The first attempt at a microscopic description of magnetism was by Wilhelm Lenz and Ernst Ising through the Ising model that described magnetic materials as consisting of a periodic lattice of quantum spins that collectively acquired magnetization.

It holds that quantum mechanics does not yield a description of an objective reality but deals only with probabilities of observing, or measuring, various aspects of energy quanta, entities which fit neither the classical idea of particles nor the classical idea of waves.

# The quantum mechanical description of large systems will closely approximate the classical description.

In 1927, the first mathematically complete quantum description of a simple chemical bond, i. e. that produced by one electron in the hydrogen molecular ion, H < sub > 2 </ sub >< sup >+</ sup >, was derived by the Danish physicist Oyvind Burrau.

In this approach the physical vacuum is viewed as the quantum superfluid which is essentially non-relativistic whereas the Lorentz symmetry is not an exact symmetry of nature but rather the approximate description valid only for the small fluctuations of the superfluid background.

The holographic principle is a property of quantum gravity and string theories which states that the description of a volume of space can be thought of as encoded on a boundary to the region — preferably a light-like boundary like a gravitational horizon.

The space-time in quantum gravity should emerge as an effective description of the theory of oscillations of a lower dimensional black-hole horizon.

The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics.

At the heart of the description are ideas of quantum state and quantum observable which are radically different from those used in previous models of physical reality.

The von Neumann description of quantum measurement of an observable A, when the system is prepared in a pure state ψ is the following ( note, however, that von Neumann's description dates back to the 1930s and is based on experiments as performed during that time – more specifically the Compton – Simon experiment ; it is not applicable to most present-day measurements within the quantum domain ):

These can be chosen appropriately in order to obtain a quantitative description of a quantum system.

The first complete quantum field theory, quantum electrodynamics, provides a fully quantum description of the electromagnetic interaction.

They are also used in the description of critical phenomena and quantum phase transitions, such as in the BCS theory of superconductivity.

Although it is often claimed that the photoelectric and Compton effects require a quantum description of the EM field, this is now understood to be untrue, and proper proof of the quantum nature of radiation is now taken up into modern quantum optics as in the antibunching effect.

It consists of 100 five-option multiple-choice questions covering subject areas including classical mechanics, electromagnetism, wave phenomena and optics, thermal physics, relativity, atomic and nuclear physics, quantum mechanics, laboratory techniques, and mathematical methods.

For example, one can talk about motion of a wave or a quantum particle ( or any other field ) where the configuration consists of probabilities of occupying specific positions.

For example, the stability of bulk matter ( which consists of atoms and molecules which would quickly collapse under electric forces alone ), the rigidity of solids, and the mechanical, thermal, chemical, optical and magnetic properties of matter are all results of the interaction of electric charges under the rules of quantum mechanics.

Then, in 1905, to explain the photoelectric effect ( 1839 ), i. e., that shining light on certain materials can function to eject electrons from the material, Albert Einstein postulated, based on Planck ’ s quantum hypothesis, that light itself consists of individual quantum particles, which later came to be called photons ( 1926 ).

Non-adiabatic dynamics consists of taking the interaction between several coupled potential energy surface ( corresponding to different electronic quantum states of the molecule ).

* Quarks, first posited in the 1960s, have quantized charge, but the charge is quantized into multiples of e. However, quarks cannot be seen as isolated particles ; they exist only in groupings, and stable groupings of quarks ( such as a proton, which consists of three quarks ) all have charges that are integer multiples of e. For this reason, either 1 e or e can be justifiably considered to be " the quantum of charge ", depending on the context.

A quantum solvation shell consists of a region of non-superfluid helium-4 atoms that surround the molecule ( s ) and exhibit adiabatic following around the centre of gravity of the solute.

The laser resonator consists of two distributed Bragg reflector ( DBR ) mirrors parallel to the wafer surface with an active region consisting of one or more quantum wells for the laser light generation in between.

# The observable is a Hermitian ( self-adjoint ) operator mapping a Hilbert space ( namely, the state space, which consists of all possible quantum states ) into itself.

Each Principal quantum number | n-level consists of n-1 Degenerate energy level | degenerate sublevels ; application of an electric field breaks the degeneracy.

A quantum circuit consists of simple quantum gates which act on at most a fixed number of qubits, usually 2 or 3.

It consists of a collection of particles, called superpartners, corresponding to operators in a quantum field theory which in superspace are represented by superfields.

In that book, which consists of several volumes, the Lorentz transformation was accepted as well as quantum theory.

The quantum algebra of a symplectic manifold consists of the operators of functions whose Hamiltonian vector fields satisfiy the condition.

Initially, de Broglie proposed a double solution approach, in which the quantum object consists of a physical wave ( u-wave ) in real space which has a spherical singular region that gives rise to particle-like behaviour ; in this initial form of his theory he did not have to postulate the existence of a quantum particle.

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.

A typical rotational spectrum consists of a series of peaks that correspond to transitions between levels with different values of the angular momentum quantum number ().

It was described by Tony Skyrme and consists of a quantum superposition of baryons and resonance states.

The simplest non-dissipative quantum circuit consists simply of an inductor and capacitor, with the metallic wires connecting them being superconducting.

In quantum mechanics and classical mechanics, a few-body system consists of a small number of well-defined structures or point particles.

* Wave functions and other quantum states can be represented as vectors in a complex Hilbert space.

( Technically, the quantum states are rays of vectors in the Hilbert space, as corresponds to the same state for any nonzero complex number c .)

* Measurements are associated with linear operators ( called observables ) on the Hilbert space of quantum states.

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.

His work was a key aspect of Hermann Weyl and John von Neumann's work on the mathematical equivalence of Werner Heisenberg's matrix mechanics and Erwin Schrödinger's wave equation and his namesake Hilbert space plays an important part in quantum theory.

The spin degree of freedom for an electron is associated with a two-dimensional complex Hilbert space H, with each quantum state corresponding to a vector in that space.

In contrast all finite-dimensional inner product spaces over or, such as those used in quantum computation, are automatically metrically complete and hence Hilbert spaces.

* Physical symmetries act on the Hilbert space of quantum states unitarily or antiunitarily due to Wigner's theorem ( supersymmetry is another matter entirely ).

A systematic understanding of its consequences has led to the phase space formulation of quantum mechanics, which works in full phase space instead of Hilbert space, so then with a more intuitive link to the classical limit thereof.

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

In more technical terms, they are described by quantum state vectors in a Hilbert space, which is also treated in quantum field theory.

In the language of quantum mechanics, hermitian matrices are observables, so the Pauli matrices span the space of observables of the 2-dimensional complex Hilbert space.

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.

In the mathematically rigorous formulation of quantum mechanics developed by Paul Dirac David Hilbert, and John von Neumann, the possible states of a quantum mechanical system are represented by unit vectors ( called " state vectors ").

In its most general formulation, quantum mechanics is a theory of abstract operators ( observables ) acting on an abstract state space ( Hilbert space ), where the observables represent physically observable quantities and the state space represents the possible states of the system under study.

As mentioned above, a state of a quantum system is given by a unit vector in a Hilbert space.

The general formulation of the Heisenberg uncertainty principle is derived using the Cauchy – Schwarz inequality in the Hilbert space of quantum observables.

In the standard quantum formalism, measuring observables is generally thought of as measuring operators on the Hilbert space.

This relationship between physical measurements and Hilbert space operators is, for standard quantum mechanics, an additional axiom of the theory.

Firstly, the accepted mathematical structure of quantum mechanics is based on fairly abstract mathematics, such as Hilbert spaces and operators on those spaces.

In quantum physics, a general model for this convex set is the set of density operators on a Hilbert space.