( quantum chemistry ) and spectroscopy.

Applied quantum chemistry, specifically, orbital hybridization best describes chemical bonding in nanotubes.

Fields of specialization include biochemistry, nuclear chemistry, organic chemistry, inorganic chemistry, polymer chemistry, analytical chemistry, theoretical chemistry, quantum chemistry, environmental chemistry, and physical chemistry.

Important areas of study include chemical thermodynamics, chemical kinetics, electrochemistry, quantum chemistry, statistical mechanics, and spectroscopy.

In particular the application of quantum mechanics to chemistry is called quantum chemistry.

* COLUMBUS, ab initio quantum chemistry software

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 book helped experimental chemists to understand the impact of quantum theory on chemistry.

Modern calculations in quantum chemistry usually start from ( but ultimately go far beyond ) a molecular orbital rather than a valence bond approach, not because of any intrinsic superiority in the former but rather because the MO approach is more readily adapted to numerical computations.

Building on the founding discoveries and theories in the history of quantum mechanics, the first theoretical calculations in chemistry were those of Walter Heitler and Fritz London in 1927.

The books that were influential in the early development of computational quantum chemistry include Linus Pauling and E. Bright Wilson's 1935 Introduction to Quantum Mechanics – with Applications to Chemistry, Eyring, Walter and Kimball's 1944 Quantum Chemistry, Heitler's 1945 Elementary Wave Mechanics – with Applications to Quantum Chemistry, and later Coulson's 1952 textbook Valence, each of which served as primary references for chemists in the decades to follow.

A notable exception are certain approaches called direct quantum chemistry, which treat electrons and nuclei on a common footing.

These methods are called quantum chemistry composite methods.

Semi-empirical quantum chemistry methods are based on the Hartree – Fock formalism, but make many approximations and obtain some parameters from empirical data.

If a particle and antiparticle are in the appropriate quantum states, then they can annihilate each other and produce other particles.

Lemaître, however, thought thatIf the world has begun with a single quantum, the notions of space and time would altogether fail to have any meaning at the beginning ; they would only begin to have a sensible meaning when the original quantum had been divided into a sufficient number of quanta.

Cold atoms in optical lattices are used as " quantum simulators ", that is, they act as controllable systems that can model behavior of more complicated systems, such as frustrated magnets.

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.

: Copenhagenists claim that interpretations of quantum mechanics where the wave function is regarded as real have problems with EPR-type effects, since they imply that the laws of physics allow for influences to propagate at speeds greater than the speed of light.

This does not imply that the solution is an exact one ; they are all approximate quantum mechanical calculations.

Ever since the origination of quantum mechanics, some theorists have searched for ways to incorporate additional determinants or " hidden variables " that, were they to become known, would account for the location of each individual impact with the target.

Albert Einstein and his colleagues Boris Podolsky and Nathan Rosen ( known collectively as EPR ) designed a thought experiment intended to reveal what they believed to be inadequacies of quantum mechanics.

To that end they pointed to a consequence of quantum mechanics that its supporters had not noticed.

The conclusion they drew was that quantum mechanics is not a complete theory.

However, quantum field theories that are " local " in this sense appear to violate the principle of locality as defined by EPR, but they nevertheless do not violate locality in a more general sense.

In quantum computation, entangled quantum states are used to perform computations in parallel, which may allow certain calculations to be performed much more quickly than they ever could be with classical computers.

While the diagrams are applied primarily to quantum field theory, they can also be used in other fields, such as solid-state theory.

String theories are quantum theories of gravity in the sense that they reduce to classical general relativity plus field theory at low energies, but are fully quantum mechanical, contain a graviton, and are believed to be mathematically consistent.

In technical terms, they are vector gauge bosons that mediate strong interactions of quarks in quantum chromodynamics ( QCD ).

Modifying the integer-charged quark model of Han and Nambu, Fritzsch and Gell-Mann were the first to write down the modern accepted theory of quantum chromodynamics, although they did not anticipate asymptotic freedom.

In 1972 he and Harald Fritzsch introduced the conserved quantum number " color charge ", and later along with Heinrich Leutwyler, they introduced quantum chromodynamics ( QCD ) as the gauge theory of the strong interaction ( cf.

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.

In Level III they live on another quantum branch in infinite-dimensional Hilbert space.

This is a very important fact in nuclear physics: Protons and neutrons in an atomic nucleus cannot all be in the same quantum state, but instead they spread out into nuclear shells analogous to electron shells in chemistry.

* Induced gravity ( 1967 ), a proposal by Andrei Sakharov according to which general relativity might arise from quantum field theories of matter

The probabilities discovered in quantum mechanics do nevertheless arise from measurement ( of the perceived path of the particle ).

Such discrete values arise from a quantum mechanical restraint on the number of electrons that can travel through the wire at the nanometer scale.

Another accurate method for measuring the elementary charge is by inferring it from measurements of two effects in quantum mechanics: The Josephson effect, voltage oscillations that arise in certain superconducting structures ; and the quantum Hall effect, a quantum effect of electrons at low temperatures, strong magnetic fields, and confinement into two dimensions.

In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence that arise in probability, such as the Edgeworth series ; in combinatorics, as an example of an Appell sequence, obeying the umbral calculus ; in numerical analysis as Gaussian quadrature ; in finite element methods as Shape Functions for beams ; and in physics, where they give rise to the eigenstates of the quantum harmonic oscillator.

The quantum linear harmonic oscillator and hence, the coherent states, arise in the quantum theory of a wide range of physical systems.

Functional integrals arise in probability, in the study of partial differential equations and in Feynman's approach to the quantum mechanics of particles and fields.

If quantum mechanics does not demonstrate an exponential sensitivity to initial conditions, how can exponential sensitivity to initial conditions arise in classical chaos, which must be the correspondence principle limit of quantum mechanics?

In Drinfeld's approach, quantum groups arise as Hopf algebras depending on an auxiliary parameter q or h, which become universal enveloping algebras of a certain Lie algebra, frequently semisimple or affine, when q

He has recently coauthored a series of papers on the Ponzano-Regge model, a spin foam model of 3-dimensional Riemannian quantum gravity, in which he explains how point particles naturally arise in this theory.

Another generalization defines a complex Hadamard matrix to be a matrix in which the entries are complex numbers of unit modulus and which satisfies H H < sup >*</ sup >= n I < sub > n </ sub > where H < sup >*</ sup > is the conjugate transpose of H. Complex Hadamard matrices arise in the study of operator algebras and the theory of quantum computation.

* Confinement in quantum dots can also arise from electrostatic potentials ( generated by external electrodes, doping, strain, or impurities ).

The mixing between photons and mirror photons could be present in tree level Feynman diagrams or arise as a consequence of quantum corrections due to the presence of particles that carry both ordinary and mirror charges.

* Composite fermions arise in a two-dimensional system subject to a large magnetic field, most famously those systems that exhibit the fractional quantum Hall effect.

Anomalies in quantum theory arise from one-loop diagrams, with a chiral fermion in the loop and gauge fields, Ricci tensors, or global symmetry currents as the external legs.

The Laguerre polynomials arise in quantum mechanics, in the radial part of the solution

In physics, the Clebsch – Gordan coefficients are sets of numbers that arise in angular momentum coupling under the laws of quantum mechanics.

For system s with two-Elementary particle | particle interaction s, the above Feynman diagram s arise at first order in the Perturbation theory ( quantum mechanics ) | perturbation expansion of both Z and E. The perturbation expansion for Z consists of all diagrams which are closed, while the perturbation expansion for E consists of all diagrams which are both closed and connected.

In Mathematics and Physics the CCR and CAR algebras arise from the study of canonical commutation relations in Bosonic and Fermionic quantum mechanics.

The initial conditions for the universe are thought to arise from the scale invariant quantum mechanical fluctuations of cosmic inflation.