# Category:Werner Heisenberg

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## Pages in category "Werner Heisenberg"

The following 14 pages are in this category, out of 14 total. This list may not reflect recent changes (learn more).

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## Pages in category "Werner Heisenberg"

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Wikimedia Commons has media related to .Werner Heisenberg |

The following 14 pages are in this category, out of 14 total. This list may not reflect recent changes (learn more).

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1. Werner Heisenberg – Werner Karl Heisenberg was a German theoretical physicist and one of the key pioneers of quantum mechanics. He published his work in 1925 in a breakthrough paper, in the subsequent series of papers with Max Born and Pascual Jordan, during the same year, this matrix formulation of quantum mechanics was substantially elaborated. In 1927 he published his uncertainty principle, upon which he built his philosophy, Heisenberg was awarded the Nobel Prize in Physics for 1932 for the creation of quantum mechanics. He was a principal scientist in the Nazi German nuclear weapon project during World War II and he travelled to occupied Copenhagen where he met and discussed the German project with Niels Bohr. Following World War II, he was appointed director of the Kaiser Wilhelm Institute for Physics and he was director of the institute until it was moved to Munich in 1958, when it was expanded and renamed the Max Planck Institute for Physics and Astrophysics. He studied physics and mathematics from 1920 to 1923 at the Ludwig-Maximilians-Universität München, at Munich, he studied under Arnold Sommerfeld and Wilhelm Wien. At Göttingen, he studied physics with Max Born and James Franck and he received his doctorate in 1923, at Munich under Sommerfeld. He completed his Habilitation in 1924, at Göttingen under Born, at the event, Bohr was a guest lecturer and gave a series of comprehensive lectures on quantum atomic physics. There, Heisenberg met Bohr for the first time, and it had a significant, Heisenbergs doctoral thesis, the topic of which was suggested by Sommerfeld, was on turbulence, the thesis discussed both the stability of laminar flow and the nature of turbulent flow. The problem of stability was investigated by the use of the Orr–Sommerfeld equation and he briefly returned to this topic after World War II. Heisenbergs paper on the anomalous Zeeman effect was accepted as his Habilitationsschrift under Max Born at Göttingen, in his youth he was a member and Scoutleader of the Neupfadfinder, a German Scout association and part of the German Youth Movement. In August 1923 Robert Honsell and Heisenberg organized a trip to Finland with a Scout group of this association from Munich, Heisenberg arrived at Munich in 1919 as a member of Freikorps to fight the Bavarian Soviet Republic established a year earlier. Five decades later he recalled those days as youthful fun, like playing cops and robbers and so on, from 1924 to 1927, Heisenberg was a Privatdozent at Göttingen. His seminal paper, Über quantentheoretischer Umdeutung was published in September 1925 and he returned to Göttingen and with Max Born and Pascual Jordan, over a period of about six months, developed the matrix mechanics formulation of quantum mechanics. On 1 May 1926, Heisenberg began his appointment as a university lecturer and it was in Copenhagen, in 1927, that Heisenberg developed his uncertainty principle, while working on the mathematical foundations of quantum mechanics. On 23 February, Heisenberg wrote a letter to fellow physicist Wolfgang Pauli, in his paper on the uncertainty principle, Heisenberg used the word Ungenauigkeit. In 1927, Heisenberg was appointed ordentlicher Professor of theoretical physics and head of the department of physics at the Universität Leipzig, in his first paper published from Leipzig, Heisenberg used the Pauli exclusion principle to solve the mystery of ferromagnetism. Slater, Edward Teller, John Hasbrouck van Vleck, Victor Frederick Weisskopf, Carl Friedrich von Weizsäcker, Gregor Wentzel, in early 1929, Heisenberg and Pauli submitted the first of two papers laying the foundation for relativistic quantum field theory

2. Uncertainty principle – The formal inequality relating the standard deviation of position σx and the standard deviation of momentum σp was derived by Earle Hesse Kennard later that year and by Hermann Weyl in 1928. Heisenberg offered such an effect at the quantum level as a physical explanation of quantum uncertainty. Thus, the uncertainty principle actually states a fundamental property of quantum systems, since the uncertainty principle is such a basic result in quantum mechanics, typical experiments in quantum mechanics routinely observe aspects of it. Certain experiments, however, may deliberately test a particular form of the uncertainty principle as part of their research program. These include, for example, tests of number–phase uncertainty relations in superconducting or quantum optics systems, applications dependent on the uncertainty principle for their operation include extremely low-noise technology such as that required in gravitational wave interferometers. The uncertainty principle is not readily apparent on the scales of everyday experience. So it is helpful to demonstrate how it applies to more easily understood physical situations, two alternative frameworks for quantum physics offer different explanations for the uncertainty principle. The wave mechanics picture of the uncertainty principle is more visually intuitive, a nonzero function and its Fourier transform cannot both be sharply localized. In matrix mechanics, the formulation of quantum mechanics, any pair of non-commuting self-adjoint operators representing observables are subject to similar uncertainty limits. An eigenstate of an observable represents the state of the wavefunction for a certain measurement value, for example, if a measurement of an observable A is performed, then the system is in a particular eigenstate Ψ of that observable. According to the de Broglie hypothesis, every object in the universe is a wave, the position of the particle is described by a wave function Ψ. The time-independent wave function of a plane wave of wavenumber k0 or momentum p0 is ψ ∝ e i k 0 x = e i p 0 x / ℏ. In the case of the plane wave, | ψ |2 is a uniform distribution. In other words, the position is extremely uncertain in the sense that it could be essentially anywhere along the wave packet. The figures to the right show how with the addition of many plane waves, in mathematical terms, we say that ϕ is the Fourier transform of ψ and that x and p are conjugate variables. Adding together all of these plane waves comes at a cost, namely the momentum has become less precise, One way to quantify the precision of the position and momentum is the standard deviation σ. Since | ψ |2 is a probability density function for position, the precision of the position is improved, i. e. reduced σx, by using many plane waves, thereby weakening the precision of the momentum, i. e. increased σp. Another way of stating this is that σx and σp have a relationship or are at least bounded from below

3. Heisenberg group – In mathematics, the Heisenberg group, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form under the operation of matrix multiplication. Elements a, b and c can be taken from any commutative ring with identity, the continuous Heisenberg group arises in the description of one-dimensional quantum mechanical systems. More generally, one can consider Heisenberg groups associated to n-dimensional systems, in the three-dimensional case, the product of two Heisenberg matrices is given by, =. The neutral element of the Heisenberg group is the identity matrix and it is a subgroup of 2-dimensional affine group A f f. corresponds to the affine transform x +. There are several prominent examples of the three-dimensional case, if a, b, c, are real numbers then one has the continuous Heisenberg group H3. It is a nilpotent real Lie group of dimension 3, in addition to the representation as real 3x3 matrices, the continuous Heisenberg group also has several different representations in terms of function spaces. By Stone–von Neumann theorem, there is, up to isomorphism and this representation has several important realizations, or models. In the Schrödinger model, the Heisenberg group acts on the space of integrable functions. In the theta representation, it acts on the space of functions on the upper half-plane. If a, b, c, are integers then one has the discrete Heisenberg group H3 and it is a non-abelian nilpotent group. It has two generators, x =, y = and relations z = x y x −1 y −1, x z = z x, y z = z y, by Basss theorem, it has a polynomial growth rate of order 4. One can generate any element through = y b z c x a, if one takes a, b, c in Z/p Z for an odd prime p, then one has the Heisenberg group modulo p. Analogues of Heisenberg groups over fields of odd prime order p are called extra special groups, or more properly. More generally, if the subgroup of a group G is contained in the center Z of G. If G is extra special but does not have exponent p, the Heisenberg group modulo 2 is of order 8 and is isomorphic to the dihedral group D4. Observe that if x =, y =, the elements x and y correspond to reflections, whereas xy and yx correspond to rotations by 90°. The other reflections are xyx and yxy, and rotation by 180° is xyxy, more general Heisenberg groups Hn may be defined for higher dimensions in Euclidean space, and more generally on symplectic vector spaces. The simplest general case is the real Heisenberg group of dimension 2n+1 and this is indeed a group, as is shown by the multiplication, ⋅ = and ⋅ =

4. Heisenberg's microscope – In particular, it provided an argument for the uncertainty principle on the basis of the principles of classical optics. Recent theoretical and experimental developments have argued that Heisenbergs intuitive explanation of his mathematical result are misleading, while the act of measurement does lead to uncertainty, the loss of precision is less than that predicted by Heisenbergs argument when measured at the level of an individual state. Heisenberg begins by supposing that an electron is like a particle, moving in the x direction along a line below the microscope. Let the cone of light leaving the microscope lens and focusing on the electron make an angle ε with the electron. Let λ be the wavelength of the light rays, then, according to the laws of classical optics, the microscope can only resolve the position of the electron up to an accuracy of Δ x = λ sin ε. When an observer perceives an image of the particle, its because the light strike the particle. However, we know from experimental evidence that when a photon strikes an electron, the latter has a Compton recoil with momentum proportional to h / λ and it is at this point that Heisenberg introduces objective indeterminacy into the thought experiment. He writes that the recoil cannot be known, since the direction of the scattered photon is undetermined within the bundle of rays entering the microscope. In particular, the momentum in the x direction is only determined up to Δ p x ≈ h λ sin ε. Combining the relations for Δ x and Δ p x, we thus have that Δ x Δ p x ≈ = h, ¹ Heisenberg, excerpt giving his own description of this thought experiment in The World of Mathematics, II, p.1052. Amir D. Aezel, Entanglement, pp. 77–79, niels Bohr, Nature,121, p.580,1928. Werner Heisenberg, Physics and Philosophy, pp. 46ff, albert Messiah, Quantum Mechanics, I, p. 143f James R. Newman, ed. The World of Mathematics, II, pp