In the quantum picture of Heisenberg, Schrödinger and others, the Bohr atom number n for each orbital became known as an n-sphere in a three dimensional atom and was pictured as the mean energy of the probability cloud of the electron's wave packet which surrounded the atom.

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.

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.

The photon is the quantum of the electromagnetic interaction, and is the basic " unit " or constituent of all forms of EMR.

A quantum theory of the interaction between electromagnetic radiation and matter such as electrons is described by the theory of quantum electrodynamics.

At the quantum level, electromagnetic radiation is produced when the wavepacket of a charged particle oscillates or otherwise accelerates.

* Optics, the electromagnetic impedance of a medium, or the quantum efficiency of detectors.

From a classical perspective, the electromagnetic field can be regarded as a smooth, continuous field, propagated in a wavelike manner ; whereas from the perspective of quantum field theory, the field is seen as quantized, being composed of individual particles.

Although modern quantum optics tells us that there also is a semi-classical explanation of the photoelectric effect — the emission of electrons from metallic surfaces subjected to electromagnetic radiation — the photon was historically ( although not strictly necessarily ) used to explain certain observations.

A central feature in elementary particle theory is the early 20th century idea of " quanta ", which revolutionized the understanding of electromagnetic radiation and brought about quantum mechanics.

Proponents argue that it provides compact and intuitive descriptions in many areas including classical and quantum mechanics, electromagnetic theory and relativity.

* There are always vacuum fluctuations of the electromagnetic field, according to quantum mechanics.

An external electromagnetic field at a frequency associated with a transition can affect the quantum mechanical state of the atom.

Moreover, various phenomena occur in the world even though Maxwell's equations predicts them to be impossible, such as " nonclassical light " and quantum entanglement of electromagnetic fields ( see quantum optics ).

Just as he argues that there is no particular reason why particular macroscopic physical features in the brain should give rise to consciousness, he also thinks that there is no particular reason why a particular quantum feature, such as the electromagnetic field in the brain, should give rise to consciousness, either.

The ultimate culmination was the theory of quantum electrodynamics, which explains all optics and electromagnetic processes in general as being the result of the exchange of real and virtual photons.

Following the work of Paul Dirac in quantum field theory, George Sudarshan, Roy J. Glauber, and Leonard Mandel applied quantum theory to the electromagnetic field in the 1950s and 1960s to gain a more detailed understanding of photodetection and the statistics of light.

Light waves are now generally treated as electromagnetic waves except when quantum mechanical effects have to be considered.

A unified interpretation of antiparticles is now available in quantum field theory, which solves both these problems.

This is an example of renormalization in quantum field theory — the field theory being necessary because the number of particles changes from one to two and back again.

In quantum field theory, this process is allowed only as an intermediate quantum state for times short enough that the violation of energy conservation can be accommodated by the uncertainty principle.

These processes are important in the vacuum state and renormalization of a quantum field theory.

This section draws upon the ideas, language and notation of canonical quantization of a quantum field theory.

This technique is the most widespread method of computing amplitudes in quantum field theory today.

Practical applications are made impossible due to the no-cloning theorem, and the fact that quantum field theories preserve causality, so that quantum correlations cannot be used to transfer information.

The analogy was completed when Hawking, in 1974, showed that quantum field theory predicts that black holes should radiate like a black body with a temperature proportional to the surface gravity of the black hole.

* Canonical anticommutation relation, a concept in quantum field theory

The physical model behind cosmic inflation is extremely simple, however it has not yet been confirmed by particle physics, and there are difficult problems reconciling inflation and quantum field theory.

* An introduction including more on general relativity and quantum field theory than most.

After World War II, several ideas from quantum field theory were applied to condensed matter problems.

These ideas were unified by Kenneth Wilson in 1972, under the formalism of the renormalization group in the context of quantum field theory.

Theoretical models have also been developed to study the physics of phase transitions, such as the Landau-Ginzburg theory, Critical exponents and the use of mathematical techniques of quantum field theory and the renormalization group.

Goldstone's theorem in quantum field theory states that in a system with broken continuous symmetry, there may exist excitations with arbitrarily low energy, called the Goldstone bosons.

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

The problem of thinking in terms of classical measurements of a quantum system becomes particularly acute in the field of quantum cosmology, where the quantum system is the universe.

The general concept of a chemical reaction has been extended to non-chemical reactions between entities smaller than atoms, including nuclear reactions, radioactive decays, and reactions between elementary particles as described by quantum field theory.

In quantum field theory, the Casimir effect and the Casimir – Polder force are physical forces arising from a quantized field.