The atoms in molecules model developed by Richard Bader was developed in order to effectively link the quantum mechanical picture of a molecule, as an electronic wavefunction, to chemically useful older models such as the theory of Lewis pairs and the valence bond model.

While the idea of shared electron pairs provides an effective qualitative picture of covalent bonding, quantum mechanics is needed to understand the nature of these bonds and predict the structures and properties of simple molecules.

In quantum electrodynamics, electromagnetic interactions between charged particles can be calculated using the method of Feynman diagrams, in which we picture messenger particles called virtual photons being exchanged between charged particles.

This quantum picture of the electromagnetic field ( which treats it as analogous to harmonic oscillators ) has proved very successful, giving rise to quantum electrodynamics, a quantum field theory describing the interaction of electromagnetic radiation with charged matter.

In the canonical quantum field theory the S-matrix is represented within the interaction picture by the perturbation series in the powers of the interaction Lagrangian,

Such computational chemistry methods have been used to create a quantum mechanical picture of helium electron binding which is accurate to within < 2 % of the correct value, in a few computational steps.

With the advent of quantum mechanics this picture was given more formal interpretation in the form of the free electron model and its further extension, the nearly free electron model.

* In the so-called Schrödinger picture of quantum mechanics, the dynamics is given as follows:

* The Heisenberg picture of quantum mechanics focuses on observables and instead of considering states as varying in time, it regards the states as fixed and the observables as changing.

In interacting quantum field theories, Haag's theorem states that the interaction picture does not exist.

The Heisenberg picture is the closest to classical Hamiltonian mechanics ( for example, the commutators appearing in the above equations directly translate into the classical Poisson brackets ); but this is already rather " high-browed ", and the Schrödinger picture is considered easiest to visualize and understand by most people, to judge from pedagogical accounts of quantum mechanics.

The Dirac picture is the one used in perturbation theory, and is specially associated to quantum field theory and many-body physics.

However, it leads to a fairly simple picture of the vacuum state and is not easily amenable to use in some quantum field theories, such as quantum chromodynamics which is known to have a complicated vacuum characterized by many different condensates.

Complementarity says there is no logical picture ( one obeying classical propositional logic ) that can simultaneously describe and be used to reason about all properties of a quantum system S. This is often phrased by saying that there are " complementary " propositions A and B that can each describe S, but not at the same time.

On the other hand, the consequences of LQG are radical, because they change in depth our understanding of the nature of space and time and provide a tentative but detailed physical and mathematical picture of quantum spacetime.

Proposals towards a theory of quantum gravity do away with this picture.

From a different perspective, if it is correct that the properties of a quantum black hole should correspond at a broad level more or less to a classical general-relativistic black hole, then it is believed that the appearance and effects of the Hawking radiation can be interpreted as quantum " corrections " to the classical picture, as Planck's constant is " tuned up " away from zero up to h. Outside the event horizon of an astronomical-sized black hole these corrections are tiny.

Actually, the picture of one photon being in-phase with another is not valid in quantum theory.

To find the wavefunction of the coherent state, it is easiest to employ the Heisenberg picture of the quantum harmonic oscillator for the coherent state.

Werner Heisenberg had been an assistant to Niels Bohr at his institute in Copenhagen during part of the 1920s, when they helped originate quantum mechanical theory.

In 1929, Heisenberg gave a series of invited lectures at the University of Chicago explaining the new field of quantum mechanics.

There must be some unknown mechanism acting on these variables to give rise to the observed effects of " non-commuting quantum observables ", i. e. the Heisenberg uncertainty principle.

* The Heisenberg group is a connected nilpotent Lie group of dimension 3, playing a key role in quantum mechanics.

While Maxwell's equations are consistent within special and general relativity, there are some quantum mechanical situations in which Maxwell's equations are significantly inaccurate: including extremely strong fields ( see Euler – Heisenberg Lagrangian ) and extremely short distances ( see vacuum polarization ).

In contrast to classical mechanics, where accurate measurements and predictions can be calculated about location and velocity, in the quantum mechanics of a subatomic particle, one can never specify its state, such as its simultaneous location and velocity, with complete certainty ( this is called the Heisenberg uncertainty principle ).

This limitation was first elucidated by Heisenberg through a thought experiment, and is represented mathematically in the new formalism by the non-commutativity of quantum observables.

To be more precise, already before Schrödinger, the young postdoctoral fellow Werner Heisenberg invented his matrix mechanics, which was the first correct quantum mechanics –– the essential breakthrough.

File: Bundesarchiv Bild183-R57262, Werner Heisenberg. jpg | Werner Heisenberg ( 1901-1976 ): developed method to express ideas of quantum mechanics in terms of matrices in 1925, published his famous uncertainty principle in 1927, awarded Nobel Prize in Physics in 1932

Early quantum theory was significantly reformulated in the mid-1920s by Werner Heisenberg, Erwin Schrodinger, Max Born, Wolfgang Pauli and their collaborators, and the Copenhagen interpretation of Niels Bohr became widely accepted.

The foundations of quantum mechanics were established during the first half of the 20th century by Max Planck, Niels Bohr, Werner Heisenberg, Louis de Broglie, Arthur Compton, Albert Einstein, Erwin Schrödinger, Max Born, John von Neumann, Paul Dirac, Enrico Fermi, Wolfgang Pauli, David Hilbert, Wilhelm Wien, Satyendra Nath Bose, Arnold Sommerfeld and others.

One of the oldest and most commonly used formulations is the " transformation theory " proposed by the late Cambridge theoretical physicist Paul Dirac, which unifies and generalizes the two earliest formulations of quantum mechanics-matrix mechanics ( invented by Werner Heisenberg ) and wave mechanics ( invented by Erwin Schrödinger ).

Especially since Werner Heisenberg was awarded the Nobel Prize in Physics in 1932 for the creation of quantum mechanics, the role of Max Born in the development of QM has become somewhat confused and overlooked.

In the following years, with contributions from Wolfgang Pauli, Eugene Wigner, Pascual Jordan, Werner Heisenberg and an elegant formulation of quantum electrodynamics due to Enrico Fermi, physicists came to believe that, in principle, it would be possible to perform any computation for any physical process involving photons and charged particles.

* Second, it is not clear how to determine the gravitational field of a particle, since under the Heisenberg uncertainty principle of quantum mechanics its location and velocity cannot be known with certainty.

Heisenberg offered such an observer effect at the quantum level ( see below ) as a physical " explanation " of quantum uncertainty.

In his work on formulating quantum mechanics, Werner Heisenberg postulated his uncertainty principle, which states:

It is well known from the theory of Fourier analysis, or from the Heisenberg uncertainty principle ( in the case of quantum mechanics ) that a narrow range of wavelengths is necessary to produce a localized wave packet, and the more localized the envelope, the larger the spread in required wavelengths.

While temporarily at the University of Copenhagen, German physicist Werner Heisenberg formulated his famous Uncertainty principle, and, with Max Born and Pascual Jordan, accomplished the first complete and correct definition of quantum mechanics, through the invention of Matrix mechanics.

The Soma cube is a solid dissection puzzle invented by Piet Hein in 1933 during a lecture on quantum mechanics conducted by Werner Heisenberg.

This molecular orbital theory represented a covalent bond as an orbital formed by combining the quantum mechanical Schrödinger atomic orbitals which had been hypothesized for electrons in single atoms.

Provided the theory is linear with respect to the wavefunction, the exact form of the quantum dynamics modelled, be it the non-relativistic Schrödinger equation, relativistic quantum field theory or some form of quantum gravity or string theory, does not alter the validity of MWI since MWI is a metatheory applicable to all linear quantum theories, and there is no experimental evidence for any non-linearity of the wavefunction in physics.

On the other hand the Bohm interpretation of quantum mechanics keeps counter-factual definiteness while introducing a conjectured non-local mechanism in form of the ' quantum potential ', defined as one of the terms of the Schrödinger equation.

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.

Within the framework of the approach a theory was proposed in which the physical vacuum is conjectured to be the quantum Bose liquid whose ground-state wavefunction is described by the logarithmic Schrödinger equation.

For example, as the only neutral atom with an analytic solution to the Schrödinger equation, the study of the energetics and bonding of the hydrogen atom played a key role in the development of quantum mechanics.

Schrödinger himself initially did not understand the fundamental probabilistic nature of quantum mechanics, as he thought that the absolute square of the wave function of an electron should be interpreted as the charge density of an object smeared out over an extended, possibly infinite, volume of space.

Finally, some of the originators of quantum theory ( notably Einstein and Schrödinger ) were unhappy with what they thought were the philosophical implications of quantum mechanics.

File: Schrodinger. jpg | Erwin Schrödinger ( 1887-1961 ): formulated the Schrödinger equation in 1926 describing how the quantum state of a physical system changes with time, awarded the Nobel Prize in Physics in 1933, two years later proposed the thought experiment known as Schrödinger's cat

File: Broglie Big. jpg | Louis de Broglie ( 1892-1987 ): researched quantum theory, discovered the wave nature of electrons, awarded the 1929 Nobel Prize in Physics, ideas on the wave-like behavior of particles used by Erwin Schrödinger in his formulation of wave mechanics.

File: Dirac 4. jpg | Paul Dirac ( 1902-1984 ): made fundamental contributions to the early development of quantum mechanics and quantum electrodynamics, formulated the Dirac equation describing the behavior of fermions, predicted the existence of antimatter, shared the1933 Nobel Prize in Physics with Erwin Schrödinger,

Some trajectories of a particle in a box according to Newton's laws of classical mechanics ( A ), and according to the Schrödinger equation of quantum mechanics ( B-F ).

This model is described by a quantum nonlinear Schrödinger equation.