Multiphoton localization and propagating quantum gap solitons in a
... soliton in the space of frequencies [13], ωj = Ω + i γ2 (N + 1 − 2j) with E = ΩN. This also has a string structure. To avoid a possible confusion in what follows we use the term “string” for solutions of BAE in the h-space and the term “soliton” to refer to string’s images in the ω- and k-spaces. I ...
... soliton in the space of frequencies [13], ωj = Ω + i γ2 (N + 1 − 2j) with E = ΩN. This also has a string structure. To avoid a possible confusion in what follows we use the term “string” for solutions of BAE in the h-space and the term “soliton” to refer to string’s images in the ω- and k-spaces. I ...
Spectral And Dynamical Properties Of Strongly Correlated Systems
... variance with bosons, often uncontrolled approximations need to be introduced in their simulation. Fermions do however constitute a substantial part of the matter surrounding us and the description of most phenomena due to strong correlation suffers from the approximations needed in their numerical ...
... variance with bosons, often uncontrolled approximations need to be introduced in their simulation. Fermions do however constitute a substantial part of the matter surrounding us and the description of most phenomena due to strong correlation suffers from the approximations needed in their numerical ...
Supmech: the Geometro-statistical Formalism Underlying Quantum
... a symplectic superalgebra and operates essentially as noncommutative Hamiltonian mechanics with an extra condition of ‘compatible completeness’ between observables and pure states incorporated. A noncommutative analogue of the Poincaré-Cartan form is introduced. It is shown that interactions betwee ...
... a symplectic superalgebra and operates essentially as noncommutative Hamiltonian mechanics with an extra condition of ‘compatible completeness’ between observables and pure states incorporated. A noncommutative analogue of the Poincaré-Cartan form is introduced. It is shown that interactions betwee ...
Charge Transport in Semiconductors Contents
... the presence of an electric field. This happens since the electrons move within the bands in endless cycles, and a sum of the velocities due to all electrons is zero, since for every electron with a positive velocity, there is one with an equivalent negative velocity. Strange things happen if one con ...
... the presence of an electric field. This happens since the electrons move within the bands in endless cycles, and a sum of the velocities due to all electrons is zero, since for every electron with a positive velocity, there is one with an equivalent negative velocity. Strange things happen if one con ...
Quantum computation and cryptography: an overview
... alternatives wj ehSj ; whose action gap DS ¼ Sj Sk is of the order of the Planck constant h; coexist in some sort of quantum superposition with complex weights, like (2). These ...
... alternatives wj ehSj ; whose action gap DS ¼ Sj Sk is of the order of the Planck constant h; coexist in some sort of quantum superposition with complex weights, like (2). These ...
C.P. Boyer y K.B. Wolf, Canonical transforms. III. Configuration and
... harmonic oscillator. Although the complete dynamical groups for the two systems are different (the symplectic group Sp(n, R) for the oscillator and O(n, 2) for the Coulomb system), the representations of the SL(2, R) subgroup are isomorphic ally related and appear to playa fundamental role in both s ...
... harmonic oscillator. Although the complete dynamical groups for the two systems are different (the symplectic group Sp(n, R) for the oscillator and O(n, 2) for the Coulomb system), the representations of the SL(2, R) subgroup are isomorphic ally related and appear to playa fundamental role in both s ...
introduction to quantum field theory
... described by some sort of field theory. Almost all systems in nature involve an extremely large number of degrees of freedom. For instance, a droplet of water contains of the order of 1026 molecules and while each water molecule can in many applications be described as a point particle, each molecul ...
... described by some sort of field theory. Almost all systems in nature involve an extremely large number of degrees of freedom. For instance, a droplet of water contains of the order of 1026 molecules and while each water molecule can in many applications be described as a point particle, each molecul ...
Adiabatic Continuation of Fractional Chern Insulators to Fractional
... q1 ; q2 Þ, and mutatis mutandis for ¼0 Ax ð~ Any ðq1 ; q2 Þ. This resolution of (2) yields a unitary transformation of the original single particle basis. As angles, the values of Anx ðq1 ; q2 Þ are defined modulo 2, and we ensure that the shift in x position x ¼ ðky Þ=2 satisfies 0 x < 1. Th ...
... q1 ; q2 Þ, and mutatis mutandis for ¼0 Ax ð~ Any ðq1 ; q2 Þ. This resolution of (2) yields a unitary transformation of the original single particle basis. As angles, the values of Anx ðq1 ; q2 Þ are defined modulo 2, and we ensure that the shift in x position x ¼ ðky Þ=2 satisfies 0 x < 1. Th ...
Schroedinger`s Model of Hydrogen Atom
... by classical Newtonian particle laws. We will start to look at this new situation in the next subsection, but first we want to investigate how the new situation gives an insight into the existence of restrictions on the allowed orbits in an atom. In the case of the Bohr model, these were postulated. ...
... by classical Newtonian particle laws. We will start to look at this new situation in the next subsection, but first we want to investigate how the new situation gives an insight into the existence of restrictions on the allowed orbits in an atom. In the case of the Bohr model, these were postulated. ...
Foundations and Measures of Quantum Non
... The standard approach to the theoretical description and analysis of dissipation and decoherence processes in open quantum systems presupposes a time evolution without memory. Employing the concept of a quantum Markov process which is given by a semigroup of completely positive dynamical maps one ob ...
... The standard approach to the theoretical description and analysis of dissipation and decoherence processes in open quantum systems presupposes a time evolution without memory. Employing the concept of a quantum Markov process which is given by a semigroup of completely positive dynamical maps one ob ...
Wave function
A wave function in quantum mechanics describes the quantum state of an isolated system of one or more particles. There is one wave function containing all the information about the entire system, not a separate wave function for each particle in the system. Its interpretation is that of a probability amplitude. Quantities associated with measurements, such as the average momentum of a particle, can be derived from the wave function. It is a central entity in quantum mechanics and is important in all modern theories, like quantum field theory incorporating quantum mechanics, while its interpretation may differ. The most common symbols for a wave function are the Greek letters ψ or Ψ (lower-case and capital psi).For a given system, once a representation corresponding to a maximal set of commuting observables and a suitable coordinate system is chosen, the wave function is a complex-valued function of the system's degrees of freedom corresponding to the chosen representation and coordinate system, continuous as well as discrete. Such a set of observables, by a postulate of quantum mechanics, are Hermitian linear operators on the space of states representing a set of physical observables, like position, momentum and spin that can, in principle, be simultaneously measured with arbitrary precision. Wave functions can be added together and multiplied by complex numbers to form new wave functions, and hence are elements of a vector space. This is the superposition principle of quantum mechanics. This vector space is endowed with an inner product such that it is a complete metric topological space with respect to the metric induced by the inner product. In this way the set of wave functions for a system form a function space that is a Hilbert space. The inner product is a measure of the overlap between physical states and is used in the foundational probabilistic interpretation of quantum mechanics, the Born rule, relating transition probabilities to inner products. The actual space depends on the system's degrees of freedom (hence on the chosen representation and coordinate system) and the exact form of the Hamiltonian entering the equation governing the dynamical behavior. In the non-relativistic case, disregarding spin, this is the Schrödinger equation.The Schrödinger equation determines the allowed wave functions for the system and how they evolve over time. A wave function behaves qualitatively like other waves, such as water waves or waves on a string, because the Schrödinger equation is mathematically a type of wave equation. This explains the name ""wave function"", and gives rise to wave–particle duality. The wave of the wave function, however, is not a wave in physical space; it is a wave in an abstract mathematical ""space"", and in this respect it differs fundamentally from water waves or waves on a string.For a given system, the choice of which relevant degrees of freedom to use are not unique, and correspondingly the domain of the wave function is not unique. It may be taken to be a function of all the position coordinates of the particles over position space, or the momenta of all the particles over momentum space, the two are related by a Fourier transform. These descriptions are the most important, but they are not the only possibilities. Just like in classical mechanics, canonical transformations may be used in the description of a quantum system. Some particles, like electrons and photons, have nonzero spin, and the wave function must include this fundamental property as an intrinsic discrete degree of freedom. In general, for a particle with half-integer spin the wave function is a spinor, for a particle with integer spin the wave function is a tensor. Particles with spin zero are called scalar particles, those with spin 1 vector particles, and more generally for higher integer spin, tensor particles. The terminology derives from how the wave functions transform under a rotation of the coordinate system. No elementary particle with spin 3⁄2 or higher is known, except for the hypothesized spin 2 graviton. Other discrete variables can be included, such as isospin. When a system has internal degrees of freedom, the wave function at each point in the continuous degrees of freedom (e.g. a point in space) assigns a complex number for each possible value of the discrete degrees of freedom (e.g. z-component of spin). These values are often displayed in a column matrix (e.g. a 2 × 1 column vector for a non-relativistic electron with spin 1⁄2).In the Copenhagen interpretation, an interpretation of quantum mechanics, the squared modulus of the wave function, |ψ|2, is a real number interpreted as the probability density of measuring a particle as being at a given place at a given time or having a definite momentum, and possibly having definite values for discrete degrees of freedom. The integral of this quantity, over all the system's degrees of freedom, must be 1 in accordance with the probability interpretation, this general requirement a wave function must satisfy is called the normalization condition. Since the wave function is complex valued, only its relative phase and relative magnitude can be measured. Its value does not in isolation tell anything about the magnitudes or directions of measurable observables; one has to apply quantum operators, whose eigenvalues correspond to sets of possible results of measurements, to the wave function ψ and calculate the statistical distributions for measurable quantities.The unit of measurement for ψ depends on the system, and can be found by dimensional analysis of the normalization condition for the system. For one particle in three dimensions, its units are [length]−3/2, because an integral of |ψ|2 over a region of three-dimensional space is a dimensionless probability.