush/?page_id=484 "Hydrodynamic quantum analogues"] articles by John W. M. Bush (MIT) and coworkers on hydrodynamic quantum analogues and hydrodynamic pilot-wave theory. 2012, 2013. |
| | |
| | * [http://plato.stanford.edu/entries/qm-bohm/ More complete HTML encyclopedic page about the subject]. |
| | |
| | {{DEFAULTSORT:Pilot wave}} |
| | [[Category:Interpretations of quantum mechanics]] |
| | [[Category:Quantum measurement]] |
In theoretical physics, the pilot wave theory was the first known example of a hidden variable theory, presented by Louis de Broglie in 1927. Its more modern version, the Bohm interpretation,
remains a controversial attempt to interpret quantum mechanics as a deterministic theory, avoiding
troublesome notions such as instantaneous wavefunction collapse and the paradox of Schrödinger's cat.
The pilot wave theory
The pilot wave theory is one of several interpretations of quantum mechanics. It uses the same mathematics as other interpretations of quantum mechanics; consequently, it is
also supported by the current experimental evidence to the same extent as the other interpretations.
Principles
The pilot wave theory is a hidden variable theory. Consequently:
- the theory has realism (meaning that its concepts exist independently of the observer);
- the theory has determinism.
The positions and momenta of the particles are considered to be the hidden variables. However, the observer not only doesn't know the precise value of these variables, but more importantly, cannot know them precisely because any measurement disturbs them – as stipulated by the Heisenberg uncertainty principle.
A collection of particles has an associated matter wave, which evolves according to the Schrödinger Equation. Each particle follows a deterministic trajectory, which is guided by the wave function; collectively, the density of the particles conforms to the magnitude of the wave function. The wave function is not influenced by the particle and can exist also as an empty wave function.[1]
The theory brings to light nonlocality that is implicit in the non-relativistic formulation of quantum mechanics and uses it to satisfy Bell's theorem. Interestingly, these nonlocal effects are compatible with no-communication theorem, which prevents us from using them for faster-than-light communication.
Consequences
The pilot wave theory shows that it is possible to have a realistic and deterministic hidden variable theory, which reproduces the experimental results of ordinary quantum mechanics. The price which has to be paid for this is manifest nonlocality.
Mathematical formulation for a single particle
The matter wave of de Broglie is described by the time-dependent Schrödinger equation:
The complex wavefunction can be represented as:
By plugging this into the Schrödinger-equation, one can derive two new equations for the real variables.
The first is the continuity equation for the probability density :[2]
where the velocity field is defined by the guidance equation
According to pilot wave theory, the point particle and the matter wave are both real and distinct physical entities. ( Unlike standard quantum mechanics, where particles and waves are considered to be the same entities, connected by wave-particle duality. ) The pilot wave guides the motion of the point particles as described by the guidance equation.
Ordinary quantum mechanics and pilot wave theory are based on the same partial differential equation.
The main difference is that in ordinary quantum mechanics, the Schrödinger-equation is connected to reality by the Born postulate, which states that the probability density of the particle's position is given by . Pilot wave theory considers the guidance equation to be the fundamental law, and sees the Born rule as a derived concept.
The second equation is a modified Hamilton–Jacobi equation for the action :
where Q is the quantum potential defined by
By neglecting Q, our equation is reduced to the Hamilton-Jacobi equation of a classical point particle. ( Stricly speaking, this is only a semiclassical limit Template:Clarify, because the superposition principle still holds and one needs a decoherence mechanism to get rid of it. Interaction with the environment can provide this mechanism.) So, the quantum potential is responsible for all the mysterious effects of quantum mechanics.
One can also combine the modified Hamilton-Jacobi equation with the guidance equation to derive a quasi-Newtonian equation of motion
where the hydrodynamic time derivative is defined as
Mathematical formulation for multiple particles
The Schrödinger equation for the many-body wavefunction is given by
The complex wavefunction can be represented as:
The pilot wave guides the motion of the particles. The guidance equation for the jth particle is:
The velocity of the jth particle explicitly depends on the positions of the other particles.
This means that the theory is nonlocal.
Empty wave function
Lucien Hardy[3] and J. S. Bell[1] have emphasized that in the de Broglie–Bohm picture of quantum mechanics there can exist empty waves, represented by wave functions propagating in space and time but not carrying energy or momentum,[4] and not associated with a particle. The same concept was called ghost waves (or "Gespensterfelder", ghost fields) by Albert Einstein.[5]
The empty wave function notion has been discussed controversially.[6][7][8] In contrast, the many-worlds interpretation of quantum mechanics does not call for empty wave functions.[1]
History
In his 1926 paper,[9] Max Born suggested that the wave function of Schrödinger's wave equation represents the probability density of finding a particle.
From this idea, de Broglie developed the pilot wave theory, and worked out a function for the guiding wave. Initially, de Broglie proposed a double solution approach, in which the quantum object consists of a physical wave (u-wave) in real space which has a spherical singular region that gives rise to particle-like behaviour; in this initial form of his theory he did not have to postulate the existence of a quantum particle.[10] He later formulated it as a theory in which a particle is accompanied by a pilot wave. He presented the pilot wave theory at the 1927 Solvay Conference.[11] However, Wolfgang Pauli raised an objection to it at the conference, saying that it did not deal properly with the case of inelastic scattering. De Broglie was not able to find a response to this objection, and he and Born abandoned the pilot-wave approach. Unlike David Bohm, de Broglie did not complete his theory to encompass the many-particle case.[10]
Later, in 1932, John von Neumann published a paper claiming to prove that all hidden variable theories were impossible.[12] (A result found to be flawed by Grete Hermann three years later, though this went unnoticed by the physics community for over fifty years). However, in 1952, David Bohm, dissatisfied with the prevailing orthodoxy, rediscovered de Broglie's pilot wave theory. Bohm developed pilot wave theory into what is now called the De Broglie-Bohm theory.[13]
The de Broglie-Bohm theory itself might have gone unnoticed by most physicists, if it had not been championed by John Bell, who also countered the objections to it. In 1987, John Bell[14] rediscovered Grete Hermann's work, and thus showed the physics community that Pauli's and von Neumann's objections really only showed that the pilot wave theory did not have locality.
The de Broglie-Bohm theory is now considered by some to be a valid challenge to the prevailing orthodoxy of the Copenhagen Interpretation, but it remains controversial.
Yves Couder and co-workers recently discovered a macroscopic pilot wave system in the form of walking droplets. This system exhibits behaviour of a pilot wave, heretofore considered to be reserved to microscopic phenomena.[15]
References
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External links
- ↑ 1.0 1.1 1.2 J. S. Bell: Six possible worlds of quantum mechanics, Foundations of Physics, vol. 22, no. 10, Part I. Invited Papers Dedicated To Louis De Broglie, 1992, pp. 1201-1215, 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park., p. 1212
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A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang (full text)
- ↑ Lucien Hardy: On the existence of empty waves in quantum theory, Physics Letters A, vol. 167, no. 1, 6 July 1992, pp. 11-16, 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park. (abstract)
- ↑ Franco Selleri, Alwyn Van der Merwe: Quantum paradoxes and physical reality, p. 86
- ↑ Franco Selleri, Alwyn Van der Merwe: Quantum paradoxes and physical reality, Fundamental Theories of Physics, Kluwer Academic, 1990, ISBN 0-7923-0253-2, p. 85–86
- ↑ Marek Zukowski: "On the existence of empty waves in quantum theory": a comment, Physics Letters A, vol. 175, no. 3-4, 12 April 1993, pp. 257-258, 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park. (abstract)
- ↑ H. D. Zeh: Why Bohm's Quantum Theory?, Found. Phys. Lett. 12 (1999) pp. 197-200, quant-ph/9812059v2
- ↑ L. Vaidman: The Reality in Bohmian Quantum Mechanics or Can You Kill with an Empty Wave Bullet?, quant-ph/0312227 (submitted on 31 Dec 2003)
- ↑ Born M. 1926, Z Phys. 38;803. Wave Mechanics
- ↑ 10.0 10.1 C. Dewdney, G. Horton, M. M. Lam, Z. Malik, M. Schmidt: Wave-particle dualism and the interpretation of quantum mechanics, Foundations of Physics, vol. 22, no. 10, pp. 1217-1265, 1992, 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park., therein: page 1221.
- ↑ Solvay Conference, 1928, Electrons et Photons: Rapports et Discussions du Cinquieme Conseil de Physique tenu a Bruxelles du 24 au 29 Octobre 1927 sous les auspices de l'Institut International Physique Solvay.
- ↑ von Neumann J. 1932, Mathematische Grundlagen der Quantenmechanik.
- ↑ Bohm, David (1952). "A suggested Interpretation of the Quantum Theory in Terms of Hidden Variables, I and II, Physical Review 85.
- ↑ Bell, John S, Speakable and Unspeakable in Quantum Mechanics, Cambridge University Press
- ↑ Y. Couder, A. Boudaoud, S. Protière, Julien Moukhtar, E. Fort: Walking droplets: a form of wave-particle duality at macroscopic level? , 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park., (PDF)