In Fig. 1: The wavefunctions of an electron in a hydrogen atom possessing definite energy (increasing downward: , , and zero everywhere else. If we perform a position measurement on such a wavefunction, we will obtain the result , where , neither an eigenstate of position nor of momentum. When we measure the position of... quantum physics, the Heisenberg uncertainty principle, sometimes called the Heisenberg indeterminacy principle, expresses a limitation on accuracy of (nearly) simultaneous measurement of In physics, particularly in quantum physics a system observable is a property of the system state that can be determined by some sequence of physical operations. These operations might involve submitting the system to various electromagnetic fields and eventually reading a value off some gauge. In systems governed by classical... observables such as the position and the In physics, momentum is a physical quantity related to the velocity and mass of an object. Momentum is the Noether charge of translational invariance. As such, even fields as well as other things can have momentum, not just particles. However, in curved spacetime which is not asymptotically Minkowski, momentum isn... momentum of a particle. It furthermore precisely quantifies the imprecision by providing a lower bound (greater than zero) for the product of the In descriptive statistics, statistical dispersion (also called statistical variability) is quantifiable variation of measurements of differing members of a population within the scale on which they are measured. Measures of statistical dispersion A measure of statistical dispersion is a real number that is zero if all the data are identical... dispersions of the measurements. For instance, consider repeated trials of the following experiment: By an An operational definition of a quantity is a specific process whereby it is measured. For example, the weight of an object can be operationally defined by using a balance and standard weights. Operational definitions are also used in defining system states by a specific preparation process. For example, 100 degrees... operational process, a particle is prepared in a definite state and two successive measurements are performed on the particle. The first one measures its position and the second immediately after measures its momentum. Suppose furthermore that the operational process of preparing the state is such that on every trial the first measurement yields the same value, or at least a distribution of values with a very small dispersion d_p around a value p. Then the second measurement will have a distribution of values whose dispersion d_q is at least inversely proportional to d_p.

In quantum mechanical terminology, the operational process has produced a particle in a possibly The term mixed state refers to a concept in physics, particularly quantum mechanics. In quantum mechanics a state E of a quantum mechanical ensemble is represented by a density operator which can be decomposed as a randomization of two statistically different statistical ensembles, or a linear combination of pure states... mixed state with definite position. Any momentum measurement on the particle will necessarily yield a dispersion of values on repeated trials. Moreover, if we follow the momentum measurement by a measurement of position, we will get a dispersion of position values.

More generally, an uncertainty relation arises between any two observable quantities defined by non-commuting operators. It is one of the cornerstones of Fig. 1: The wavefunctions of an electron in a hydrogen atom possessing definite energy (increasing downward: , , and zero everywhere else. If we perform a position measurement on such a wavefunction, we will obtain the result , where , neither an eigenstate of position nor of momentum. When we measure the position of... quantum mechanics and was discovered by Werner Heisenberg Werner Karl Heisenberg (December 5, 1901 – February 1, 1976) was a celebrated German physicist and Nobel laureate, one of the founders of quantum mechanics. He was born in Würzburg, Germany and died in Munich. Heisenberg was the head of Nazi Germanys nuclear energy program, though... Werner Heisenberg in Events January 7 - First transatlantic telephone call - New York City to London January 9 - Military rebellion crushed in Lisbon January 14 - Paul Doumer elected president of France January 19 - Britain sends troops to China February 12 - First British troops lad on Shanghai February 14 - Earthquake in Yugoslavia - 700 dead February... 1927.

Overview

The uncertainty principle in Fig. 1: The wavefunctions of an electron in a hydrogen atom possessing definite energy (increasing downward: , , and zero everywhere else. If we perform a position measurement on such a wavefunction, we will obtain the result , where , neither an eigenstate of position nor of momentum. When we measure the position of... quantum mechanics is sometimes explained by claiming that the measurement of position necessarily disturbs a particle's momentum. Heisenberg himself may have offered explanations which suggest this view, at least initially. That the role of disturbance is not essential can be seen as follows: Consider an ensemble of (non-interacting) particles all prepared in the same state; for each particle in the ensemble we measure the momentum or the position (but not both). From the measurement results, we will obtain probability distributions of values for both these quantities and the uncertainty relations still hold for the dispersions d_p, d_q of the values.

The Heisenberg uncertainty relations are a theoretical bound over all measurements. They hold for so-called ideal measurements, sometimes called John von Neumann in the 1940s. John von Neumann (Neumann János) (December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician who made important contributions in quantum physics, set theory, computer science, economics and virtually all mathematical fields. The oldest of three children, von Neumann was born... von Neumann measurements. They hold This page includes English translations of several Latin phrases and abbreviations such as . Some of these are themselves translations from Greek. For a list of more formal proverbs, see: List of Latin proverbs. Note that the difference between phrases and proverbs is often subjective. Please use this test to see... a fortiori for non-ideal or Lev Davidovich Landau (Ле́в Дави́дович Ланда́у) (January 22, 1908 – April 1, 1968) was a prominent Soviet physicist and winner of the Nobel Prize for Physics whose broad field of work included... Landau measurements.

Correspondingly, any one particle (in the general sense, e. g. carrying discrete Electric charge is a fundamental property of some subatomic particles, which determines their electromagnetic interactions. It is one of the quantum numbers. Matter that possesses a charge is influenced by, and produces, electromagnetic fields. The interaction between charge and an electromagnetic field is the source of one of the four... electric charge) cannot be described simultaneously as a "classic point particle" and as a A wave is a disturbance that propagates. Apart from electromagnetic radiation, and probably gravitational radiation, which can travel through vacuum, waves exist in a medium (which on deformation is capable of producing elastic restoring forces) through which they travel and can transfer energy from one place to another without any... wave. (The fact itself that either one of these descriptions can be appropriate at least in separate cases is called In modern physics, duality most often refers to the paradigm underlying quantum mechanics, according to which matter or energy can exhibit properties associated with wave physics as well as classical particle mechanics. These two sets of phenomena are mutually exclusive in classical physics, but nevertheless are both needed in order... wave-particle duality; a change of appropriate descriptions according to measured values is known as In certain interpretations of quantum mechanics, wavefunction collapse is one of two processes by which quantum systems apparently evolve according to the laws of quantum mechanics. It is also called , the probability of collapsing to a particular eigenstate of is a fundamental problem in the interpretation of quantum mechanics known... wavefunction collapse.) The uncertainty principle (as initially considered by Heisenberg) is concerned with cases in which neither of these two descriptions is fully and exclusively appropriate, such as a In physics, the particle in a box (or the square well) is a simple idealized system that can be completely solved within quantum mechanics. It is the situation of a particle confined within a finite region of space (the box) by an infinite potential that exists at the walls of... particle in a box with a particular energy value; i. e. systems which are characterized neither by one unique "position" (one particular value of distance from a potential wall) nor by one unique value of momentum (incl. its direction).

There is a precise, quantitative analogy between the Heisenberg uncertainty relations and properties of waves or signals. Consider a time-varying signal such as a sound wave. It is meaningless to ask about the frequency spectrum of the signal at a moment in time. In order to determine the frequencies accurately, one needs to sample the signal for some time, thereby losing time precision. In other words, a sound cannot have both a precise time, as in a short pulse, and a precise frequency, as in a continuous pure tone. The time and frequency of a wave in time are analogous to the position and momentum of a wave in space.

Definition

The statement is as follows. If several identical copies of a system in a given state are prepared, measurements of position and momentum will vary according to known In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. In technical terms, a probability distribution is a probability measure whose domain is the Borel algebra on the reals. A probability distribution is a special case of the... probability distributions; this is the fundamental postulate of quantum mechanics. We could measure the In probability and statistics, the standard deviation is the most commonly used measure of statistical dispersion. Standard deviation is defined as the square root of the variance. It is defined this way in order to give us a measure of dispersion that is (1) a non-negative number, and (2... standard deviation Δx of the position measurements and the standard deviation Δp of the momentum measurements. Then we will find that

where h is Plancks constant, denoted ): where π is the constant pi. This constant is pronounced as h-bar. The figures cited here are the 2002 CODATA-recommended values for the constants and their uncertainties. The 2002 CODATA results were made available in December 2003 and represent the best-known, internationally-accepted... Planck's constant and π is . The minuscule, or lower-case, pi The mathematical constant π (written as pi when the Greek letter is not available) is ubiquitous in mathematics and physics. In Euclidean plane geometry, π may be defined as either the ratio of a circles circumference to its diameter, or as the area... Archimedes' constant (pi). (In some treatments, the "uncertainty" of a variable is taken to be the smallest width of a range which contains 50% of the values, which, in the case of Probability density function of Gaussian distribution (bell curve). The normal distribution, also called Gaussian distribution, is an extremely important probability distribution in many fields, especially in physics and engineering. It is actually a family of distributions of the same general form, differing only in their , 1738) in the context of... normally distributed variables, leads to a larger lower bound of h/2π for the product of the uncertainties.) Note that this inequality allows for several possibilities: the state could be such that x can be measured with high precision, but then p will only approximately be known, or conversely p could be sharply defined while x cannot be precisely determined. In yet other states, both x and p can be measured with "reasonable" (but not arbitrarily high) precision.

In everyday life, we don't observe these uncertainties because the value of h is extremely small.

Expression of finite available amount of Fisher information

The uncertainty principle alternatively derives as an expression of the In statistics, the Cramér-Rao inequality, named in honor of Harald Cramér and Calyampudi Radhakrishna Rao, states that the reciprocal of the Fisher information about a parameter θ, I(θ), is a lower bound on the variance of an unbiased estimator. In some cases, no unbiased estimator actually... Cramer-Rao inequality of classical measurement theory. This is in the case where a particle position is measured. See Stam (1959). The mean-squared particle momentum enters as the In statistics, the Fisher information is the probability density function of random variable has been averaged out. The concept of information is useful when comparing two methods of observation of some random process. Information as defined above may be written as and is thus the expection of log of the... Fisher information in the inequality. See also Extreme physical information (EPI) is a principle for discovering laws of science. The laws are in the form of differential equations or distribution functions. Examples are the Schrodinger wave equation and the Maxwell-Boltzmann distribution law. The EPI principle builds on the well known idea that the observation of a... extreme physical information.

Generalized uncertainty principle

The uncertainty principle does not just apply to position and momentum. In its general form, it applies to every pair of conjugate variables. An example of a pair of conjugate variables is the x-component of In physics, angular momentum intuitively measures how much the linear momentum is directed around a certain point called the origin; the moment of momentum. Since angular momentum depends upon the origin of choice, one must be careful when discussing angular momentum to specify the origin and not to combine angular... angular momentum ( In physics, spin is an intrinsic angular momentum associated with microscopic particles. It is a purely quantum mechanical phenomenon without any analogy in classical mechanics. Whereas classical angular momentum arises from the rotation of an extended object, spin is not associated with any rotating internal masses, but is intrinsic to... spin) vs. the y-component of angular momentum. In general, and unlike the case of position versus momentum discussed above, the lower bound for the product of the uncertainties of two conjugate variables depends on the system state. The uncertainty principle becomes then a theorem in the theory of operators which we now state

Theorem. For arbitrary On a finite-dimensional inner product space, a self-adjoint operator is one that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose. By the finite-dimensional spectral theorem such operators have an orthonormal... symmetric operators A: H → H and B: H → H, and any element x of H such that A B x and B A x are both defined (so that in particular, A x and B x are also defined), then

This is an immediate consequence of the In mathematics, the Cauchy-Schwarz inequality, also known as the Schwarz inequality, or the Cauchy-Bunyakovski-Schwarz inequality, is a useful inequality encountered in many different settings, such as linear algebra applied to vectors, in analysis applied to infinite series and integration of products, and in probability theory, applied to... Cauchy-Bunyakovski-Schwarz inequality.

Consequently, the following general form of the uncertainty principle, first pointed out in 1930 is a common year starting on Wednesday. Events January-February January 6 - The first diesel-engine automobile trip is completed (Indianapolis, Indiana, to New York City). January 27 - Miguel Primo de Rivera resigns January 30 - General Damaso Berenquer becomes the new prime minister of Spain February 18 - While studying... 1930 by Howard Percy Robertson (January 27, 1903 - August 26, 1961) was a scientist known for contributions related to cosmology and the uncertainty principle. One of Robertsons first landmark papers, a brief note in The Annals of Mathematics, series II, Vol 39, pp 101-104 (1938) entitled Note on the preceding... Howard Percy Robertson and (independently) by Erwin Schrödinger, as depicted on the former Austrian 1000 Schilling bank note. Erwin Rudolf Josef Alexander Schrödinger ( August 12, 1887 – January 4, 1961), an Austrian physicist, achieved fame for his contributions to quantum mechanics, especially the Schrödinger equation, for which he won the... Erwin Schrödinger, holds:

This inequality is called the Robertson-Schrödinger relation.

The operator A B - B A is called the commutator of A, B and is denoted [A, B]. It is defined on those x for which A B x and B A x are both defined.

From the Robertson-Schrödinger relation, the following Heisenberg uncertainty relation is immediate:

Suppose A and B are two observables which are identified to self-adjoint (and in particular symmetric) operators. If B A ψ and A B ψ are defined then

where

is the operator In statistics, . As well as statistics, means are often used in geometry and analysis; a wide range of means have been developed for these purposes, which are not much used in statistics. See the Other Means section below for a list of means. Sample mean is often used as an... mean of observable X in the system state ψ and

is the operator In probability and statistics, the standard deviation is the most commonly used measure of statistical dispersion. Standard deviation is defined as the square root of the variance. It is defined this way in order to give us a measure of dispersion that is (1) a non-negative number, and (2... standard deviation of observable X in the system state ψ

The above definitions of mean and standard deviation are defined formally in purely operator-theoretic terms. The statement becomes more meaningful however, once we note that these actually are the mean and standard deviation for the measured distribution of values. See Quantum statistical mechanics is the study of statistical ensembles of quantum mechanical systems. A statistical ensemble is described by a density operator is completely determined by its distribution D be an observable of a quantum mechanical system. defined by uniquely determines is a boolean homomorphism from the Borel subsets of... quantum statistical mechanics.

It may be evaluated not only for pairs of conjugate operators (e.g. those defining measurements of distance and of momentum, or of A duration is an amount of time or a particular time interval. For example, an event in the common sense has a duration greater than zero (but not very long), but in certain specialised senses, a duration of zero. It is often cited as one of the fundamental aspects of... duration and of Energy is a fundamental quantity that every physical system possesses; it allows us to predict how much work the system could be made to do, or how much heat it can exchange. In the past, energy was discussed in terms of easily observable effects it has on the properties of... energy) but generally for any pair of A number of mathematical entities are named Hermitian, after the mathematician Charles Hermite: Hermitian matrix Hermitian operator Hermitian adjoint Hermitian form Hermitian metric See also: self-adjoint Categories: Disambiguation ... Hermitian operators. There is also an uncertainty relation between the field strength and the number of particles which is responsible for the phenomenon of In the description of the interaction between elementary particles in quantum field theory, a virtual particle is a temporary elementary particle, used to describe an intermediate stage in the interaction. A virtual particle is never the end result of such a process. Using the language of Feynman diagrams, a virtual... virtual particles.

Note that it is possible to have two non-commuting self-adjoint operators A and B which share an In linear algebra, the eigenvectors (from the German -> , the eigenspace for the eigenvalue . Identifying eigenvectors For example, consider the matrix which represents a linear operator R3 -> R3. One can check that and therefore 2 is an eigenvalue of A and we have found a corresponding eigenvector. The characteristic... eigenvector ψ in this case ψ represents a pure state which is simultaneously measurable for A and B.

Generalizations

Other forms of the uncertainty principle can be formulated for the The Fourier transform, named for Jean Baptiste Joseph Fourier, is an integral transform that re-expresses a function in terms of sinusoidal basis functions, i.e. as a sum or integral of sinusoidal functions multiplied by some coefficients (amplitudes). There are many closely-related variations of this transform, summarized below... Fourier transform on general locally compact groups or for Fourier integral operators on manifolds. For example, Hirschman proved in 1957 a form of the uncertainty principle which is stronger than the Weyl form stated above.

Common observables which obey the uncertainty principle

The previous mathematical results suggest how to find uncertainty relations between physical observables. Specifically, locate pairs of observables A and B whose commutator has certain analytic properties.

• The most common one is the uncertainty relation between position and momentum of a particle in space:

• The uncertainty relation between two orthogonal components of the In physics, angular momentum intuitively measures how much the linear momentum is directed around a certain point called the origin; the moment of momentum. Since angular momentum depends upon the origin of choice, one must be careful when discussing angular momentum to specify the origin and not to combine angular... total angular momentum operator of a particle is as follows:

where i, j, k are distinct and J_i denotes angular momentum along the x_i axis.

• The following uncertainty relation between energy and time is often presented in physics textbooks, although its interpretation requires more care because there is no operator representing time:

Interpretations

Main article: In a nontechnical sense, an interpretation of quantum mechanics is an attempt to answer the question: interpretation, other than very minimal of measurement plays an apparently essential role in the theory. It relates the abstract elements of the theory, such as the wavefunction, to operationally definable values, such as probabilities... Interpretation of quantum mechanics

has become a byword for great intelligence or even genius. His face remains one of the most recognizable in the world. This popularity has also led to widespread use of Einstein in advertising and merchandising, eventually including the registration of Albert Einstein as a trademark. Biography Youth and college Young... Albert Einstein was not happy with the uncertainty principle, and he challenged Niels Bohr Niels Henrik David Bohr (October 7, 1885 – November 18, 1962) was a Danish physicist who made essential contributions to understanding atomic structure and quantum mechanics. Bohrs contributions to physics Bohrs model of atomic structure. The electrons orbital angular momentum is quantized; L=nħ... Niels Bohr with a famous In philosophy, physics, and other fields, a thought experiment (from the German Gedankenexperiment) is an attempt to solve a problem using the power of human imagination. These experiments are used to attempt to understand something about the universe. Thought experiments have been used to pose questions in philosophy at least... thought experiment (See the The Bohr-Einstein debates on foundational aspects on quantum mechanics happened during the Solvay conferences. They consisted of analyses of thought experiments. Put simply, they were an attempt by Einstein to explain away the aspects of Bohrs interpretation of Quantum Mechanics that he disliked. Bohr attempted (and, most scholars... Bohr-Einstein debates for more details): we fill a box with a radioactive material which randomly emits radiation. The box has a shutter, which is opened and immediately thereafter shut by a clock at a precise time, thereby allowing some radiation to escape. So the time is already known with precision. We still want to measure the conjugate variable energy precisely. Einstein proposed doing this by weighing the box before and after. The equivalence between mass and energy from Special relativity (SR) or the special theory of relativity is the physical theory published in 1905 by Albert Einstein. It replaced Newtonian notions of space and time, and incorporated electromagnetism as represented by Maxwells equations. The theory is called special because it is a special case of Einsteins... special relativity will allow you to determine precisely how much energy was left in the box. Bohr countered as follows: should energy leave, then the now lighter box will rise slightly on the scale. That changes the position of the clock. Thus the clock deviates from our stationary A frame of reference in physics is a set of axes which enable an observer to measure the aspect, position and motion of all points in a system relative to the reference frame. Two observers may choose to use different frames of reference to investigate a common system. In essence... reference frame, and again by special relativity, its measurement of time will be different from ours, leading to some unavoidable margin of error. In fact, a detailed analysis shows that the imprecision is correctly given by Heisenberg's relation.

Within the widely but not universally accepted The Copenhagen interpretation is an interpretation of quantum mechanics formulated by Niels Bohr and Werner Heisenberg while collaborating in Copenhagen around 1927. Bohr and Heisenberg extended the probabilistic interpretation of the wavefunction, proposed by Max Born. Their interpretation attempts to answer some perplexing questions which arise as a result of... Copenhagen interpretation quantum mechanics, the uncertainty principle is taken to mean that on an elementary level, the physical universe does not exist in a Determinism is the philosophical conception which claims that every physical event, including human cognition and action, is causally determined by an unbroken chain of prior occurrences. No mysterious miracles or totally random events occur. Philosophy of determinism The principal consequences of deterministic philosophy are that free will (except as defined... deterministic form—but rather as a collection of probabilities, or potentials. For example, the pattern (probability distribution) produced by millions of photons passing through a diffraction slit can be calculated using quantum mechanics, but the exact path of each photon cannot be predicted by any known method. The The Copenhagen interpretation is an interpretation of quantum mechanics formulated by Niels Bohr and Werner Heisenberg while collaborating in Copenhagen around 1927. Bohr and Heisenberg extended the probabilistic interpretation of the wavefunction, proposed by Max Born. Their interpretation attempts to answer some perplexing questions which arise as a result of... Copenhagen interpretation holds that it cannot be predicted by any method.

It is this interpretation that Einstein was questioning when he said "I cannot believe that God would choose to play dice with the universe." Bohr, who was one of the authors of the Copenhagen interpretation responded, "Einstein, don't tell God what to do."

Einstein was convinced that this interpretation was in error. His reasoning was that all previously known probability distributions arose from deterministic events. The distribution of a flipped coin or a rolled dice can be described with a probability distribution (50% heads, 50% tails). But this does not mean that their physical motions are unpredictable. Ordinary mechanics can be used to calculate exactly how each coin will land, if the forces acting on it are known. And the heads/tails distribution will still line up with the probability distribution (given random initial forces).

Einstein assumed that there are similar In physics, a hidden variable theory is urged by a minority of physicists who argue that the statistical nature of quantum mechanics implies that quantum mechanics is incomplete; it is really applicable only to ensembles of particles; new physical phenomena beyond quantum mechanics are needed to explain an individual event... hidden variables in quantum mechanics which underlie the observed probabilities.

Neither Einstein nor anyone since has been able to construct a satisfying hidden variable theory, and the In quantum mechanics, the EPR paradox is a thought experiment which demonstrates that the result of a measurement performed on one part of a quantum system can have an instantaneous effect on the result of a measurement performed on another part, regardless of the distance separating the two parts. This... Bell inequality illustrates some very thorny issues in trying to do so. Although the behavior of an individual particle is random, it is also correlated with the behavior of other particles. Therefore, if the uncertainty principle is the result of some deterministic process, it must be the case that particles at great distances instantly transmit information to each other to ensure that the correlations in behavior between particles occur.

Fiction

In Science fiction is a form of speculative fiction principally dealing with the impact of imagined science and technology, or both, upon society and persons as individuals. There are exceptions (or, at least, some unusual examples) to this general definition. Scope In defining the scope of the science fiction genre, we... science fiction, a device to circumvent the uncertainty principle is called a In the fictional Star Trek universe, the Heisenberg compensators are part of the Transporter system. In quantum physics, the Heisenberg uncertainty principle states that one cannot know both the position of a subatomic particle and its momentum to arbitrary precision. The more you know about one, the less you can... Heisenberg compensator.

In the animated series Futurama was an exhibit/ride at the 1939-40 New York Worlds Fair designed by Norman Bel Geddes that showed the world 30 years into the future, including automated highways and vast suburbs. The exhibit was sponsored by General Motors. An updated version, Futurama II, appeared at the 1964... Futurama, there is an uncertainty principle based joke in the episode "Luck of the Fryrish," rumoured to be the only joke in pop culture concerning this subject. Professor Farnsworth, watching a horse race that ends in a quantum finish (the (30th century - 31st century - 32nd century - more centuries) The 31st century (Gregorian Calendar) comprises the years 3001-3100. The TV show , the latest novel in Arthur C. Clarkes Space Odyssey series takes place in 3001. The next movie by director Mike Judge (Creator of the animated series King of... 31st century equivalent of a photo finish) exclaims "No fair, you changed the outcome by measuring it!"

See also

• Quantum Mechanical indeterminacy, or often just quantum indeterminacy refers to the same fundamental physics phenomenon as does the more frequently used Heisenberg uncertainty principle. Quantum uncertainty is usually described in terms of a particle with a definitely measured momentum for which there must be a fundamental limit to how precisely... Quantum indeterminacy

References

• G. Folland and A. Sitaram, The Uncertainty Principle: A Mathematical Survey, Journal of Fourier Analysis and Applications, 1997 pp 207-238.
• W. Heisenberg, Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Zeitschrift für Physik, 43 1927, pp 172-198. English translatation: J. A. Wheeler and H. Zurek, Quantum Theory and Measurement Princeton Univ. Press, 1983, pp. 62-84.
• I Hirschman, A Note on Entropy, American Journal of Mathematics, 1957. This paper formulates a general uncertainty principle using a measure of uncertainty based on entropy.
• J. John von Neumann in the 1940s. John von Neumann (Neumann János) (December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician who made important contributions in quantum physics, set theory, computer science, economics and virtually all mathematical fields. The oldest of three children, von Neumann was born... von Neumann, Mathematical Foundations of Quantum Mechanics, Princeon University Press, 1955. This book has been reissued in paprpack and is widely available
• R. Omnès, Understanding Quantum Mechanics, Princeton University Press, 1999.
• A. J. Stam, Information and Control, vol. 2, 1959, p. 101.
• H. Hermann Weyl (November 9, 1885 - December 8, 1955) was a German mathematician. Although much of his working life was spent in Zürich and then Princeton, he is closely identified with the University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski. His research has had... Weyl, The Theory of Groups and Quantum Mechanics, Dover Publications 1950. Was originally published in German in 1928.

External links

• aip.org: Quantum meachanics 1925-1927 - The uncertainty principle (http://www.aip.org/history/heisenberg/p08.htm)
• Eric Weisstein's World of Physics - Uncertainty principle (http://scienceworld.wolfram.com/physics/UncertaintyPrinciple.html)
• John Baez on the time-energy uncertainty relation (http://math.ucr.edu/home/baez/uncertainty.html)