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Method Of Continued Fractions
The method of continued fractions is a method developed specifically for solution of integral equations of quantum scattering theory like Lippmann–Schwinger equation or Faddeev equations. It was invented by Jiří Horáček (physicist), Horáček and Sasakawa in 1983. The goal of the method is to solve the integral equation : , \psi\rangle = , \phi\rangle + G_0 V, \psi\rangle iteratively and to construct convergent continued fraction for the T-matrix : T= \langle \phi , V , \psi\rangle . The method has two variants. In the first one (denoted as MCFV) we construct approximations of the potential energy operator V in the form of finite rank operator, separable function of rank 1, 2, 3 ... The second variant (MCFG method) constructs the finite rank approximations to Green's operator. The approximations are constructed within Krylov subspace constructed from vector , \phi\rangle with action of the operator A=G_0 V. The method can thus be understood as resummation of (in general diverge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Scattering Theory
In physics, a quantum (: quanta) is the minimum amount of any physical entity (physical property) involved in an fundamental interaction, interaction. The fundamental notion that a property can be "quantized" is referred to as "the hypothesis of quantization (physics), quantization". This means that the Magnitude (mathematics), magnitude of the physical property can take on only Wiktionary:discrete, discrete values consisting of Multiple (mathematics), integer multiples of one quantum. For example, a photon is a single quantum of light of a specific frequency (or of any other form of electromagnetic radiation). Similarly, the energy of an electron bound within an atom is quantized and can exist only in certain discrete values. Atoms and matter in general are stable because electrons can exist only at discrete energy levels within an atom. Quantization is one of the foundations of the much broader physics of quantum mechanics. Quantization of energy and its influence on how energy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schwinger Variational Principle
Schwinger variational principle is a variational principle which expresses the scattering T-matrix method, T-matrix as a Functional (mathematics), functional depending on two unknown wave functions. The functional attains stationary value equal to actual scattering T-matrix. The functional is stationary if and only if the two functions satisfy the Lippmann-Schwinger equation. The development of the variational formulation of the scattering theory can be traced to works of L. Hultén and J. Schwinger in 1940s.R.G. Newton, Scattering Theory of Waves and Particles Linear form of the functional The T-matrix expressed in the form of stationary value of the functional reads : \langle\phi', T(E), \phi\rangle = T[\psi',\psi] \equiv \langle\psi', V, \phi\rangle + \langle\phi', V, \psi\rangle - \langle\psi', V-VG_0^(E)V, \psi\rangle , where \phi and \phi' are the initial and the final states respectively, V is the interaction potential and G_0^(E) is the retarded Green's operator for collis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Molecular Physics
Molecular physics is the study of the physical properties of molecules and molecular dynamics. The field overlaps significantly with physical chemistry, chemical physics, and quantum chemistry. It is often considered as a sub-field of atomic, molecular, and optical physics. Research groups studying molecular physics are typically designated as one of these other fields. Molecular physics addresses phenomena due to both molecular structure and individual atomic processes within molecules. Like atomic physics, it relies on a combination of classical and quantum mechanics to describe interactions between electromagnetic radiation and matter. Experiments in the field often rely heavily on techniques borrowed from atomic physics, such as spectroscopy and scattering. Molecular structure In a molecule, both the electrons and nuclei experience similar-scale forces from the Coulomb interaction. However, the nuclei remain at nearly fixed locations in the molecule while the electrons ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nuclear Physics
Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions, in addition to the study of other forms of nuclear matter. Nuclear physics should not be confused with atomic physics, which studies the atom as a whole, including its electrons. Discoveries in nuclear physics have led to applications in many fields such as nuclear power, nuclear weapons, nuclear medicine and magnetic resonance imaging, industrial and agricultural isotopes, ion implantation in materials engineering, and radiocarbon dating in geology and archaeology. Such applications are studied in the field of nuclear engineering. Particle physics evolved out of nuclear physics and the two fields are typically taught in close association. Nuclear astrophysics, the application of nuclear physics to astrophysics, is crucial in explaining the inner workings of stars and the origin of the chemical elements. History The history of nuclear physics as a discipline ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite-rank Operator
In functional analysis, a branch of mathematics, a finite-rank operator is a bounded linear operator between Banach spaces whose range is finite-dimensional. Finite-rank operators on a Hilbert space A canonical form Finite-rank operators are matrices (of finite size) transplanted to the infinite dimensional setting. As such, these operators may be described via linear algebra techniques. From linear algebra, we know that a rectangular matrix, with complex entries, M \in \mathbb^ has rank 1 if and only if M is of the form :M = \alpha \cdot u v^*, \quad \mbox \quad \, u \, = \, v\, = 1 \quad \mbox \quad \alpha \geq 0 . The same argument and Riesz' lemma show that an operator T on a Hilbert space H is of rank 1 if and only if :T h = \alpha \langle h, v\rangle u \quad \mbox \quad h \in H , where the conditions on \alpha, u, v are the same as in the finite dimensional case. Therefore, by induction, an operator T of finite rank n takes the form :T h = \sum _ ^n \alpha_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scattering Cross-section
In physics, the cross section is a measure of the probability that a specific process will take place in a collision of two particles. For example, the Rutherford cross-section is a measure of probability that an alpha particle will be deflected by a given angle during an interaction with an atomic nucleus. Cross section is typically denoted (sigma) and is expressed in units of area, more specifically in barns. In a way, it can be thought of as the size of the object that the excitation must hit in order for the process to occur, but more exactly, it is a parameter of a stochastic process. When two discrete particles interact in classical physics, their mutual cross section is the area transverse to their relative motion within which they must meet in order to scatter from each other. If the particles are hard inelastic spheres that interact only upon contact, their scattering cross section is related to their geometric size. If the particles interact through some action-a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hydrogen Atom
A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral hydrogen atom contains a single positively charged proton in the nucleus, and a single negatively charged electron bound to the nucleus by the Coulomb force. Atomic hydrogen constitutes about 75% of the baryonic mass of the universe. In everyday life on Earth, isolated hydrogen atoms (called "atomic hydrogen") are extremely rare. Instead, a hydrogen atom tends to combine with other atoms in compounds, or with another hydrogen atom to form ordinary (diatomic) hydrogen gas, H2. "Atomic hydrogen" and "hydrogen atom" in ordinary English use have overlapping, yet distinct, meanings. For example, a water molecule contains two hydrogen atoms, but does not contain atomic hydrogen (which would refer to isolated hydrogen atoms). Atomic spectroscopy shows that there is a discrete infinite set of states in which a hydrogen (or any) atom can exist, contrary to the predictions of classical physics. At ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electron
The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up quark, up and down quark, down quarks. Electrons are extremely lightweight particles that orbit the positively charged atomic nucleus, nucleus of atoms. Their negative charge is balanced by the positive charge of protons in the nucleus, giving atoms their overall electric charge#Charge neutrality, neutral charge. Ordinary matter is composed of atoms, each consisting of a positively charged nucleus surrounded by a number of orbiting electrons equal to the number of protons. The configuration and energy levels of these orbiting electrons determine the chemical properties of an atom. Electrons are bound to the nucleus to different degrees. The outermost or valence electron, valence electrons are the least tightly bound and are responsible for th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Null Space
In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the part of the domain which is mapped to the zero vector of the co-domain; the kernel is always a linear subspace of the domain. That is, given a linear map between two vector spaces and , the kernel of is the vector space of all elements of such that , where denotes the zero vector in , or more symbolically: \ker(L) = \left\ = L^(\mathbf). Properties The kernel of is a linear subspace of the domain .Linear algebra, as discussed in this article, is a very well established mathematical discipline for which there are many sources. Almost all of the material in this article can be found in , , and Strang's lectures. In the linear map L : V \to W, two elements of have the same image in if and only if their difference lies in the kernel of , that is, L\left(\mathbf_1\right) = L\left(\mathbf_2\right) \quad \text \quad L\left(\mathbf_1-\mathbf_2\right) = \mathbf. From this, it follows ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite-rank Operator
In functional analysis, a branch of mathematics, a finite-rank operator is a bounded linear operator between Banach spaces whose range is finite-dimensional. Finite-rank operators on a Hilbert space A canonical form Finite-rank operators are matrices (of finite size) transplanted to the infinite dimensional setting. As such, these operators may be described via linear algebra techniques. From linear algebra, we know that a rectangular matrix, with complex entries, M \in \mathbb^ has rank 1 if and only if M is of the form :M = \alpha \cdot u v^*, \quad \mbox \quad \, u \, = \, v\, = 1 \quad \mbox \quad \alpha \geq 0 . The same argument and Riesz' lemma show that an operator T on a Hilbert space H is of rank 1 if and only if :T h = \alpha \langle h, v\rangle u \quad \mbox \quad h \in H , where the conditions on \alpha, u, v are the same as in the finite dimensional case. Therefore, by induction, an operator T of finite rank n takes the form :T h = \sum _ ^n \alpha_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Padé Approximant
In mathematics, a Padé approximant is the "best" approximation of a function near a specific point by a rational function of given order. Under this technique, the approximant's power series agrees with the power series of the function it is approximating. The technique was developed around 1890 by Henri Padé, but goes back to Georg Frobenius, who introduced the idea and investigated the features of rational approximations of power series. The Padé approximant often gives better approximation of the function than truncating its Taylor series, and it may still work where the Taylor series does not converge. For these reasons Padé approximants are used extensively in computer calculations. They have also been used as auxiliary functions in Diophantine approximation and transcendental number theory, though for sharp results ad hoc methods—in some sense inspired by the Padé theory—typically replace them. Since a Padé approximant is a rational function, an artificial sin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
