Spontaneous symmetry breaking of solitons trapped in a double
... on the total norm, n. The way the figure was generated (through direct simulations converging to stationary states) made is possible to display only stable branches of the solutions, both symmetric and asymmetric ones. Although the unstable branches are missing, there is little doubt that the full S ...
... on the total norm, n. The way the figure was generated (through direct simulations converging to stationary states) made is possible to display only stable branches of the solutions, both symmetric and asymmetric ones. Although the unstable branches are missing, there is little doubt that the full S ...
Nonequilibrium Quantum Magnetism in a Dipolar Lattice Gas
... this Heisenberg-like Hamiltonian: a chromium gas loaded in a 3D lattice provides an interesting platform for the analysis of quantum magnetism. In our experiment, we first create a chromium BoseEinstein condensate in a crossed-beam optical dipole trap as described in Ref. [10]. The condensate, compr ...
... this Heisenberg-like Hamiltonian: a chromium gas loaded in a 3D lattice provides an interesting platform for the analysis of quantum magnetism. In our experiment, we first create a chromium BoseEinstein condensate in a crossed-beam optical dipole trap as described in Ref. [10]. The condensate, compr ...
Motion in a Straight Line
... Just for Quantum Mechanics course The expectation value is interpreted as the average value of x that we would expect to obtain from a large number of measurements. ...
... Just for Quantum Mechanics course The expectation value is interpreted as the average value of x that we would expect to obtain from a large number of measurements. ...
propagation methods for quantum molecular dynamics
... space, such as the potential, are calculated on the grid points while nonlocal operators, such as the kinetic energy operator, are calculated in the functional space (27). They are then recast into the same grid points. The discretization schemedetermines the domainof values of the operator I~. For ...
... space, such as the potential, are calculated on the grid points while nonlocal operators, such as the kinetic energy operator, are calculated in the functional space (27). They are then recast into the same grid points. The discretization schemedetermines the domainof values of the operator I~. For ...
Chapter 14
... pulling the chains at the same point in each swing, or a friend gives you repeated pushes at just the right times. Resonance occurs when small forces are applied at regular intervals to a vibrating or oscillating object and the amplitude of the vibration increases. The time interval between applicat ...
... pulling the chains at the same point in each swing, or a friend gives you repeated pushes at just the right times. Resonance occurs when small forces are applied at regular intervals to a vibrating or oscillating object and the amplitude of the vibration increases. The time interval between applicat ...
7. The Hydrogen Atom in Wave Mechanics
... form, V (~x) = V (r). The Yl,ml (θ, φ)’s are the eigenfunctions of the angular part of the Laplacian operator, L, with l(l + 1) being its eigenvalue. Elements of Nuclear Engineering and Radiological Sciences I ...
... form, V (~x) = V (r). The Yl,ml (θ, φ)’s are the eigenfunctions of the angular part of the Laplacian operator, L, with l(l + 1) being its eigenvalue. Elements of Nuclear Engineering and Radiological Sciences I ...
Electronic Structure of Clusters and Nanocrystals
... but considerable progress has been made since the advent of reliable digit computers. A number of highly successful approximations have been made to solve for the both the ground state and excited state energies. For the most part, these approximations used are to remove as many “irrelevant” degrees ...
... but considerable progress has been made since the advent of reliable digit computers. A number of highly successful approximations have been made to solve for the both the ground state and excited state energies. For the most part, these approximations used are to remove as many “irrelevant” degrees ...
Thesis - Institut für Physik
... MCTDHB(M ) method reaches its limits in form of computational time and memory even for single digit values of M on typical workstations because of the exponential growth of the Hilbert space. In order to overcome these limitations and especially to take into account four and more modes in our simula ...
... MCTDHB(M ) method reaches its limits in form of computational time and memory even for single digit values of M on typical workstations because of the exponential growth of the Hilbert space. In order to overcome these limitations and especially to take into account four and more modes in our simula ...
Condensed Matter Approaches to Quantum Gases
... Here âν and â†ν are (Schrödinger) annihilation and creation operators of an excitation characterized by a set of quantum numbers ν. The Hamiltonian Ĥµ takes the form (18) if the functions uν , vν ...
... Here âν and â†ν are (Schrödinger) annihilation and creation operators of an excitation characterized by a set of quantum numbers ν. The Hamiltonian Ĥµ takes the form (18) if the functions uν , vν ...
Two-body momentum correlations in a weakly interacting one
... [3, 24–28], scale invariance of two-dimensional (2D) systems [29], the phase diagram of the 1D Bose gas [30, 31], entanglement and spin squeezing in two-component and double-well systems [32–36], sub-Poissonian relative atom number statistics [35, 37, 38], and violation of the Cauchy-Schwarz inequal ...
... [3, 24–28], scale invariance of two-dimensional (2D) systems [29], the phase diagram of the 1D Bose gas [30, 31], entanglement and spin squeezing in two-component and double-well systems [32–36], sub-Poissonian relative atom number statistics [35, 37, 38], and violation of the Cauchy-Schwarz inequal ...
Spin-valley lifetimes in a silicon quantum dot with tunable valley
... j2i ¼ jv ; "i, j3i ¼ jv þ ; #i, j4i ¼ jv þ ; "i. These states are considered to be only very weakly affected by higher excitations, such as orbital levels that are at least 8 meV above the ground state in our device41. In Supplementary Note 3 we detail how mixing to a 2p-like orbital state leads t ...
... j2i ¼ jv ; "i, j3i ¼ jv þ ; #i, j4i ¼ jv þ ; "i. These states are considered to be only very weakly affected by higher excitations, such as orbital levels that are at least 8 meV above the ground state in our device41. In Supplementary Note 3 we detail how mixing to a 2p-like orbital state leads t ...
FEYNMANWS PATH INTEGRAL APPROACH TO QUANTUM FIELD
... function 3 (x00 x0 ) (t00 t0 ). In fact, for t00 = t0 it is precisely the delta function 3 (x00 x0 ). However, when t00 6= t0 we give it the name propagator. The propagator hx00 ; t00 jx0 ; t0 i is actually a Green’s function that determines how the wave function develops in time and space (it’s oft ...
... function 3 (x00 x0 ) (t00 t0 ). In fact, for t00 = t0 it is precisely the delta function 3 (x00 x0 ). However, when t00 6= t0 we give it the name propagator. The propagator hx00 ; t00 jx0 ; t0 i is actually a Green’s function that determines how the wave function develops in time and space (it’s oft ...
Statistical Physics Notes
... One question worth thinking about is how we end up with probabilities. We wouldn’t need probability theory if we carry out Newton’s plan exactly. Note that the first thing we drop to come over the obstacles is to drop initial condition because it’s impossible to write down the initial condition of a ...
... One question worth thinking about is how we end up with probabilities. We wouldn’t need probability theory if we carry out Newton’s plan exactly. Note that the first thing we drop to come over the obstacles is to drop initial condition because it’s impossible to write down the initial condition of a ...
Spin-orbit coupling in superconductor-normal metal
... The spin-orbit (SO) coupling gives rise to a large splitting of the subband energy levels in semiconducting heterostructures. Both theoretical and experimental interest towards SO interactions in superconductors and superconducting heterostructures has been on the rise due to new experimental findin ...
... The spin-orbit (SO) coupling gives rise to a large splitting of the subband energy levels in semiconducting heterostructures. Both theoretical and experimental interest towards SO interactions in superconductors and superconducting heterostructures has been on the rise due to new experimental findin ...
Quantum Field Theory I
... and leptons, the massless vector propagator describes photon and gluons, the scalar propagator describes the Higgs boson. The only missing propagator is the massive vector one, describing the W ± and Z 0 bosons. This can be, however, ...
... and leptons, the massless vector propagator describes photon and gluons, the scalar propagator describes the Higgs boson. The only missing propagator is the massive vector one, describing the W ± and Z 0 bosons. This can be, however, ...
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.