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Model in Quantum Physics
In quantum mechanics, the particle in a one-dimensional lattice is a problem that occurs in the model of a periodic crystal lattice. The potential is
Particle in a one-dimensional lattice
Particle_in_a_one-dimensional_lattice
Theoretical electronic band structure model in which the potential is periodic and weak
what their signs are or how limited their sizes are. For a particle in a one-dimensional lattice, like the Kronig–Penney model, it is possible to calculate
Empty_lattice_approximation
Theory of quantum gauge fields on a lattice
In physics, lattice gauge theory is the study of gauge theories on a spacetime that has been discretized into a lattice. Gauge theories are important
Lattice_gauge_theory
Quasiparticle of mechanical vibrations
phonons are vibrations in a lattice of atoms. Phonons have both wave and particle-like properties, in a way related to the wave–particle duality of quantum
Phonon
Mathematical model in quantum mechanics
form of the particle in a box model considers a one-dimensional system. Here, the particle may only move backwards and forwards along a straight line
Particle_in_a_box
Fundamental theorem in condensed matter physics
worked out in a specific situation, see the article Particle in a one-dimensional lattice (periodic potential). Bloch's theorem—For electrons in a perfect
Bloch's_theorem
Type of cellular automaton
model is a two-dimensional model of fluid particle interactions. In this model, the lattice is square, and the particles travel independently at a unit speed
Lattice_gas_automaton
Mathematical model of ferromagnetism in statistical mechanics
The one-dimensional Ising model was solved by Ising (1925) alone in his 1924 thesis; it has no phase transition. The two-dimensional square-lattice Ising
Ising_model
Threshold of percolation theory models
thresholds are only known for certain two-dimensional lattices that can be broken up into a self-dual array, such that under a triangle-triangle transformation
Percolation_threshold
a ring particle in a spherically symmetric potential quantum harmonic oscillator hydrogen atom ring wave guide particle in a one-dimensional lattice (periodic
List of mathematical topics in quantum theory
List_of_mathematical_topics_in_quantum_theory
Toy model for electronic localization
developed by Serge Aubry and Gilles André in 1980. The Aubry–André model describes a one-dimensional lattice with hopping between nearest-neighbor sites
Aubry–André_model
Class of computational fluid dynamics methods
consisting of fictive particles, and such particles perform consecutive propagation and collision processes over a discrete lattice. Due to its particulate
Lattice_Boltzmann_methods
Quantum mechanical model
group. The notation of a harmonic oscillator can be extended to a one-dimensional lattice of many particles. Consider a one-dimensional quantum mechanical
Quantum_harmonic_oscillator
Theory of subatomic structure
In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called
String_theory
In statistical mechanics, the hard hexagon model is a 2-dimensional lattice model of a gas, where particles are allowed to be on the vertices of a triangular
Hard_hexagon_model
List of particles in matter including fermions and bosons
This is a list of known and hypothesized molecular, atomic, and subatomic particles in particle physics, condensed matter physics and cosmology. Elementary
List_of_particles
Ordered arrangement of atoms, ions, or molecules in a crystalline material
particles to form symmetric patterns that repeat along the principal directions of three-dimensional space in matter. The smallest group of particles
Crystal_structure
Process in materials science
higher dimensions is not known. For k {\displaystyle k} -mers on a one-dimensional lattice, we have for the fraction of vertices covered, θ k = k ∫ 0 ∞ exp
Random_sequential_adsorption
Model of an energy potential in quantum mechanics
oscillator Hydrogen atom or hydrogen-like atom Ring wave guide Particle in a one-dimensional lattice (periodic potential) Hydrogen molecular ion Holstein–Herring
Delta_potential
Concept in molecular modelling
a common form of PBC particle bookkeeping in which each particle in the simulation interacts with the closest image of the remaining particles. One example
Periodic_boundary_conditions
Model for topological superconductors in physics
Alexei Kitaev in 2000. The tight binding Hamiltonian of a Kitaev chain considers a one dimensional lattice with N site and spinless particles at zero temperature
Kitaev_chain
Pictorial representation of the behavior of subatomic particles
calculation of probability amplitudes in theoretical particle physics requires the use of large, complicated integrals over a large number of variables. Feynman
Feynman_diagram
Process of particles clustering together
particles that are unrestricted by a lattice, however computer simulation of DLA on a lattice will change the fractal dimension slightly for a DLA in
Diffusion-limited_aggregation
Analysis of the dimensions of different physical quantities
In engineering and science, dimensional analysis of different physical quantities is the analysis of their physical dimension or quantity dimension, defined
Dimensional_analysis
Simple model for one-dimensional crystal in solid-state physics
lattice, introduced by Morikazu Toda (1967), is a simple model for a one-dimensional crystal in solid state physics. It is famous because it is one of
Toda_lattice
Description of a quantum-mechanical system
momentum in classical mechanics. The quantum expectation values satisfy the Ehrenfest theorem. For a one-dimensional quantum particle moving in a potential
Schrödinger_equation
Distortion of the periodic lattice of a one-dimensional crystal
A Peierls distortion, named after Rudolf Peierls, is a distortion of the periodic lattice of a one-dimensional crystal which causes interatomic distances
Peierls_transition
Duality between theories of gravity on anti-de Sitter space and conformal field theories
the one-dimensional diagram representing the path of a point particle by a two-dimensional surface representing the motion of a string. Unlike in quantum
AdS/CFT_correspondence
Electronic states at the surface of materials
This state is either a band-edge state or a surface state in the band gap(see, Particle in a one-dimensional lattice, Particle in a box). Numerical calculations
Surface_states
Arrangement of spheres within a space
in n-dimensional Euclidean space, needs only n vectors to be defined. Lattice arrangements are periodic, and have the property that when the lattice is
Sphere_packing
Quantum physics terminology
Sanders, Jerome C. (Apr 2010). "Dimer of two bosons in a one-dimensional optical lattice". Phys. Rev. A. 81 (4) 043609. arXiv:1004.5118. Bibcode:2010PhRvA
Bound_state
Theoretical subdimensional particle
lineon (also known as the one-dimensional particle). If a quantum state is such that the eigenvalue of A c = − 1 {\displaystyle A_{c}=-1} for some unit cube
Fracton (subdimensional particle)
Fracton_(subdimensional_particle)
Process forming a path from many random steps
random walk on the d-dimensional integer lattice (sometimes called the hypercubic lattice) Z d {\displaystyle \mathbb {Z} ^{d}} . If, in addition, the state
Random_walk
Type of model in quantum statistical physics
example of a spin chain is the Heisenberg model, described by Werner Heisenberg in 1928. This models a one-dimensional lattice of fixed particles with spin
Spin_chain
Transient quantum fluctuation (physics)
A virtual particle is a theoretical transient particle that exhibits some of the characteristics of an ordinary particle, while having its existence limited
Virtual_particle
Fractal describing electrons in a magnetic field
derivation: a charged quantum particle in a two-dimensional square lattice, with a lattice spacing a {\displaystyle a} , is described by a periodic Schrödinger
Hofstadter's_butterfly
American-Argentinian physicist (born 1950)
to two-dimensional quantum spin liquids and other topologically ordered phases in two-dimensional lattices. He has done many important works in the area
Eduardo_Fradkin
Application of theoretical physics to experimental data
space-time) and the results of the high-energy particle experiments. It is sometimes used in other fields such as in condensed matter physics and plasma physics
Phenomenology_(physics)
Simplified model in condensed matter physics
("hopping") of particles between lattice sites and a potential term reflecting on-site interaction. The particles can either be fermions, as in Hubbard's original
Hubbard_model
52-dimensional exceptional simple Lie group
The F4 lattice is a four-dimensional body-centered cubic lattice (i.e. the union of two hypercubic lattices, each lying in the center of the
F4_(mathematics)
Ionizing radiation that presents as free neutrons
particles bombard the lattice, creating collision cascades and additional vacancies, which migrate towards sinks. The main effect of irradiation in a
Neutron_radiation
Dissipative particle dynamics (DPD) is an off-lattice mesoscopic simulation technique which involves a set of particles moving in continuous space and
Dissipative_particle_dynamics
No spontaneous symmetry breaking in two-dimensional systems at finite temperature
point, so that a one-dimensional or two-dimensional scalar does not have a well defined average value. If the field is an angle, θ, as it is in the Mexican
Mermin–Wagner_theorem
Geometric arrangements of points, foundational to Lie theory
systems) may be modeled in the Zometool construction set. In general, the An root lattice is the vertex arrangement of the n-dimensional simplicial honeycomb
Root_system
Model of hadrons
In particle physics, the parton model is a model of hadrons, such as protons and neutrons, proposed by Richard Feynman. It is useful for interpreting the
Parton_(particle_physics)
Proposed higher dimensions of space and time
all fields propagate universally in extra dimensions. Dimensional deconstruction is a lattice description of compactified extra dimensions that maintains
Extra_dimensions
Mathematical description in crystallography
reciprocal lattice is easily constructed in one dimension: for particles on a line with a period a {\displaystyle a} , the reciprocal lattice is an infinite
Structure_factor
Analysis and solving of problems that involve fluid flows
fictive particles, and such particles perform consecutive propagation and collision processes over a discrete lattice mesh. In this method, one works with
Computational_fluid_dynamics
State of matter
bosons in a one-dimensional bichromatic optical lattice in the regime of the pinning transition: A worm- algorithm Monte Carlo study". Physical Review A. 94
Bose–Einstein_condensate
Branch of physics
carriers: phonons (lattice vibration waves), electrons, fluid particles, and photons. Heat is defined as thermal energy in transit resulting from a spatial temperature
Heat_transfer_physics
Geometric space with six dimensions
necessarily Euclidean ones, is six-dimensional. One example is the surface of the 6-sphere, S6. This is the set of all points in seven-dimensional space (Euclidean)
Six-dimensional_space
potential The particle in a lattice The particle in a lattice of finite length The Pöschl–Teller potential The quantum pendulum The three-dimensional potentials
List of quantum-mechanical systems with analytical solutions
List_of_quantum-mechanical_systems_with_analytical_solutions
Cellular automaton that can be run backwards
automata", One Dimensional Cellular Automata, Luniver Press, pp. 205–246. Meyer, David A. (1996), "From quantum cellular automata to quantum lattice gases"
Reversible_cellular_automaton
Formula in X-ray diffraction and crystallography
to the broadening of a peak in a diffraction pattern. It is often referred to, incorrectly, as a formula for particle size measurement or analysis. It
Scherrer_equation
Atomic-scale structure formed through the Stark shift by opposing beams of light
provide a possible platform for direct tuning of wavelength in optical lattice systems. Continuous control of the periodicity of a one-dimensional optical
Optical_lattice
movement of gases and fluid. In this model, the lattice takes the form of a two-dimensional square grid, with particles capable of moving to any of the
HPP_model
Periodic optical nanostructure that affects the motion of photons
Two-dimensional ones can be made by photolithography, or by drilling holes in a suitable substrate. Fabrication methods for three-dimensional ones include
Photonic_crystal
Fermion path integral approach in 1+1 dimensions
for a free spin-1/2 particle moving in one spatial dimension. It provides a representation of solutions of the Dirac equation in (1+1)-dimensional spacetime
Feynman_checkerboard
Putting fermions on a lattice with chiral symmetry results in more fermions than expected
In lattice field theory, fermion doubling occurs when naively putting fermionic fields on a lattice, resulting in more fermionic states than expected
Fermion_doubling
Abstract model of quantum computation
Bruce M.; Taylor, Washington (1998). "Quantum lattice-gas model for the many-particle Schrödinger equation in d {\displaystyle d} dimensions". Physical Review
Quantum_cellular_automaton
Quasiparticle which is a bound state of an electron and an electron hole
describing free propagation of the electron-hole pair as a composite particle in the crystalline lattice in agreement with the Bloch theorem. The exciton energy
Exciton
Solid (crystalline) phase of electrons
energy, the electrons form a bcc (body-centered cubic) lattice in 3D, a triangular lattice in 2D and an evenly spaced lattice in 1D. Most experimentally
Wigner_crystal
Attractor in dynamical systems theory
cyclically symmetric in the x, y, and z variables and can be viewed as the trajectory of a frictionally dampened particle moving in a 3D lattice of forces. The
Thomas' cyclically symmetric attractor
Thomas'_cyclically_symmetric_attractor
Phenomenon from solid state physics
oscillation is a phenomenon from solid state physics. It describes the oscillation of a particle (e.g. an electron) confined in a periodic potential when a constant
Bloch_oscillation
Quantum variations of random walks
measurement has been explored on both one-dimensional and two-dimensional lattices, where the standard deviation grows in direct proportion to the evolution
Quantum_walk
Quantum-mechanical vector property in solid-state physics
In solid-state physics, crystal momentum or quasimomentum is a momentum-like vector associated with electrons in a crystal lattice. It is defined by the
Crystal_momentum
Concept in three-dimensional geometry
multiple particles with generally different orientations per repeating unit), and thus they showed that the best tetrahedron packing cannot be a lattice packing
Tetrahedron_packing
Extended physical object in string theory
generalizes the notion of a zero-dimensional point particle, a one-dimensional string, or a two-dimensional membrane to higher-dimensional objects. Branes are
Brane
Screening phenomenon in metals
Jacques Friedel, who predicted the effect in 1952. As a simple model, consider one-dimensional electron gas in a half-space x > 0 {\displaystyle x>0} . The
Friedel_oscillations
Concept in statistical physics
one of the models used to model ferromagnetism and other phenomena. The classical Heisenberg model can be formulated as follows: take a d-dimensional
Classical_Heisenberg_model
78-dimensional exceptional simple Lie group
{\mathfrak {e}}_{6}} , all of which have dimension 78; the same notation E6 is used for the corresponding root lattice, which has rank 6. The designation E6
E6_(mathematics)
Condensed matter system
fermionic quasi-particles occur in systems where there is a hexagonal crystal lattice; so bosonic quasiparticles on an hexagonal lattice are the natural
Dirac_matter
Theory of logic to account for observations from quantum theory
measurement of whether a disjunction holds does not measure which of the disjuncts is true. For example, consider a simple one-dimensional particle with position
Quantum_logic
Unit of information in a quantum computer
introduced a new resource efficient high dimensional protocol that dramatically reduces the resources needed to transmit information via high-dimensional quantum
Qudit
Concept in condensed matter physics
a 3×1018-dimensional vector space—one dimension for each coordinate (x, y, z) of each particle. Directly and straightforwardly trying to solve such a
Quasiparticle
Phenomenon of deformation due to structural stress
particle}}}}} In these formulas, r particle {\displaystyle r_{\text{particle}}\,} is the particle radius, γ particle-matrix {\displaystyle
Yield_(engineering)
No-go theorem concerning chirality of regularized fermions
In lattice field theory, the Nielsen–Ninomiya theorem is a no-go theorem about placing chiral fermions on a lattice. In particular, under very general
Nielsen–Ninomiya_theorem
State of matter with insulating bulk but conductive boundary
Rahul (2009-05-21). "Three dimensional topological invariants for time reversal invariant Hamiltonians and the three dimensional quantum spin Hall effect"
Topological_insulator
Two-dimensional state of matter
hexatic phase is a state of matter that is between the solid and the isotropic liquid phases in two dimensional systems of particles. It is characterized
Hexatic_phase
Millennium Prize Problem
programmatically. [...] one does not yet have a mathematically complete example of a quantum gauge theory in four-dimensional space-time, nor even a precise definition
Yang–Mills existence and mass gap
Yang–Mills_existence_and_mass_gap
Particle in a box Particle in a one-dimensional lattice Particle in a ring Particle in a spherically symmetric potential Particle number Particle number operator
Index_of_physics_articles_(P)
Elementary cellular automaton
through a one-dimensional medium. When two such particles collide, they annihilate each other, so that at each step the number of particles remains unchanged
Rule_184
Crystalline materials consisting of a single layer of atoms
solid supports. Antimonene is a two-dimensional allotrope of antimony, with its atoms arranged in a buckled honeycomb lattice. Theoretical calculations predicted
Single-layer_materials
American physicist
contributions in field theory and particle physics. Samuel graduated from Princeton University with a Bachelor of Arts in mathematics in 1975, and in 1979, he
Stuart_Samuel_(physicist)
Quantization of cyclotron orbits
spectrum of charged particles in a two dimensional infinite lattice is known to be self-similar and fractal, as demonstrated in Hofstadter's butterfly
Landau_levels
State of matter
one particle on each lattice site on average. Alternatively, a supersolid can also emerge from a superfluid. In this situation, which is realised in the
Supersolid
In physics, a Super Bloch oscillation describes a certain type of motion of a particle in a lattice potential under external periodic driving. The term
Super_Bloch_oscillations
Mathematical mapping in quantum mechanics
by Pascual Jordan and Eugene Wigner in 1928 for one-dimensional lattice models, but now two-dimensional analogues of the transformation have also been
Jordan–Wigner_transformation
Electronic structure imaging method
superconductors. Because QPI only images a two-dimensional projection of the electronic structure and due to the three-dimensionality of the electronic structure of
Quasiparticle interference imaging
Quasiparticle_interference_imaging
Numerical technique for solving quantum Hamiltonians
(2024-10-28). "Exploring two-dimensional coherent spectroscopy with exact diagonalization: Spinons and confinement in one-dimensional quantum magnets". Physical
Exact_diagonalization
Type of atomic clock
lattice clock in which strontium-87 atoms are packed into a tiny three-dimensional (3-D) cube at 1,000 times the density of previous one-dimensional (1-D)
Optical_clock
Method for finding the exact solution of certain quantum mechanics models
spatial dimension. This ansatz was introduced by Hans Bethe in 1931 to obtain the exact eigenvalues and eigenvectors of the one-dimensional antiferromagnetic
Bethe_ansatz
Problem in applied mathematics
multi-particle states involves an integral over a function that is highly oscillatory, hence hard to evaluate numerically, particularly in high dimension.
Numerical_sign_problem
Mathematical theory on behavior of connected clusters in a random graph
be asked for any lattice dimension. As is quite typical, it is actually easier to examine infinite networks than just large ones. In this case the corresponding
Percolation_theory
Composite material
the oxide particles within the lattice of the material. Coherent particles have a continuous lattice plane from the matrix to the particles, whereas incoherent
Oxide dispersion-strengthened alloy
Oxide_dispersion-strengthened_alloy
Symmetry breaking through the vacuum state
symmetry in different directions, leading to topological defects, such as two-dimensional domain walls, one-dimensional cosmic strings, zero-dimensional monopoles
Spontaneous_symmetry_breaking
Condition for ferromagnetism
a lattice model in mind, N {\textstyle N} is the number of lattice sites and N ↑ {\displaystyle N_{\uparrow }} is the number of spin-up electrons in the
Stoner_criterion
Model of electrical resistance
superconductivity in cuprate superconductors which arise from doped antiferromagnets, particularly in the case where the lattice considered is the two-dimensional lattice
T-J_model
System exhibiting quantum mechanical effects at the macroscopic level
of a particle is greater than the spacing between the particles in the lattice that comprises the matter. The de Broglie wavelength associated with a massive
Quantum_fluid
Matrices named after Élie Cartan
lattice and root lattice, respectively. In modular representation theory, and more generally in the theory of representations of finite-dimensional associative
Cartan_matrix
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