Floquet topological insulator in semiconductor
... where ± (k) are the energies corresponding to P̌± (k). In the ‘rotating’ picture, the two bands cross if ω is larger than the gap M . HI is solved by the eigenstates |ψI ± (k)i, which for the values of momenta, k, away from the resonance ring are only weakly modified compared to the equilibrium, V̌ ...
... where ± (k) are the energies corresponding to P̌± (k). In the ‘rotating’ picture, the two bands cross if ω is larger than the gap M . HI is solved by the eigenstates |ψI ± (k)i, which for the values of momenta, k, away from the resonance ring are only weakly modified compared to the equilibrium, V̌ ...
Experimental verification of the Heisenberg uncertainty principle for
... To trace out the uncertainty relation we varied the width of the second slit from about 20 m down to roughly 50 nm and record the molecular-beam width in the detection plane. In Fig. 2, we show the measured molecular-beam profiles as full circles for two different widths of the second collimation ...
... To trace out the uncertainty relation we varied the width of the second slit from about 20 m down to roughly 50 nm and record the molecular-beam width in the detection plane. In Fig. 2, we show the measured molecular-beam profiles as full circles for two different widths of the second collimation ...
Backup of MajorFileds070805jrv.wbk
... Potential energies are associated with systems of interacting entities. Potential energies are associated with pairs of things (at least) while a kinetic energy can be owned by a single entity. The potential energy can be shifted by an additive constant. For SHM, it is often set to zero for a partic ...
... Potential energies are associated with systems of interacting entities. Potential energies are associated with pairs of things (at least) while a kinetic energy can be owned by a single entity. The potential energy can be shifted by an additive constant. For SHM, it is often set to zero for a partic ...
Contradiction within Paraxial Wave Optics and its - LAS
... photon is only valid if the phase fronts are represented by real mirrors, since otherwise no real force is exerted on the photon. Concerning this objection two arguments can be indicated. First, the properties of a propagating wave are not changed, if it is assumed that this wave is bouncing between ...
... photon is only valid if the phase fronts are represented by real mirrors, since otherwise no real force is exerted on the photon. Concerning this objection two arguments can be indicated. First, the properties of a propagating wave are not changed, if it is assumed that this wave is bouncing between ...
Theory and simulations of quantum glass forming liquids
... slowdown of dynamics when a liquid transforms into a glass is still a subject of great debate.1–4 Essentially all discussion of the glass transition has focused on the strictly classical regime of liquid state behavior, namely where the de Broglie wavelength is significantly smaller than the particl ...
... slowdown of dynamics when a liquid transforms into a glass is still a subject of great debate.1–4 Essentially all discussion of the glass transition has focused on the strictly classical regime of liquid state behavior, namely where the de Broglie wavelength is significantly smaller than the particl ...
A simple experiment for discussion of quantum interference and
... measurement variable. In other words, if we follow a first, nontrivial measurement 共nontrivial meaning it can yield more than one possible result兲 with a second measurement which reproduces the original measured value with 100% certainty, then a measurement must have occurred. Issues relating to qua ...
... measurement variable. In other words, if we follow a first, nontrivial measurement 共nontrivial meaning it can yield more than one possible result兲 with a second measurement which reproduces the original measured value with 100% certainty, then a measurement must have occurred. Issues relating to qua ...
Essay Review of Quantum State Diffusion by Ian Percival
... to an eigenstate of an operator being measured. This localization is very important and it is discussed in detail later on in the book. The construction of a state evolution which on average reproduces a particular master equation is often referred to as an unravelling of that master equation. Altho ...
... to an eigenstate of an operator being measured. This localization is very important and it is discussed in detail later on in the book. The construction of a state evolution which on average reproduces a particular master equation is often referred to as an unravelling of that master equation. Altho ...
Quantum walk as a generalized measuring device
... state discrimination problem. Generation of arbitrary rank 1 POVM elements. We focus on rank 1 POVMs, since higher rank POVMs can be constructed as a convex combination of rank 1 elements. We will come back to this problem at the end of the next section. In case of a single qubit the orthogonality r ...
... state discrimination problem. Generation of arbitrary rank 1 POVM elements. We focus on rank 1 POVMs, since higher rank POVMs can be constructed as a convex combination of rank 1 elements. We will come back to this problem at the end of the next section. In case of a single qubit the orthogonality r ...
Understanding degenerate ground states of a protected
... θ = π/2. Nevertheless, the wave functions shown in Fig. 3 are qualitatively similar between the toy model and the actual potential. Which of these two types is formed generally depends on the parameters EL , EC , and magnetic flux. As long as the magnetic flux is away from half-integer flux quanta, ...
... θ = π/2. Nevertheless, the wave functions shown in Fig. 3 are qualitatively similar between the toy model and the actual potential. Which of these two types is formed generally depends on the parameters EL , EC , and magnetic flux. As long as the magnetic flux is away from half-integer flux quanta, ...
The de Broglie Wave as Evidence of a Deeper Wave Structure
... In the rest frame of the particle, this underlying structure has the form of a standing wave2 . So regarded, the de Broglie wave is not itself, strictly speaking, the matter wave of quantum mechanics, but evidences the existence of a deeper wave structure more deserving of that title. If, consisten ...
... In the rest frame of the particle, this underlying structure has the form of a standing wave2 . So regarded, the de Broglie wave is not itself, strictly speaking, the matter wave of quantum mechanics, but evidences the existence of a deeper wave structure more deserving of that title. If, consisten ...
Lecture Notes on Statistical Mechanics and Thermodynamics
... 1. Introduction and Historical Overview As the name suggests, thermodynamics historically developed as an attempt to understand phenomena involving heat. This notion is intimately related to irreversible processes involving typically many, essentially randomly excited, degrees of freedom. The prope ...
... 1. Introduction and Historical Overview As the name suggests, thermodynamics historically developed as an attempt to understand phenomena involving heat. This notion is intimately related to irreversible processes involving typically many, essentially randomly excited, degrees of freedom. The prope ...
Rigorous results in electronic structure calculations
... ×Ψ(x10 , x20 , x3 , . . . , xN ) dx3 · · · xN . The Born-Oppenheimer approximation reduces the problem to find the ground state energy of an N-electron system to the problem of searching the two-electron density matrix which gives the lowest energy expectation value. This problem is also known under ...
... ×Ψ(x10 , x20 , x3 , . . . , xN ) dx3 · · · xN . The Born-Oppenheimer approximation reduces the problem to find the ground state energy of an N-electron system to the problem of searching the two-electron density matrix which gives the lowest energy expectation value. This problem is also known under ...
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.