• mechanical
  • The shortest summary of quantum mechanical reality I know of would be this: The dimensionality of the wave packet is not always negligible, although it often is. (principiadiscordia.com)
  • Sommerfeld's model was much closer to the modern quantum mechanical picture than Bohr's. (wikipedia.org)
  • In 1926 Erwin Schrödinger found a completely quantum mechanical wave-equation, which reproduced all the successes of the old quantum theory without ambiguities and inconsistencies. (wikipedia.org)
  • See also list of functional analysis topics, list of Lie group topics, list of quantum-mechanical systems with analytical solutions. (wikipedia.org)
  • Dirac supplied a systematic procedure for transferring the characteristic quantum phenomenon of discreteness of physical quantities from the quantum-mechanical treatment of particles to a corresponding treatment of fields. (wikipedia.org)
  • The methods of QFT could be applied to derive equations resulting from the quantum-mechanical (field-like) treatment of particles, e.g. the Dirac equation, the Klein-Gordon equation and the Maxwell equations. (wikipedia.org)
  • For example, by adding a perturbative electric potential to the quantum mechanical model of the hydrogen atom, tiny shifts in the spectral lines of hydrogen caused by the presence of an electric field (the Stark effect) can be calculated. (wikipedia.org)
  • spacetime
  • The action (which determines the path integral) is S = ∫ M B F {\displaystyle S=\int _{M}BF} The spacetime metric does not appear anywhere in the theory, so the theory is explicitly topologically invariant. (wikipedia.org)
  • One discovery of the theory, that can be related in non-technical terms, is that the dimension d of the spacetime involved is crucial. (wikipedia.org)
  • special relativity
  • Pascual Jordan and Wolfgang Pauli showed in 1928 that quantum fields could be made to behave in the way predicted by special relativity during coordinate transformations (specifically, they showed that the field commutators were Lorentz invariant). (wikipedia.org)
  • photons
  • Bose and Einstein gave the correct quantum statistics for photons. (wikipedia.org)
  • Since photons have rest mass zero, and correspondingly travel in the vacuum at the speed c, a non-relativistic theory such as ordinary QM cannot give even an approximate description. (wikipedia.org)
  • The idea was that not only the quanta of the electromagnetic field, photons, but also other particles might emerge and disappear as a result of their interaction with other particles. (wikipedia.org)
  • Dirac
  • Paul Dirac later proved in 1926 that both methods can be obtained from a more general method called transformation theory. (wikipedia.org)
  • This work was performed first by Dirac himself with the invention of hole theory in 1930 and by Wendell Furry, Robert Oppenheimer, Vladimir Fock, and others. (wikipedia.org)
  • procedure
  • The main tool of the old quantum theory was the Bohr-Sommerfeld quantization condition, a procedure for selecting out certain states of a classical system as allowed states: the system can then only exist in one of the allowed states and not in any other state. (wikipedia.org)
  • superposition
  • Since we do not know, the cat is both dead and alive, according to quantum law - in a superposition of states. (techtarget.com)
  • It differs from classical game theory in three primary ways: Superposed initial states, Quantum entanglement of initial states, Superposition of strategies to be used on the initial states. (wikipedia.org)
  • In the quantum version of the game, the bit is replaced by the qubit, which is a quantum superposition of two or more base states. (wikipedia.org)
  • entanglement
  • If you've done any reading on quantum theory -- particularly the double-slit experiment and entanglement -- you've probably realized the same thing as Albert "Spooky" Einstein: That sh*t is craaazy. (comicsalliance.com)
  • gauge
  • An alternative approach using lattice approximations is covered in (Wick rotated) lattice gauge theory. (wikipedia.org)
  • It has been generalized to theories with gauge invariance and was a central tool in the study of a conjectured deconfining phase transition of Yang-Mills theory. (wikipedia.org)
  • invariance
  • In the next few years Arnold Sommerfeld extended the quantum rule to arbitrary integrable systems making use of the principle of adiabatic invariance of the quantum numbers introduced by Lorentz and Einstein. (wikipedia.org)
  • Relativistic invariance can however be retained in the sense of twisted Poincaré invariance of the theory. (wikipedia.org)
  • oscillators
  • The ideas of QM were thus extended to systems having an infinite number of degrees of freedom, so an infinite array of quantum oscillators. (wikipedia.org)
  • Planck
  • In 1900, physicist Max Planck presented his quantum theory to the German Physical Society. (techtarget.com)
  • Planck wrote a mathematical equation involving a figure to represent these individual units of energy, which he called quanta . (techtarget.com)
  • Planck assumed there was a theory yet to emerge from the discovery of quanta, but, in fact, their very existence implied a completely new and fundamental understanding of the laws of nature. (techtarget.com)
  • In 1900, Planck made the assumption that energy was made of individual units, or quanta. (techtarget.com)
  • The old quantum theory was sparked by the 1900 work of Max Planck on the emission and absorption of light, and began in earnest after the work of Albert Einstein on the specific heats of solids. (wikipedia.org)
  • Einstein's
  • These are but a few of the extraordinary consequences of Einstein's theory of relativity. (audible.com)
  • This theory became known as the uncertainty principle , which prompted Albert Einstein's famous comment, 'God does not play dice. (techtarget.com)
  • It was evident from the beginning that a proper quantum treatment of the electromagnetic field had to somehow incorporate Einstein's relativity theory, which had grown out of the study of classical electromagnetism. (wikipedia.org)
  • multiplayer
  • In addition to single player campaign, Quantum Theory also supports four modes of online multiplayer, complete with voice chat for real-time communication. (worthplaying.com)
  • Introducing quantum information into multiplayer games allows a new type of "equilibrium strategy" which is not found in traditional games. (wikipedia.org)
  • coordinates
  • In order for the old quantum condition to make sense, the classical motion must be separable, meaning that there are separate coordinates q i {\displaystyle q_{i}} in terms of which the motion is periodic. (wikipedia.org)
  • quantization
  • The final chapters introduce readers who are familiar with the theory of manifolds to more advanced topics, including geometric quantization. (springer.com)
  • Sommerfeld made a crucial contribution by quantizing the z-component of the angular momentum, which in the old quantum era was called space quantization (Richtungsquantelung). (wikipedia.org)
  • endpoints
  • They argued in the context of string theory that the coordinate functions of the endpoints of open strings constrained to a D-brane in the presence of a constant Neveu-Schwarz B-field-equivalent to a constant magnetic field on the brane-would satisfy the noncommutative algebra set out above. (wikipedia.org)
  • displaystyle
  • The quantum numbers n i {\displaystyle n_{i}} are integers and the integral is taken over one period of the motion at constant energy (as described by the Hamiltonian). (wikipedia.org)
  • fields
  • The existence theorems for quantum fields can be expected to be very difficult to find, if indeed they are possible at all. (wikipedia.org)