|
Second Quantization
Second quantization, also referred to as occupation number representation, is a formalism used to describe and analyze quantum many-body systems. In quantum field theory, it is known as canonical quantization, in which the fields (typically as the wave functions of matter) are thought of as field operators, in a manner similar to how the physical quantities (position, momentum, etc.) are thought of as operators in first quantization. The key ideas of this method were introduced in 1927 by Paul Dirac, and were later developed, most notably, by Pascual Jordan and Vladimir Fock. In this approach, the quantum many-body states are represented in the Fock state basis, which are constructed by filling up each single-particle state with a certain number of identical particles. The second quantization formalism introduces the creation and annihilation operators to construct and handle the Fock states, providing useful tools to the study of the quantum many-body theory. Quantum many-bo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, quantum field theory, quantum technology, and quantum information science. Classical physics, the collection of theories that existed before the advent of quantum mechanics, describes many aspects of nature at an ordinary ( macroscopic) scale, but is not sufficient for describing them at small (atomic and subatomic) scales. Most theories in classical physics can be derived from quantum mechanics as an approximation valid at large (macroscopic) scale. Quantum mechanics differs from classical physics in that energy, momentum, angular momentum, and other quantities of a bound system are restricted to discrete values ( quantization); objects have characteristics of both particles and waves ( wave–particle duality); and there are lim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Antisymmetrizer
In quantum mechanics, an antisymmetrizer \mathcal (also known as antisymmetrizing operatorP.A.M. Dirac, ''The Principles of Quantum Mechanics'', 4th edition, Clarendon, Oxford UK, (1958) p. 248) is a linear operator that makes a wave function of ''N'' identical fermions antisymmetric under the exchange of the coordinates of any pair of fermions. After application of \mathcal the wave function satisfies the Pauli exclusion principle. Since \mathcal is a projection operator, application of the antisymmetrizer to a wave function that is already totally antisymmetric has no effect, acting as the identity operator. Mathematical definition Consider a wave function depending on the space and spin coordinates of ''N'' fermions: : \Psi(1,2, \ldots, N)\quad\text \quad i \leftrightarrow (\mathbf_i, \sigma_i), where the position vector r''i'' of particle ''i'' is a vector in \mathbb^3 and σi takes on 2''s''+1 values, where ''s'' is the half-integral intrinsic spin of the fermion. Fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spin Structure
In differential geometry, a spin structure on an orientable Riemannian manifold allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry. Spin structures have wide applications to mathematical physics, in particular to quantum field theory where they are an essential ingredient in the definition of any theory with uncharged fermions. They are also of purely mathematical interest in differential geometry, algebraic topology, and K theory. They form the foundation for spin geometry. Overview In geometry and in field theory, mathematicians ask whether or not a given oriented Riemannian manifold (''M'',''g'') admits spinors. One method for dealing with this problem is to require that ''M'' has a spin structure. This is not always possible since there is potentially a topological obstruction to the existence of spin structures. Spin structures will exist if and only if the second Stiefel–Whitney class ''w''2(''M'') ∈ H2(''M' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jordan–Wigner Transformation
The Jordan–Wigner transformation is a transformation that maps spin operators onto fermionic creation and annihilation operators. It was proposed by Pascual Jordan and Eugene Wigner for one-dimensional lattice models, but now two-dimensional analogues of the transformation have also been created. The Jordan–Wigner transformation is often used to exactly solve 1D spin-chains such as the Ising and XY models by transforming the spin operators to fermionic operators and then diagonalizing in the fermionic basis. This transformation actually shows that the distinction between spin-1/2 particles and fermions is nonexistent. It can be applied to systems with an arbitrary dimension. Analogy between spins and fermions In what follows we will show how to map a 1D spin chain of spin-1/2 particles to fermions. Take spin-1/2 Pauli operators acting on a site j of a 1D chain, \sigma_^, \sigma_^, \sigma_^. Taking the anticommutator of \sigma_^ and \sigma_^, we find \ = I, as would be exp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kronecker Delta
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 &\text i=j. \end or with use of Iverson brackets: \delta_ = =j, where the Kronecker delta is a piecewise function of variables and . For example, , whereas . The Kronecker delta appears naturally in many areas of mathematics, physics and engineering, as a means of compactly expressing its definition above. In linear algebra, the identity matrix has entries equal to the Kronecker delta: I_ = \delta_ where and take the values , and the inner product of vectors can be written as \mathbf\cdot\mathbf = \sum_^n a_\delta_b_ = \sum_^n a_ b_. Here the Euclidean vectors are defined as -tuples: \mathbf = (a_1, a_2, \dots, a_n) and \mathbf= (b_1, b_2, ..., b_n) and the last step is obtained by using the values of the Kronecker del ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tensor Algebra
In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', in the sense of being left adjoint to the forgetful functor from algebras to vector spaces: it is the "most general" algebra containing ''V'', in the sense of the corresponding universal property (see below). The tensor algebra is important because many other algebras arise as quotient algebras of ''T''(''V''). These include the exterior algebra, the symmetric algebra, Clifford algebras, the Weyl algebra and universal enveloping algebras. The tensor algebra also has two coalgebra structures; one simple one, which does not make it a bialgebra, but does lead to the concept of a cofree coalgebra, and a more complicated one, which yields a bialgebra, and can be extended by giving an antipode to create a Hopf algebra structure. ''Note'': In this article, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Harmonic Oscillator
量子調和振動子 は、 古典調和振動子 の 量子力学 類似物です。任意の滑らかな ポテンシャル は通常、安定した 平衡点 の近くで 調和ポテンシャル として近似できるため、最も量子力学における重要なモデル系。さらに、これは正確な 解析解法が知られている数少ない量子力学系の1つである。 author=Griffiths, David J. , title=量子力学入門 , エディション=2nd , 出版社=プレンティス・ホール , 年=2004 , isbn=978-0-13-805326-0 , author-link=David Griffiths (物理学者) , URL アクセス = 登録 , url=https://archive.org/details/introductiontoel00grif_0 One-dimensional harmonic oscillator Hamiltonian and energy eigenstates 粒子の ハミルトニアン は次のとおりです。 \hat H = \frac + \frac k ^2 = \frac + \frac m \omega^2 ^2 \, , ここで、 は粒子の質量、 は力定数、\omega = \sqrt は ��動子の [角周波数 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vacuum State
In quantum field theory, the quantum vacuum state (also called the quantum vacuum or vacuum state) is the quantum state with the lowest possible energy. Generally, it contains no physical particles. The word zero-point field is sometimes used as a synonym for the vacuum state of a quantized field which is completely individual. According to present-day understanding of what is called the vacuum state or the quantum vacuum, it is "by no means a simple empty space". According to quantum mechanics, the vacuum state is not truly empty but instead contains fleeting electromagnetic waves and particles that pop into and out of the quantum field. The QED vacuum of quantum electrodynamics (or QED) was the first vacuum of quantum field theory to be developed. QED originated in the 1930s, and in the late 1940s and early 1950s it was reformulated by Feynman, Tomonaga, and Schwinger, who jointly received the Nobel prize for this work in 1965. For a historical discussion, see for e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that defines a distance function for which the space is a complete metric space. The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics). John von Neumann coined the term ''Hilbert space'' for the abstract concept that u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fock Space
The Fock space is an algebraic construction used in quantum mechanics to construct the quantum states space of a variable or unknown number of identical particles from a single particle Hilbert space . It is named after V. A. Fock who first introduced it in his 1932 paper "Konfigurationsraum und zweite Quantelung" (" Configuration space and second quantization"). M.C. Reed, B. Simon, "Methods of Modern Mathematical Physics, Volume II", Academic Press 1975. Page 328. Informally, a Fock space is the sum of a set of Hilbert spaces representing zero particle states, one particle states, two particle states, and so on. If the identical particles are bosons, the -particle states are vectors in a symmetrized tensor product of single-particle Hilbert spaces . If the identical particles are fermions, the -particle states are vectors in an antisymmetrized tensor product of single-particle Hilbert spaces (see symmetric algebra and exterior algebra respectively). A general state in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pauli Exclusion Principle
In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formulated by Austrian physicist Wolfgang Pauli in 1925 for electrons, and later extended to all fermions with his spin–statistics theorem of 1940. In the case of electrons in atoms, it can be stated as follows: it is impossible for two electrons of a poly-electron atom to have the same values of the four quantum numbers: ''n'', the principal quantum number; ', the azimuthal quantum number; ''m'', the magnetic quantum number; and ''ms'', the spin quantum number. For example, if two electrons reside in the same orbital, then their ''n'', ', and ''m'' values are the same; therefore their ''ms'' must be different, and thus the electrons must have opposite half-integer spin projections of 1/2 and −1/2. Particles with an integer spin, or boso ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Slater Determinant
In quantum mechanics, a Slater determinant is an expression that describes the wave function of a multi-fermionic system. It satisfies anti-symmetry requirements, and consequently the Pauli principle, by changing sign upon exchange of two electrons (or other fermions).Molecular Quantum Mechanics Parts I and II: An Introduction to QUANTUM CHEMISTRY (Volume 1), P. W. Atkins, Oxford University Press, 1977, . Only a small subset of all possible fermionic wave functions can be written as a single Slater determinant, but those form an important and useful subset because of their simplicity. The Slater determinant arises from the consideration of a wave function for a collection of electrons, each with a wave function known as the spin-orbital \chi(\mathbf), where \mathbf denotes the position and spin of a single electron. A Slater determinant containing two electrons with the same spin orbital would correspond to a wave function that is zero everywhere. The Slater determinant is name ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
