The configuration of these electrons follows from the principles of quantum mechanics.

The principles of quantum mechanics were used to successfully model the atom.

The study of these lines led to the Bohr atom model and to the birth of quantum mechanics.

With the development of quantum mechanics, it was found that the orbiting electrons around a nucleus could not be fully described as particles, but needed to be explained by the wave-particle duality.

Specifically, in quantum mechanics, the state of an atom, i. e. an eigenstate of the atomic Hamiltonian, is approximated by an expansion ( see configuration interaction expansion and basis set ) into linear combinations of anti-symmetrized products ( Slater determinants ) of one-electron functions.

Explaining the behavior of these electron " orbits " was one of the driving forces behind the development of quantum mechanics.

Still, the Bohr model's use of quantized angular momenta and therefore quantized energy levels was a significant step towards the understanding of electrons in atoms, and also a significant step towards the development of quantum mechanics in suggesting that quantized restraints must account for all discontinuous energy levels and spectra in atoms.

In the end, this was solved by the discovery of modern quantum mechanics and the Pauli Exclusion Principle.

In quantum mechanics, where all particle momenta are associated with waves, it is the formation of such a wave packet which localizes the wave, and thus the particle, in space.

In quantum mechanics, as a particle is localized to a smaller region in space, the associated compressed wave packet requires a larger and larger range of momenta, and thus larger kinetic energy.

The new quantum mechanics did not give exact results, but only the probabilities for the occurrence of a variety of possible such results.

In modern quantum mechanics however, n determines the mean distance of the electron from the nucleus ; all electrons with the same value of n lie at the same average distance.

Classically, it is forbidden to escape, but according to the ( then ) newly-discovered principles of quantum mechanics, it has a tiny ( but non-zero ) probability of " tunneling " through the barrier and appearing on the other side to escape the nucleus.

He discovered that the so-called Weil representation, previously introduced in quantum mechanics by Irving Segal and Shale, gave a contemporary framework for understanding the classical theory of quadratic forms.

Further investigation and theoretical work showed that the effect was a radiationless effect more than an internal conversion effect by use of elementary quantum mechanics and transition rate and transition probability calculations.

Angular momentum in quantum mechanics differs in many profound respects from angular momentum in classical mechanics.

The classical definition of angular momentum as can be carried over to quantum mechanics, by reinterpreting r as the quantum position operator and p as the quantum momentum operator.

In quantum mechanics, angular momentum is quantized – that is, it cannot vary continuously, but only in " quantum leaps " between certain allowed values.

All of this cosmic evolution after the inflationary epoch can be rigorously described and modeled by the ΛCDM model of cosmology, which uses the independent frameworks of quantum mechanics and Einstein's General Relativity.

In quantum mechanics, Bra-ket notation is a standard notation for describing quantum states, composed of angle brackets and vertical bars.

This concept is similar to quantum teleportation due to entanglement, although even that does not imply a possibility of faster-than-light communication.

* Canonical anticommutation relation, a concept in quantum field theory

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.

The commutator of two operators acting on a Hilbert space is a central concept in quantum mechanics, since it quantifies how well the two observables described by these operators can be measured simultaneously.

* Electron localization function, a concept in quantum mechanics

With the advent of quantum physics, some scientists believed the concept of matter had merely changed, while others believed the conventional position could no longer be maintained.

The photon concept has led to momentous advances in experimental and theoretical physics, such as lasers, Bose – Einstein condensation, quantum field theory, and the probabilistic interpretation of quantum mechanics.

Study of the photoelectric effect led to important steps in understanding the quantum nature of light and electrons and influenced the formation of the concept of wave – particle duality.

Fundamental to both Newtonian and to quantum theoretical mechanics is the concept of velocity.

Newer interpretations of quantum mechanics have been formulated that do away with the concept of " wavefunction collapse " ( see, for example, the relative state interpretation ).

One attempt to overcome these limitations is to replace ordinary quantum field theory, which is based on the classical concept of a point particle, with a quantum theory of one-dimensional extended objects: string theory.

It is important to recognise that entanglement is more commonly viewed as an algebraic concept, noted for being a precedent to non-locality as well as to quantum teleportation and to superdense coding, whereas non-locality is defined according to experimental statistics and is much more involved with the foundations and interpretations of quantum mechanics.

The STM is based on the concept of quantum tunneling.

Tunneling is a functioning concept that arises from quantum mechanics.

* On shell and off shell, a concept in quantum field theory

Sokal wrote " Transgressing the Boundaries: Towards a Transformative Hermeneutics of Quantum Gravity ", an article proposing that quantum gravity has progressive political implications, and that the " morphogenetic field " ( characterized by Sokal as " a bizarre New Age concept due to Rupert Sheldrake ") could be a cutting-edge theory of quantum gravity.

A central concept of quantum mechanics, this duality addresses the inability of classical concepts like " particle " and " wave " to fully describe the behavior of quantum-scale objects.

In quantum mechanics, there is a mathematically analogous concept of the quantum coherence length of a wave function.

There are quantum mechanics references as well ; phenomena of non-locality, as in the EPR paradox, make appearances, and the concept of Schrödinger's Cat plays a part in helping Dirk determine Richard's mental state, Richard producing clear and rational arguments for why the experiment proposed in the theory cannot be carried out in reality.

In this context a measurement operation is a transformation which turns a ket-vector into a probability distribution ( for a formalization of this concept see quantum operations ).

Under such conditions, a large fraction of the bosons occupy the lowest quantum state of the external potential, at which point quantum effects become apparent on a macroscopic scale.

Under such conditions, a large fraction of the bosons occupy the lowest quantum state, at which point quantum effects become apparent on a macroscopic scale.

Most quantitative calculations in modern quantum chemistry use either valence bond or molecular orbital theory as a starting point, although a third approach, Density Functional Theory, has become increasingly popular in recent years.

This experimental fact is highly reproducible, and the mathematics of quantum mechanics ( see below ) allows us to predict the exact probability of an electron striking the screen at any particular point.

In the EPR paper ( 1935 ) the authors realised that quantum mechanics was inconsistent with their assumptions, but Einstein nevertheless thought that quantum mechanics might simply be augmented by hidden variables ( i. e. variables which were, at that point, still obscure to him ), without any other change, to achieve an acceptable theory.

In particular, in the quantum theory a smeared supertube is present that cuts the spacetime in such a way that, although in the full spacetime a closed timelike curve passed through every point, no complete curves exist on the interior region bounded by the tube.

Since classical general relativity and quantum mechanics seem to be incompatible at such energies, from a theoretical point of view this situation is not tenable.

This is a direct effect of quantum mechanics: specifically, the zero point energy of the system is too high to allow freezing.

Such devices, produced by standard lithography techniques, may point the way to scalable quantum computing tools.

At this point, the quantum mechanical system has become a quantum field in the sense we described above.

String theory can be seen as a generalization of quantum field theory where instead of point particles, string-like objects propagate in a fixed spacetime background, although the interactions among closed strings give rise to space-time in a dynamical way.

One suggested starting point is ordinary quantum field theories which, after all, are successful in describing the other three basic fundamental forces in the context of the standard model of elementary particle physics.

In order to explain spontaneous transitions, quantum mechanics must be extended to a " second-quantized " theory, wherein the electromagnetic field is quantized at every point in space.

Fermat's principle is the main principle of quantum electrodynamics where it states that any particle ( e. g. a photon or an electron ) propagates over all available ( unobstructed ) paths and the interference ( sum, or superposition ) of its wavefunction over all those paths ( at the point of observer or detector ) gives the correct probability of detection of this particle ( at this point ).

As a general rule the engineering and quantum physics community use the first convention where the wave is observed from the point of view of the source.

It attempts to distinguish between the Copenhagen interpretation of quantum mechanics and the Everett many-worlds interpretation by means of a variation of the Schrödinger's cat thought experiment, from the cat's point of view.

Because an arbitrary potential can be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics.

According to Holland, failure to appreciate key points of the de Broglie – Bohm theory has led to confusion, the key point being " that the trajectories of a many-body quantum system are correlated not because the particles exert a direct force on one another ( à la Coulomb ) but because all are acted upon by an entity – mathematically described by the wavefunction or functions of it – that lies beyond them.