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Fock State
In quantum mechanics, a Fock state or number state is a quantum state that is an element of a Fock space with a well-defined number of particles (or quanta). These states are named after the Soviet physicist Vladimir Fock. Fock states play an important role in the second quantization formulation of quantum mechanics. The particle representation was first treated in detail by Paul Dirac for bosons and by Pascual Jordan and Eugene Wigner for fermions. The Fock states of bosons and fermions obey useful relations with respect to the Fock space creation and annihilation operators. Definition One specifies a multiparticle state of ''N'' non-interacting identical particles by writing the state as a sum of tensor products of ''N'' one-particle states. Additionally, depending on the integrality of the particles' spin, the tensor products must be alternating (anti-symmetric) or symmetric products of the underlying one-particle Hilbert spaces. Specifically: * Fermions, having half- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Mechanics
Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is the foundation of all quantum physics, which includes quantum chemistry, quantum field theory, quantum technology, and quantum information science. Quantum mechanics can describe many systems that classical physics cannot. Classical physics can describe many aspects of nature at an ordinary (macroscopic and Microscopic scale, (optical) microscopic) scale, but is not sufficient for describing them at very small submicroscopic (atomic and subatomic) scales. Classical mechanics can be derived from quantum mechanics as an approximation that is valid at ordinary scales. Quantum systems have Bound state, bound states that are Quantization (physics), quantized to Discrete mathematics, discrete values of energy, momentum, angular momentum, and ot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Tensor
In mathematics, a symmetric tensor is an unmixed tensor that is invariant under a permutation of its vector arguments: :T(v_1,v_2,\ldots,v_r) = T(v_,v_,\ldots,v_) for every permutation ''σ'' of the symbols Alternatively, a symmetric tensor of order ''r'' represented in coordinates as a quantity with ''r'' indices satisfies :T_ = T_. The space of symmetric tensors of order ''r'' on a finite-dimensional vector space ''V'' is naturally isomorphic to the dual of the space of homogeneous polynomials of degree ''r'' on ''V''. Over fields of characteristic zero, the graded vector space of all symmetric tensors can be naturally identified with the symmetric algebra on ''V''. A related concept is that of the antisymmetric tensor or alternating form. Symmetric tensors occur widely in engineering, physics and mathematics. Definition Let ''V'' be a vector space and :T\in V^ a tensor of order ''k''. Then ''T'' is a symmetric tensor if :\tau_\sigma T = T\, for the braiding ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Identical Particles
In quantum mechanics, indistinguishable particles (also called identical or indiscernible particles) are particles that cannot be distinguished from one another, even in principle. Species of identical particles include, but are not limited to, elementary particles (such as electrons), composite subatomic particles (such as atomic nuclei), as well as atoms and molecules. Although all known indistinguishable particles only exist at the quantum scale, there is no exhaustive list of all possible sorts of particles nor a clear-cut limit of applicability, as explored in quantum statistics. They were first discussed by Werner Heisenberg and Paul Dirac in 1926. There are two main categories of identical particles: bosons, which can share quantum states, and fermions, which cannot (as described by the Pauli exclusion principle). Examples of bosons are photons, gluons, phonons, helium-4 nuclei and all mesons. Examples of fermions are electrons, neutrinos, quarks, protons, neutrons, an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Operator (physics)
An operator is a function over a space of physical states onto another space of states. The simplest example of the utility of operators is the study of symmetry (which makes the concept of a group useful in this context). Because of this, they are useful tools in classical mechanics. Operators are even more important in quantum mechanics, where they form an intrinsic part of the formulation of the theory. They play a central role in describing observables (measurable quantities like energy, momentum, etc.). Operators in classical mechanics In classical mechanics, the movement of a particle (or system of particles) is completely determined by the Lagrangian L(q, \dot, t) or equivalently the Hamiltonian H(q, p, t), a function of the generalized coordinates ''q'', generalized velocities \dot = \mathrm q / \mathrm t and its conjugate momenta: :p = \frac If either ''L'' or ''H'' is independent of a generalized coordinate ''q'', meaning the ''L'' and ''H'' do not change when ''q' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Variance
In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers is spread out from their average value. It is the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by \sigma^2, s^2, \operatorname(X), V(X), or \mathbb(X). An advantage of variance as a measure of dispersion is that it is more amenable to algebraic manipulation than other measures of dispersion such as the expected absolute deviation; for example, the variance of a sum of uncorrelated random variables is equal to the sum of their variances. A disadvantage of the variance for practical applications is that, unlike the standard deviation, its units differ from the random variable, which is why the standard devi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Superposition
Quantum superposition is a fundamental principle of quantum mechanics that states that linear combinations of solutions to the Schrödinger equation are also solutions of the Schrödinger equation. This follows from the fact that the Schrödinger equation is a linear differential equation in time and position. More precisely, the state of a system is given by a linear combination of all the eigenfunctions of the Schrödinger equation governing that system. An example is a qubit used in quantum information processing. A qubit state is most generally a superposition of the basis states , 0 \rangle and , 1 \rangle: : , \Psi \rangle = c_0, 0\rangle + c_1, 1\rangle, where , \Psi \rangle is the quantum state of the qubit, and , 0 \rangle, , 1 \rangle denote particular solutions to the Schrödinger equation in Dirac notation weighted by the two probability amplitudes c_0 and c_1 that both are complex numbers. Here , 0 \rangle corresponds to the classical 0 bit, and , 1 \r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Basis (linear Algebra)
In mathematics, a Set (mathematics), set of elements of a vector space is called a basis (: bases) if every element of can be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as components or coordinates of the vector with respect to . The elements of a basis are called . Equivalently, a set is a basis if its elements are linearly independent and every element of is a linear combination of elements of . In other words, a basis is a linearly independent spanning set. A vector space can have several bases; however all the bases have the same number of elements, called the dimension (vector space), dimension of the vector space. This article deals mainly with finite-dimensional vector spaces. However, many of the principles are also valid for infinite-dimensional vector spaces. Basis vectors find applications in the study of crystal structures and frame of reference, frames of reference. De ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Particle Number Operator
In quantum mechanics, for systems where the total number of particles may not be preserved, the number operator is the observable that counts the number of particles. The following is in bra–ket notation: The number operator acts on Fock space. Let , \Psi\rangle_\nu=, \phi_1,\phi_2,\cdots,\phi_n\rangle_\nu be a Fock state, composed of single-particle states , \phi_i\rangle drawn from a basis of the underlying Hilbert space of the Fock space. Given the corresponding creation and annihilation operators a^(\phi_i) and a(\phi_i)\, we define the number operator by \hat \ \stackrel\ a^(\phi_i)a(\phi_i) and we have \hat, \Psi\rangle_\nu=N_i, \Psi\rangle_\nu where N_i is the number of particles in state , \phi_i\rangle. The above equality can be proven by noting that \begin a(\phi_i) , \phi_1,\phi_2,\cdots,\phi_,\phi_i,\phi_,\cdots,\phi_n\rangle_\nu &=& \sqrt , \phi_1,\phi_2,\cdots,\phi_,\phi_,\cdots,\phi_n\rangle_\nu \\ a^(\phi_i) , \phi_1,\phi_2,\cdots,\phi_,\phi_,\cdots,\phi_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthonormal Basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite Dimension (linear algebra), dimension is a Basis (linear algebra), basis for V whose vectors are orthonormal, that is, they are all unit vectors and Orthogonality_(mathematics), orthogonal to each other. For example, the standard basis for a Euclidean space \R^n is an orthonormal basis, where the relevant inner product is the dot product of vectors. The Image (mathematics), image of the standard basis under a Rotation (mathematics), rotation or Reflection (mathematics), reflection (or any orthogonal transformation) is also orthonormal, and every orthonormal basis for \R^n arises in this fashion. An orthonormal basis can be derived from an orthogonal basis via Normalize (linear algebra), normalization. The choice of an origin (mathematics), origin and an orthonormal basis forms a coordinate frame known as an ''orthonormal frame''. For a general inner product space V, an orthono ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Particle Number
In thermodynamics, the particle number (symbol ) of a thermodynamic system is the number of constituent particles in that system. The particle number is a fundamental thermodynamic property which is conjugate to the chemical potential. Unlike most physical quantities, the particle number is a dimensionless quantity, specifically a countable quantity. It is an extensive property, as it is directly proportional to the size of the system under consideration and thus meaningful only for closed systems. A constituent particle is one that cannot be broken into smaller pieces at the scale of energy involved in the process (where is the Boltzmann constant and is the temperature). For example, in a thermodynamic system consisting of a piston containing water vapour, the particle number is the number of water molecules in the system. The meaning of constituent particles, and thereby of particle numbers, is thus temperature-dependent. Determining the particle number The concept of pa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Direct Sum Of Modules
In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The direct sum of modules is the smallest module which contains the given modules as submodules with no "unnecessary" constraints, making it an example of a coproduct. Contrast with the direct product, which is the dual notion. The most familiar examples of this construction occur when considering vector spaces (modules over a field) and abelian groups (modules over the ring Z of integers). The construction may also be extended to cover Banach spaces and Hilbert spaces. See the article decomposition of a module for a way to write a module as a direct sum of submodules. Construction for vector spaces and abelian groups We give the construction first in these two cases, under the assumption that we have only two objects. Then we generalize to an arbitrary family of arbitrary modules. The key elements of the general construction are more clearly identified ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boson
In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0, 1, 2, ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have half odd-integer spin (1/2, 3/2, 5/2, ...). Every observed subatomic particle is either a boson or a fermion. Paul Dirac coined the name ''boson'' to commemorate the contribution of Satyendra Nath Bose, an Indian physicist. Some bosons are elementary particles occupying a special role in particle physics, distinct from the role of fermions (which are sometimes described as the constituents of "ordinary matter"). Certain elementary bosons (e.g. gluons) act as force carriers, which give rise to forces between other particles, while one (the Higgs boson) contributes to the phenomenon of mass. Other bosons, such as mesons, are composite particles made up of smaller constituents. Outside the realm of particle physics, multiple identical composite bosons ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
