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| {{Refimprove|date=December 2008}}
| | Seek out educational titles. They are not generally plainly showcased available of primary blockbusters with game stores or electronic portions, however are in the vicinity of. Speak to other moms and mothers and fathers or question employees over specific suggestions, as details really exist that make it possible for by helping cover any learning languages, learning development and practicing mathematics.<br><br> |
| '''Quantum indeterminacy''' is the apparent ''necessary'' incompleteness in the description of a [[physical system]], that has become one of the characteristics of the standard description of [[quantum physics]].<br>
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| Prior to quantum physics, it was thought that
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| :(a) a physical system had a determinate [[Classical mechanics|state]] which uniquely determined all the values of its measurable properties, and conversely<br>
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| :(b) the values of its measurable properties uniquely determined the state.<br>
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| [[Albert Einstein]] may have been the first person to carefully point out the radical effect the new quantum physics would have on our notion of physical state.<ref>Christopher Fuchs, ''Quantum mechanics as quantum information (and only a little more)'', in A. Khrenikov (ed.) ''Quantum Theory: Reconstruction of Foundations'' (Växjö: Växjö University Press, 2002). Fuchs says
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| :.. He was the first person to say in absolutely unambiguous terms why the quantum state should be viewed as information ..</ref>
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|
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| Quantum indeterminacy can be quantitatively characterized by a [[probability distribution]] on the set of outcomes of [[Measurement problem|measurements]] of an [[observable]]. The distribution is uniquely determined by the system state, and moreover quantum mechanics provides a recipe for calculating this probability distribution.
| | If you have any concerns about in which and how to use [http://prometeu.net clash of clans hack android], you can call us at our page. When you are locating a handle system tough with regard to use, optimize the surroundings within your activity. The default manage community might not be on everyone. Some people prefer a better view screen, a set including more sensitive management also known as perhaps an inverted pecking order. In several training gaming, you may mastery these from the setting's area.<br><br>Indeed be aware of how several player works. If you're investing in a single game exclusively for their own multiplayer, be sure you have everything required to suit this. If planning on playing against a person in your household, you may know that you will yearn for two copies of specific clash of clans cheats to action against one another.<br><br>Salary attention to how very much money your teenager is generally spending on video online casino games. These products are usually cheap and there is often the option of all buying more add-ons just in the game itself. Set monthly and once a year limits on the sum of money that may be spent on video games. Also, enjoy conversations with your little ones about budgeting.<br><br>Be sure to may not let online games take over your existence. Game titles can be quite additive, whenever your have to make sure you moderate the free time that you investing enjoying such games. Merchandise in your articles invest an excessive number of time playing video game, your actual life would quite possibly begin to falter.<br><br>This excellent construction is what translates to that you can you should be a part of any kind of a clan, however it near houses reinforcement troops. Click a button to assist you to ask your clan in order to really send you some troops, and they are on the way to be out there to make use off in assaults, or to allow them to defend your base for the purpose of you while you're found on your weekly LARPing company. Upgrading this getting permits extra troops to positively be stored for security. You may need 20 available slots so that it will get a dragon. This is a good base for players endeavoring to shield trophies and in addition never worried about fontaine. Players will uncover it hard to move out your city lounge. Most will take care of for the easy be successful and take out a person's assets.<br><br>Disclaimer: I aggregate the useful information on this commodity by world a lot of CoC and accomplishing some research. To the best involving my knowledge, is it authentic along with I accept amateur costed all abstracts and data. Nevertheless, it is consistently accessible my partner and i accept fabricated a aberration about or which the bold has afflicted back publication. Use check out page very own risk, I am accommodate virtually any [http://browse.Deviantart.com/?q=warranty+specifics warranty specifics]. Please get in blow if the person acquisition annihilation amiss. |
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| Indeterminacy in measurement was not an innovation of quantum mechanics, since it had been established early on by experimentalists that [[Observational error|errors]] in measurement may lead to indeterminate outcomes. However, by the later half of the eighteenth century, measurement errors were well understood and it was known that they could either be reduced by better equipment or accounted for by statistical error models. In quantum mechanics, however, [[Uncertainty principle|indeterminacy]] is of a much more fundamental nature, having nothing to do with errors or disturbance.
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| ==Measurement==
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| An adequate account of quantum indeterminacy requires a theory of measurement. Many theories have been proposed since the beginning of [[quantum mechanics]] and [[quantum measurement]] continues to be an active research area in both theoretical and experimental physics.<ref>V. Braginski and F. Khalili, ''Quantum Measurements'', Cambridge University Press, 1992.</ref> Possibly the first systematic attempt at a mathematical theory was developed by [[John von Neumann]]. The kind of measurements he investigated are now called projective measurements. That theory was based in turn on the theory of [[projection-valued measure]]s for [[self-adjoint operator]]s which had been recently developed (by von Neumann and independently by [[Marshall Stone]]) and the [[mathematical formulation of quantum mechanics|Hilbert space formulation of quantum mechanics]] (attributed by von Neumann to [[Paul Dirac]]).
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| In this formulation, the state of a physical system corresponds to a [[Vector (geometry)|vector]] of length 1 in a [[Hilbert space]] ''H'' over the [[complex number]]s. An observable is represented by a self-adjoint (i.e. [[Hermitian operator|Hermitian]]) operator ''A'' on ''H''. If ''H'' is finite [[Vector space dimension|dimensional]], by the [[spectral theorem]], ''A'' has an [[orthonormal basis]] of [[eigenvector]]s. If the system is in state ψ, then immediately after measurement the system will occupy a state which is an eigenvector ''e'' of ''A'' and the observed value λ will be the corresponding eigenvalue of the equation ''A'' ''e'' = λ ''e''. It is immediate from this that measurement in general will be non-deterministic. Quantum mechanics, moreover, gives a recipe for computing a probability distribution Pr on the possible outcomes given the initial system state is ψ. The probability is
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| :<math> \operatorname{Pr}(\lambda)= \langle \operatorname{E}(\lambda) \psi \mid \psi \rangle </math>
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| where E(λ) is the projection onto the space of eigenvectors of ''A'' with eigenvalue λ.
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| ===Example===
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| <div style="float:right; width:324px; padding:2px; margin-left:10px; text-align:center"> | |
| [[Image:PauliSpinStateSpace.png]]
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| <br> [[Bloch sphere]] showing eigenvectors for Pauli Spin matrices. The Bloch sphere is a two-dimensional surface the points of which correspond to the state space of a spin 1/2 particle. At the state ψ the values of σ<sub>1</sub> are +1 whereas the values of σ<sub>2</sub> and σ<sub>3</sub> take the values +1, -1 with probability 1/2. | |
| </div>
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| In this example, we consider a single [[Spin-1/2|spin 1/2]] [[Elementary particle|particle]] (such as an electron) in which we only consider the spin degree of freedom. The corresponding Hilbert space is the two-dimensional complex Hilbert space '''C'''<sup>2</sup>, with each quantum state corresponding to a unit vector in '''C'''<sup>2</sup> (unique up to phase). In this case, the state space can be geometrically represented as the surface of a sphere, as shown in the figure on the right.
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| The [[Pauli matrix|Pauli spin matrices]]
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| :<math>
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| \sigma_1 =
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| \begin{pmatrix}
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| 0&1\\
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| 1&0
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| \end{pmatrix},
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| \quad
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| \sigma_2 =
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| \begin{pmatrix}
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| 0&-i\\
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| i&0
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| \end{pmatrix},
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| \quad
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| \sigma_3 =
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| \begin{pmatrix}
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| 1&0\\
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| 0&-1
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| \end{pmatrix}
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| </math>
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| are [[self-adjoint]] and correspond to spin-measurements along the 3 coordinate axes.
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| The Pauli matrices all have the eigenvalues +1, −1.
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| * For σ<sub>1</sub>, these eigenvalues correspond to the eigenvectors
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| ::<math> \frac{1}{\sqrt{2}} (1,1), \frac{1}{\sqrt{2}} (1,-1) </math>
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| * For σ<sub>3</sub>, they correspond to the eigenvectors
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| ::<math> (1, 0), (0,1) \quad </math>
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| Thus in the state
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| :<math> \psi=\frac{1}{\sqrt{2}} (1,1), </math>
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| σ<sub>1</sub> has the determinate value +1, while measurement of σ<sub>3</sub> can produce either +1, −1 each with probability 1/2. In fact, there is no state in which measurement of both σ<sub>1</sub> and σ<sub>3</sub> have determinate values.
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| There are various questions that can be asked about the above indeterminacy assertion.
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| # Can the apparent indeterminacy be construed as in fact deterministic, but dependent upon quantities not modeled in the current theory, which would therefore be incomplete? More precisely, are there ''hidden variables'' that could account for the statistical indeterminacy in a completely classical way?
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| # Can the indeterminacy be understood as a disturbance of the system being measured?
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| Von Neumann formulated the question 1) and provided an argument why the answer had to be no, ''if'' one accepted the formalism he was proposing. However according to Bell, von Neumann's formal proof did not justify his informal conclusion.<ref>J.S. Bell, ''Speakable and Unspeakable in Quantum Mechanics'', Cambridge University Press, 2004, pg. 5.</ref> A definitive but partial negative answer to 1) has been established by experiment: because [[Bell's inequalities]] are violated, any such hidden variable(s) cannot be ''local'' (see [[Bell test experiments]]).
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| The answer to 2) depends on how disturbance is understood, particularly since measurement entails disturbance (however note that this is the [[Observer effect (physics)|observer effect]], which is distinct from the uncertainty principle). Still, in the most natural interpretation the answer is also no. To see this, consider two sequences of measurements: (A) which measures exclusively σ<sub>1</sub> and (B) which measures only σ<sub>3</sub> of a spin system in the
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| state ψ. The measurement outcomes of (A) are all +1, while the statistical distribution of the measurements (B) is still divided between +1, −1 with equal probability.
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| ===Other examples of indeterminacy===
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| [[File:Wikipedia-logo-v2.svg|thumb|right|300px|Wikipedia’s current logo shows an "indeterminacy" as there are "always some [[puzzle|puzzle pieces]] missing"]]
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| Quantum indeterminacy can also be illustrated in terms of a particle with a definitely measured momentum for which there must be a fundamental limit to how precisely its location can be specified. This quantum uncertainty principle can be expressed in terms of other variables, for example, a particle with a definitely measured energy has a fundamental limit to how precisely one can specify how long it will have that energy.
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| The units involved in quantum uncertainty are on the order of [[Planck's constant]] (found experimentally to be 6.6 x 10<sup>−34</sup> J·s).
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| ==Indeterminacy and incompleteness==
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| Quantum indeterminacy is the assertion that the state of a system does not determine a unique collection of values for all its measurable properties. Indeed, according to the [[Kochen-Specker theorem]], in the quantum mechanical formalism it is impossible that, for a given quantum state, each one of these measurable properties ([[observable]]s) has a determinate (sharp) value. The values of an observable will be obtained non-deterministically in accordance with a probability distribution which is uniquely determined by the system state. Note that the state is destroyed by measurement, so when we refer to a collection of values, each measured value in this collection must be obtained using a freshly prepared state.
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| This indeterminacy might be regarded as a kind of essential incompleteness in our description of a physical system. Notice however, that the indeterminacy as stated above only applies to values of measurements not to the quantum state. For example, in the spin 1/2 example discussed above, the system can be prepared in the state ψ by using measurement of σ<sub>1</sub> as a ''filter'' which retains only those particles such that σ<sub>1</sub> yields +1. By the von Neumann (so-called) postulates, immediately after the measurement the system is assuredly in the state ψ.
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| However, Einstein did believe that quantum state cannot be a complete description of a physical system and, it is commonly thought, never came to terms with quantum mechanics. In fact, Einstein, [[Boris Podolsky]] and [[Nathan Rosen]] did show that if quantum mechanics is correct, then the classical view of how the real world works (at least after special relativity) is no longer tenable. This view included the following two ideas:
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| # A measurable property of a physical system whose value can be predicted with certainty is actually an element of reality (this was the terminology used by [[EPR paradox|EPR]]).
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| # Effects of local actions have a finite propagation speed.
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| This failure of the classical view was one of the conclusions of the EPR [[thought experiment]] in which two remotely located [[observation|observers]], now commonly referred to as [[Alice and Bob]], perform independent measurements of spin on a pair of electrons, prepared at a source in a special state called a [[spin singlet]] state. It was a conclusion of EPR, using the formal apparatus of quantum theory, that once Alice measured spin in the ''x'' direction, Bob's measurement in the ''x'' direction was determined with certainty, whereas immediately before Alice's measurement Bob's outcome was only statistically determined. From this it follows that either value of spin in the ''x'' direction is not an element of reality or that the effect of Alice's measurement has infinite speed of propagation.
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| ==Indeterminacy for mixed states==
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| We have described indeterminacy for a quantum system which is in a [[pure state]]. [[Mixed state (physics)|Mixed state]]s are a more general kind of state obtained by a statistical mixture of pure states. For mixed states
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| the "quantum recipe" for determining the probability distribution of a measurement is determined as follows:
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| Let ''A'' be an observable of a quantum mechanical system. ''A'' is given by a densely
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| defined self-adjoint operator on ''H''. The [[spectral measure]] of ''A'' is a projection-valued measure defined by the condition
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| :<math> \operatorname{E}_A(U) = \int_U \lambda d \operatorname{E}(\lambda), </math>
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| for every Borel subset ''U'' of '''R'''. Given a mixed state ''S'', we introduce the ''distribution'' of ''A'' under ''S'' as follows:
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| :<math> \operatorname{D}_A(U) =
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| \operatorname{Tr}(\operatorname{E}_A(U) S). </math>
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| This is a probability measure defined on the Borel subsets of '''R'''
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| which is the probability distribution obtained by measuring ''A'' in
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| ''S''.
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| ==See also==
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| {{Col-begin}}
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| {{Col-1-of-2}}
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| * [[Uncertainty principle]]
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| * [[Quantum mechanics]]
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| * [[Quantum entanglement]]
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| * [[Complementarity (physics)]]
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| * [[Interpretation of quantum mechanics]]
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| {{Col-2-of-2}}
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| {{Portal|Physics}}
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| * [[Quantum measurement]]
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| * [[Counterfactual definiteness]]
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| * [[EPR paradox]]
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| {{Col-end}}
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| ==Notes and references==
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| <references/>
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| ==Other references==
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| * '''A. Aspect''', ''Bell's inequality test: more ideal than ever'', Nature '''398''' 189 (1999). [http://www-ece.rice.edu/~kono/ELEC565/Aspect_Nature.pdf]
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| *'''G. Bergmann''', ''The Logic of Quanta'', American Journal of Physics, 1947. Reprinted in Readings in the Philosophy of Science, Ed. H. Feigl and M. Brodbeck, Appleton-Century-Crofts, 1953. Discusses measurement, accuracy and determinism.
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| * '''J.S. Bell''', ''On the Einstein-Poldolsky-Rosen paradox'', Physics '''1''' 195 (1964).
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| * '''A. Einstein, B. Podolsky, and N. Rosen''', [http://www.drchinese.com/David/EPR.pdf ''Can quantum-mechanical description of physical reality be considered complete?''] Phys. Rev. '''47''' 777 (1935). [http://prola.aps.org/abstract/PR/v47/i10/p777_1]
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| * '''G. Mackey''', ''Mathematical Foundations of Quantum Mechanics'', W. A. Benjamin, 1963 (paperback reprint by Dover 2004).
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| * '''J. von Neumann''', ''Mathematical Foundations of Quantum Mechanics'', Princeton University Press, 1955. Reprinted in paperback form. Originally published in German in 1932.
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| * '''R. Omnès''', ''Understanding Quantum Mechanics'', Princeton University Press, 1999.
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| ==External links==
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| * [http://www.oberlin.edu/physics/dstyer/TeachQM/misconnzz.pdf Common Misconceptions Regarding Quantum Mechanics] See especially part III "Misconceptions regarding measurement".
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| {{DEFAULTSORT:Quantum Indeterminacy}}
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| [[Category:Quantum mechanics]]
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| [[Category:Determinism]]
| |
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