dict.cc

⇄

⇄

Werbung

SYNO | Schrodinger equation | Schrodinger wave equation |

Anwendungsbeispiele Englisch

weitere Beispiele ...

- The simplest atomic orbitals are those that are calculated for systems with a single electron, such as the hydrogen atom. An atom of any other element ionized down to a single electron is very similar to hydrogen, and the orbitals take the same form. In the Schrödinger
**equation**for this system of one negative and one positive particle, the atomic orbitals are the eigenstates of the Hamiltonian operator for the energy. They can be obtained analytically, meaning that the resulting orbitals are products of a polynomial series, and exponential and trigonometric functions. (see hydrogen atom). - The complex number field is intrinsic to the mathematical formulations of quantum mechanics, where complex Hilbert spaces provide the context for one such formulation that is convenient and perhaps most standard. The original foundation formulas of quantum mechanics – the Schrödinger
**equation**and Heisenberg's matrix mechanics – make use of complex numbers. - In 1925, Werner Heisenberg published the first consistent mathematical formulation of quantum mechanics (matrix mechanics). One year earlier, Louis de Broglie had proposed the de Broglie hypothesis: that all particles behave like waves to some extent, and in 1926 Erwin Schrödinger used this idea to develop the Schrödinger
**equation**, a mathematical model of the atom (wave mechanics) that described the electrons as three-dimensional waveforms rather than point particles. - In modern context Bra and Ket notation can be compared to modern row and column vectors with complex components. Matrix multiplication rules apply with a result usually of more than one row and column. Vector inside and outside products are also following modern rules. Paul Dirac invented the notation Bra and Ket before the present notation of row and column vectors was developed. The complex components are useful in deriving wave functions such as solutions to the Schrödinger
**equation**and in making probability calculations for particle location or momentum in quantum mechanics. Bra and Ket notation is still used in describing quantum mechanics. - In some simplest cases, the state of condensed particles can be described with a nonlinear Schrödinger
**equation**, also known as Gross–Pitaevskii or Ginzburg–Landau equation. The validity of this approach is actually limited to the case of ultracold temperatures, which fits well for the most alkali atoms experiments. - Quantization of angular momentum was first postulated by Niels Bohr in his model of the atom and was later predicted by Erwin Schrödinger in his Schrödinger
**equation**. - Though it is possible to derive the Schrödinger
**equation**, which describes how a state vector evolves in time, most texts assert the equation as a postulate. Common derivations include using the DeBroglie hypothesis or path integrals. - In this approach, the physical vacuum is viewed as a quantum superfluid which is essentially non-relativistic, whereas Lorentz symmetry is not an exact symmetry of nature but rather the approximate description valid only for the small fluctuations of the superfluid background. Within the framework of the approach, a theory was proposed in which the physical vacuum is conjectured to be a quantum Bose liquid whose ground-state wavefunction is described by the logarithmic Schrödinger
**equation**. It was shown that the relativistic gravitational interaction arises as the small-amplitude collective excitation mode whereas relativistic elementary particles can be described by the particle-like modes in the limit of low momenta. The important fact is that at very high velocities the behavior of the particle-like modes becomes distinct from the relativistic one - they can reach the speed of light limit at finite energy; also, faster-than-light propagation is possible without requiring moving objects to have imaginary mass. - The Dyson series, the formal solution of an explicitly time-dependent Schrödinger
**equation**by iteration, and the corresponding Dyson time-ordering operator [...] an entity of basic importance in the mathematical formulation of quantum mechanics, are also named after Dyson. - Everett's proposal was not without precedent. In 1952, Erwin Schrödinger gave a lecture in Dublin in which at one point he jocularly warned his audience that what he was about to say might "seem lunatic". He went on to assert that while the Schrödinger
**equation**seemed to be describing several different histories, they were "not alternatives but all really happen simultaneously". According to David Deutsch, this is the earliest known reference to many-worlds; Jeffrey A. Barrett describes it as indicating the similarity of "general views" between Everett and Schrödinger. Schrödinger's writings from the period also contain elements resembling the modal interpretation originated by Bas van Fraassen. Because Schrödinger subscribed to a kind of post-Machian neutral monism, in which "matter" and "mind" are only different aspects or arrangements of the same common elements, treating the wavefunction as physical and treating it as information became interchangeable. - The programs used in computational chemistry are based on many different quantum-chemical methods that solve the molecular Schrödinger
**equation**associated with the molecular Hamiltonian. Methods that do not include any empirical or semi-empirical parameters in their equations – being derived directly from theoretical principles, with no inclusion of experimental data – are called "ab initio methods". This does not imply that the solution is an exact one; they are all approximate quantum mechanical calculations. It means that a particular approximation is rigorously defined on first principles (quantum theory) and then solved within an error margin that is qualitatively known beforehand. If numerical iterative methods must be used, the aim is to iterate until full machine accuracy is obtained (the best that is possible with a finite word length on the computer, and within the mathematical and/or physical approximations made). - Erwin Rudolf Josef Alexander Schrödinger ([...] , [...]; [...]; 12 August 1887 – 4 January 1961), sometimes written as [...] or [...] , was a Nobel Prize-winning Austrian and naturalized Irish physicist who developed a number of fundamental results in quantum theory: the Schrödinger
**equation**provides a way to calculate the wave function of a system and how it changes dynamically in time. - It is expected that the "eigenvalues", i.e., the energy [...] of the box should be the same regardless of its position in space, but [...] changes. Notice that [...] represents a phase shift in the wave function. This phase shift has no effect when solving the Schrödinger
**equation**, and therefore does not affect the "eigenvalue".

Werbung