eigenvalues

In mathematics and theoretical physics, Wigner's classification is a classification of the nonnegative energy irreducible unitary representations of the Poincaré group, which have sharp mass eigenvalues. It was proposed by Eugene Wigner, for reasons coming from physics -- see the article particle physics and representation theory. The mass is a Casimir invariant of the Poincaré group. So, we can classify the irreps into whet ...

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Computational chemistry is a branch of theoretical chemistry whose major goals are to create efficient mathematical approximations and computer programs that calculate the properties of molecules (such as total energy, dipole and quadrupole moment, vibrational frequencies, reactivity and other diverse spectroscopic quantitities and cross sections for collision of molecules with diverse atomic or subatomic projectiles) and to apply these programs to concrete chemical objects. The term is also sometimes used to cover the areas of overla ...

Including:

• Computational chemistry - Introduction
• Computational chemistry - Ab initio methods
• Computational chemistry - Electronic structure
• Computational chemistry - Chemical dynamics
• Computational chemistry - Semiempirical methods
• Computational chemistry - Electronic structure
• Computational chemistry - Molecular mechanics
• Computational chemistry - Software packages

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In physics, the Casimir effect is a weak force exerted between seperate objects, which is not due to charge, gravity, or exchange of particles, but instead is due to resonance in the intervening space between the objects, of all-pervasive energy fields. The force is only measurable when the distance between the objects is extremely small, since it falls off rapidly with distance. Dutch physicist Hendrik B. G. Casimir first proposed the exitence of the force, and an experiment to detect it in 1948 whil ...

Including:

• Casimir effect - Overview
• Casimir effect - Vacuum energy
• Casimir effect - The Casimir effect
• Casimir effect - Casimir's calculation
• Casimir effect - Measurement
• Casimir effect - Regularization
• Casimir effect - Generalities
• Casimir effect - Analogies

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In probability and statistics, a Bernoulli process is a discrete-time stochastic process consisting of a sequence of independent random variables taking values over two letters. Bernoulli process - Definition. A Bernoulli process is a discrete-time stochastic process consisting of a finite or infinite sequence of independent random variables X1, X2, X3,..., such that For each i, the value of Xi is eithe ...

Including:

• Bernoulli process - Definition
• Bernoulli process - Formal definition
• Bernoulli process - Bernoulli sequence
• Bernoulli process - Bernoulli map
• Bernoulli process - Generalizations

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The Born-Oppenheimer approximation, also known as the adiabatic approximation, is a technique used in quantum chemistry and condensed matter physics in order to de-couple the motion of nuclei and electrons (i.e. to separate the variables corresponding to the nuclear and electronic coordinates in the Schrödinger equation associated to the molecular Hamiltonian). It is based upon the fact that typical electronic velocities far exceed those of nuclei. The Born-Oppenheimer approximation is commonly confused with the Born-Oppenh ...

Including:

• Born-Oppenheimer approximation - Hand-waving derivation of the approximation
• Born-Oppenheimer approximation - Beyond the Born-Oppenheimer Approximation

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Computational physics is the study and implementation of numerical algorithms in order to solve problems in physics for which a quantitative theory already exists. It is often regarded as a subdiscipline of theoretical physics. Physicists often have a very precise mathematical theory describing how a system will behave. Unfortunately, it is often the case that solving the theory's equations ab-initio in order to produce a useful prediction is not realistic. This is especially true with quantum mechanics, where only a handful of simple models can be solved ...

Including:

• Computational physics - Challenges in computational physics
• Computational physics - Applications of computational physics

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In numerical analysis, the condition number associated with a numerical problem is a measure of that quantity's amenability to digital computation, that is, how well-posed the problem is. A problem with a low condition number is said to be well-conditioned, while a problem with a high condition number is said to be ill-conditioned. Condition number - The condition number of a matrix. For example, the condition number associated with the linear equation A ...

Including:

• Condition number - The condition number of a matrix
• Condition number - The condition number in other contexts

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In mathematics, a unitary matrix is an n by n complex matrix U satisfying the condition where is the identity matrix and is the conjugate transpose (also called the Hermitian adjoint) of U. Note this condition says that a matrix U is unitary if it has an inverse which is equal to its conjugate transpose . A unitary matrix in which all entries are real is the same thing as an orthogonal matrix. Just as an orthogonal matrix G preserves the (real) inner product of two real vectors, so also a u ...

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In mathematics, an ellipse (from the Greek for absence) is a plane algebraic curve where the sum of the distances from any point on the curve to two fixed points is constant. The two fixed points are called foci (plural of focus). An ellipse is a type of conic section: if a conical surface is cut with a plane which does not intersect the cone's base, the intersection of the cone and plane is an ellipse. For a short e ...

Including:

• Ellipse - Parametrisation
• Ellipse - Eccentricity
• Ellipse - Semi-latus rectum and polar coordinates
• Ellipse - Area
• Ellipse - Circumference
• Ellipse - Stretching and Projection
• Ellipse - Reflection property
• Ellipse - Ellipses in physics
• Ellipse - Ellipses in computer graphics

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In linear algebra, a coordinate vector is an explicit representation of a vector in an abstract vector space as an ordered list of numbers or, equivalently, as an element of the coordinate space Fn. Coordinate vectors allow calculations with abstract objects to be transformed into calculations with blocks of numbers (matrices and column vectors), which we know how to do explicitly. Coordinate vector - Definition. Let V be a vector space of dimension n over a field F an ...

Including:

• Coordinate vector - Definition
• Coordinate vector - The standard representation
• Coordinate vector - Examples
• Coordinate vector - Example 1
• Coordinate vector - Example 2
• Coordinate vector - Basis transformation matrix

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In mathematics and physics, continuous spectrum is, roughly speaking, a non-countable set of eigenvalues of an operator. An operator acting on a Hilbert space is said to have a continuous spectrum if its eigenvalues can be changed continuously. If the spectrum of an operator is not continuous, we say that it is has discrete spectrum. Some of the basic questions in spectral theory are to characterise the discrete spectrum and purely continuous spectrum, just as a measure, such as a probability measure, can typically ...

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In mathematics and especially in abstract algebra, a congruence relation or simply congruence is an equivalence relation that is compatible with some algebraic operation(s). Congruence relation - Modular arithmetic. The prototypical example is modular arithmetic: for n a positive natural number, two integers a and b are called congruent modulo n if a − b is divisible by n. If and , then and . This turns the equivalence (mod ...

Including:

• Congruence relation - Modular arithmetic
• Congruence relation - Linear algebra
• Congruence relation - Universal algebra
• Congruence relation - Group theory
• Congruence relation - Ring theory
• Congruence relation - General case of kernels

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White noise (Sample ▶ (help·info)) is a random signal (or process) with a flat power spectral density. In other words, the signal's power spectral density has equal power in any band, at any centre frequency, having a given bandwidth. An infinite-bandwidth white noise signal is purely a theoretical construct. By having power at all frequencies, the total power of such a signal is infinite. In practice, a signal can be "white" with a flat spectrum over ...

Including:

• White noise - Statistical properties
• White noise - Colors of noise
• White noise - Applications
• White noise - Mathematical definition
• White noise - White random vector
• White noise - White random process white noise
• White noise - Random vector transformations
• White noise - Simulating a random vector
• White noise - Whitening a random vector
• White noise - Random signal transformations
• White noise - Simulating a continuous-time random signal
• White noise - Whitening a continuous-time random signal
• White noise - External link

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In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space. The algebra multiplication and the Banach space norm are required to be related by the following inequality: (i.e., the norm of the product is less than or equal to the product of the norms.) This ensures that the multiplication operation is continuous. A Banach algebra is called "unital" if it has an identity element for the multiplication whose norm is 1, an ...

Including:

• Banach algebra - Examples
• Banach algebra - Properties

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Almost immediately, Milnor produced a pair of 16-dimensional tori that have the same eigenvalues. However, the problem in two dimensions remained open until 1992, when Gordon, Webb and Wolpert constructed a pair of regions in the plane, which are not congruent but nevertheless have equal eigenvalues. The regions are non-convex polygons (see picture). The proof that both regions have the same eigenvalues is rather elementary and uses the symmetries of the Laplacian. This idea has been generalized by Buser et al., who constructed numerous simi ...

See also:

Hearing the shape of a drum, Hearing the shape of a drum - The answer, Hearing the shape of a drum - Weyl's formula, Hearing the shape of a drum - The Weyl-Berry conjecture

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The solution of the Schrödinger equation for the hydrogen atom uses the fact that the Coulomb potential produced by the nucleus is isotropic (it only depends on the distance to the nucleus). Although the resulting energy eigenfunctions (the "orbitals") are not necessarily isotropic themselves, their dependence on the angular coordinates follows completely generally from this isotropy of the underlying potential: The states are not only eigenstates of the Hamiltonian, but also eigenstates of the angular momentum operator. This corresponds to ...

See also:

Hydrogen atom, Hydrogen atom - Solution of Schrödinger equation: Overview of results, Hydrogen atom - Mathematical summary of eigenstates of hydrogen atom, Hydrogen atom - Visualizing the hydrogen electron orbitals, Hydrogen atom - Features going beyond the Schrödinger solution

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The model rests on four assumptions: The self-replicating entities can be represented as sequences composed of a small number of building blocks--for example, sequences of RNA consisting of the four bases adenine, guanine, cytosine, and uracil. New sequences enter the system solely as the result of a copy process, either correct or erroneous, of other sequences that are already present. The substrates, or raw materials, necessary for ongoing replication are always present in sufficient quantity. Excess sequence ...

See also:

Quasispecies model, Quasispecies model - General description, Quasispecies model - Mathematical description, Quasispecies model - Alternative formulations, Quasispecies model - A simple example

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Identical particles - Symmetrical and antisymmetrical states. We will now make the above discussion concrete, using the formalism developed in the article on the mathematical formulation of quantum mechanics. For simplicity, consider a system composed of two identical particles. As the particles possess equivalent physical properties, their state vectors occupy mathematically identical Hilbert spaces. If we denote the Hilbert space of a single particle as HSee also:

Identical particles, Identical particles - Distinguishing between particles, Identical particles - Quantum mechanical description of identical particles, Identical particles - Symmetrical and antisymmetrical states, Identical particles - Exchange symmetry, Identical particles - Fermions and bosons, Identical particles - N particles, Identical particles - Measurements of identical particles, Identical particles - Wavefunction representation, Identical particles - Statistical properties, Identical particles - Statistical effects of indistinguishability, Identical particles - Statistical properties of bosons and fermions, Identical particles - The homotopy class

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As explained in the article mathematical formulation of quantum mechanics, the physical state of a system may be characterized as a vector in an abstract Hilbert space (or, in the case of ensembles, as a countable sequence of vectors weighted by probabilities). Physically observable quantities are described by self-adjoint operators acting on these vectors. The quantum Hamiltonian H is the observable corresponding to the total energy of the system ...

See also:

Hamiltonian quantum mechanics, Hamiltonian quantum mechanics - The quantum Hamiltonian, Hamiltonian quantum mechanics - Energy eigenket degeneracy symmetry and conservation laws, Hamiltonian quantum mechanics - Hamilton's equations

Read more here: » Hamiltonian quantum mechanics: Encyclopedia II - Hamiltonian quantum mechanics - The quantum Hamiltonian

The trace is a linear map. That is, tr(A + B) = tr(A) + tr(B) tr(rA) = r tr(A) for all square matrices A and B, and all scalars r. Since the principal diagonal is not moved on transposition, a matrix and its transpose have the same trace: tr(A) = tr(AT). If A is an n×m matrix and B is an m×n matrix, then ...

See also:

Trace linear algebra, Trace linear algebra - Properties, Trace linear algebra - Eigenvalue relationships, Trace linear algebra - Other ideas and applications, Trace linear algebra - Inner product, Trace linear algebra - Generalization

Read more here: » Trace linear algebra: Encyclopedia II - Trace linear algebra - Properties