Quantum Chemistry - Eric R. Bittner
... The field of quantum chemistry seeks to provide a rigorous description of chemical processes at its most fundamental level. For ordinary chemical processes, the most fundamental and underlying theory of chemistry is given by the time-dependent and time-independent version of the Schrödinger equatio ...
... The field of quantum chemistry seeks to provide a rigorous description of chemical processes at its most fundamental level. For ordinary chemical processes, the most fundamental and underlying theory of chemistry is given by the time-dependent and time-independent version of the Schrödinger equatio ...
An Introduction to Nonequilibrium Many
... These techniques seem quite powerful, so why can’t we apply them to the more general problem of Eq. (1.1)? Let us consider the Matsubara technique first. Out of equilibrium, there is no such thing as a temperature. As a result, in general the density matrix is not of exponential form, so there is no ...
... These techniques seem quite powerful, so why can’t we apply them to the more general problem of Eq. (1.1)? Let us consider the Matsubara technique first. Out of equilibrium, there is no such thing as a temperature. As a result, in general the density matrix is not of exponential form, so there is no ...
A Theoretical Study of Atomic Trimers in the Critical Stability Region
... three particles are complex, i.e. composed of other particles. This means that the total potential of the (three-body) system can no longer be written as a sum of atom-atom or nucleon-nucleon interactions, since the total effective potential is 1 There are however so called semi-classical methods [1 ...
... three particles are complex, i.e. composed of other particles. This means that the total potential of the (three-body) system can no longer be written as a sum of atom-atom or nucleon-nucleon interactions, since the total effective potential is 1 There are however so called semi-classical methods [1 ...
On the Energy Spectrum and Ground
... Here S i,k,m is the spin operator located on i-th lattice site of the unit cell labeled by index k, m is a number of the corresponding cyclic fragment; i = 1 corresponds to the site spin s1 and i = 2, 3 correspond to the site spins s2 . The parameter α represents a relative strength of the nearest-n ...
... Here S i,k,m is the spin operator located on i-th lattice site of the unit cell labeled by index k, m is a number of the corresponding cyclic fragment; i = 1 corresponds to the site spin s1 and i = 2, 3 correspond to the site spins s2 . The parameter α represents a relative strength of the nearest-n ...
Doubly excited nonautoionizing P, D, and F states of helium with
... sets of nonlinear parameters and then checked the convergence with the increasing number of basis terms. The results presented in Table I for P, Table II for D states, and Table III for F states are obtained with 共2.84, 2.34, 0.3兲, 共1.64, 0.84, 0.12兲, and 共1.40, 0.22, 0.3兲 as optimized 共A, B, C兲 par ...
... sets of nonlinear parameters and then checked the convergence with the increasing number of basis terms. The results presented in Table I for P, Table II for D states, and Table III for F states are obtained with 共2.84, 2.34, 0.3兲, 共1.64, 0.84, 0.12兲, and 共1.40, 0.22, 0.3兲 as optimized 共A, B, C兲 par ...
Single-Site Green-Function of the Dirac Equation for Full
... termed the microelectronics era. During this time, the world witnessed unprecedented and rapid changes in communication, information processing and information storing, starting from the earliest transistor up to having impressively powerful microprocessors in our mobile phones now, which would stil ...
... termed the microelectronics era. During this time, the world witnessed unprecedented and rapid changes in communication, information processing and information storing, starting from the earliest transistor up to having impressively powerful microprocessors in our mobile phones now, which would stil ...
Universal computation by multi-particle quantum walk
... • Applies generically to multi-particle quantum walks with indistinguishable particles. • Establishes the computational power of interacting many-body systems such as the BoseHubbard model, fermions with nearest neighbour interactions, and more. Our method for performing universal computation exploi ...
... • Applies generically to multi-particle quantum walks with indistinguishable particles. • Establishes the computational power of interacting many-body systems such as the BoseHubbard model, fermions with nearest neighbour interactions, and more. Our method for performing universal computation exploi ...
On the Distribution of the Wave Function for Systems in Thermal
... are represented by probability distributions on the phase space. In quantum mechanics, ensembles are usually represented by density matrices. It is natural to regard these density matrices as arising from probability distributions on the (normalized) wave functions associated with the thermodynamica ...
... are represented by probability distributions on the phase space. In quantum mechanics, ensembles are usually represented by density matrices. It is natural to regard these density matrices as arising from probability distributions on the (normalized) wave functions associated with the thermodynamica ...
Quantum random walks without walking
... We now construct our intended graph G by simply removing all the unwanted edges (dotted lines in Fig. 1a) from its complete counterpart Gmax . In turn this has the effect of removing some of the states from the Hilbert space H (dotted circles in Fig. 1c). Removing the edge ejj ′ for example, corresp ...
... We now construct our intended graph G by simply removing all the unwanted edges (dotted lines in Fig. 1a) from its complete counterpart Gmax . In turn this has the effect of removing some of the states from the Hilbert space H (dotted circles in Fig. 1c). Removing the edge ejj ′ for example, corresp ...
e - Physlab
... (a) Assume that an “orbiting electron wave” has the same energy an orbiting particle would have if at radius r and of momentum mv. Write an expression for this energy. (b) If the electron behaves as a classical particle, it must obey F = ma. Assuming circular orbit, apply F = ma to eliminate v in fa ...
... (a) Assume that an “orbiting electron wave” has the same energy an orbiting particle would have if at radius r and of momentum mv. Write an expression for this energy. (b) If the electron behaves as a classical particle, it must obey F = ma. Assuming circular orbit, apply F = ma to eliminate v in fa ...
Time-asymptotic wave propagation in collisionless plasmas
... computations, which apply O’Neil’s general method to various more sophisticated Ansätze for the field. This latter is assumed, again, to be a monochromatic wave, but the amplitude is allowed a slow variation; the crucial point is then the solution of the resulting Newton equations via various asymp ...
... computations, which apply O’Neil’s general method to various more sophisticated Ansätze for the field. This latter is assumed, again, to be a monochromatic wave, but the amplitude is allowed a slow variation; the crucial point is then the solution of the resulting Newton equations via various asymp ...
Atom-atom interactions in ultracold gases - cours en ligne CEL
... because they limit the achievable spatial densities of atoms). Collisions are essential for reaching thermal equilibrium At very low temperatures, mean-field descriptions of degenerate quantum gases depend only on a very small number of collisional parameters. For example, the shape and the dynamics ...
... because they limit the achievable spatial densities of atoms). Collisions are essential for reaching thermal equilibrium At very low temperatures, mean-field descriptions of degenerate quantum gases depend only on a very small number of collisional parameters. For example, the shape and the dynamics ...
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.