Probability distributions in classical and quantum
... a connection between classical and quantum mechanics, we compare the probability distributions to show that both approach each other as the energy increases. Our results are compared with respect to known literature of elliptic billiards (see Refs. 6-9, and 11, and references therein). This work is ...
... a connection between classical and quantum mechanics, we compare the probability distributions to show that both approach each other as the energy increases. Our results are compared with respect to known literature of elliptic billiards (see Refs. 6-9, and 11, and references therein). This work is ...
Calculation of absolute scattering phase shifts
... approximate it by the first few terms of an asymptotic series expansion in 1/k using equation (9.2.29) of [8], yielding values denoted herein as Yae ‘ ðkrÞ. That approach usually gives satisfactory results. However, in the course of performing calculations of interaction virial and diffusion coefficie ...
... approximate it by the first few terms of an asymptotic series expansion in 1/k using equation (9.2.29) of [8], yielding values denoted herein as Yae ‘ ðkrÞ. That approach usually gives satisfactory results. However, in the course of performing calculations of interaction virial and diffusion coefficie ...
`Quantum Cheshire Cat`as Simple Quantum Interference
... make projective measurements, there are three possibilities for each photon: either the photon has an up displacement greater than the beam diameter or the photon has a positive (negative) horizontal displacement greater than the beam diameter. In these cases, if one assumes a real trajectory for th ...
... make projective measurements, there are three possibilities for each photon: either the photon has an up displacement greater than the beam diameter or the photon has a positive (negative) horizontal displacement greater than the beam diameter. In these cases, if one assumes a real trajectory for th ...
Instructions for use Title Coulomb staircase and total spin
... quantum dot1,2 and has recently been extended to roles of the spin degree of freedom in this system. A transition of a spin singlet to triplet state,3,4 a triplet state due to Hund’s rule,5 and the even larger spin polarization6,7 have been observed. The Kondo effect has been observed when the local ...
... quantum dot1,2 and has recently been extended to roles of the spin degree of freedom in this system. A transition of a spin singlet to triplet state,3,4 a triplet state due to Hund’s rule,5 and the even larger spin polarization6,7 have been observed. The Kondo effect has been observed when the local ...
A mechanistic classical laboratory situation violating the Bell
... This shows that Bell inequalities are violated, and if we compare with the quantum mechanical violation, we see that our entity violates the Bell inequalities 'more' than the quantum mechanical entity of the two spin 1/2 particles in the singlet spin state. If we present our second classical example ...
... This shows that Bell inequalities are violated, and if we compare with the quantum mechanical violation, we see that our entity violates the Bell inequalities 'more' than the quantum mechanical entity of the two spin 1/2 particles in the singlet spin state. If we present our second classical example ...
The quantum does not reduce to discrete bits
... One purpose to hidden variable theories is to give a mathematical realization of probabilities. If the observables are random variables, then it is natural to assume that they are functions of some measure-one parameter space. If some physical significance to that parameter space can be found, so mu ...
... One purpose to hidden variable theories is to give a mathematical realization of probabilities. If the observables are random variables, then it is natural to assume that they are functions of some measure-one parameter space. If some physical significance to that parameter space can be found, so mu ...
(Dynamical) quantum typicality: What is it and what are its
... Why it should occur: We see it in system we assume to be closed. Why it should not occur: There are issues with the underlying theory: Quantum Mechanics (Non-eq.) Thermodynamics autonomous dynamics of a few macrovariables attractive fixed point, equilibrium often describable by master equations, Fok ...
... Why it should occur: We see it in system we assume to be closed. Why it should not occur: There are issues with the underlying theory: Quantum Mechanics (Non-eq.) Thermodynamics autonomous dynamics of a few macrovariables attractive fixed point, equilibrium often describable by master equations, Fok ...
Continuous Matrix Product States for Quantum Fields
... be adopted to describe quantum field theories. We will define a new family of states that we call continuous MPS (CMPS) that describe field theories in 1 spatial dimension. We will also show that CMPS can be understood as the continuous limit of standard MPS. Those CMPS can be used as variational st ...
... be adopted to describe quantum field theories. We will define a new family of states that we call continuous MPS (CMPS) that describe field theories in 1 spatial dimension. We will also show that CMPS can be understood as the continuous limit of standard MPS. Those CMPS can be used as variational st ...
An amusing analogy: modelling quantum
... the state of both particles, no matter how far apart they may be. The second observation (measuring the same spin component on the other particle) just reads the already collapsed state, it does not perturb it. We thus see that the formalism predicts the occurrence of an ‘instantaneous influence’ (o ...
... the state of both particles, no matter how far apart they may be. The second observation (measuring the same spin component on the other particle) just reads the already collapsed state, it does not perturb it. We thus see that the formalism predicts the occurrence of an ‘instantaneous influence’ (o ...
Nilpotence - Nature`s Code Foundation
... since as Jessel shows such Heaviside operators are fundamental to the formalization of Huygens’ Principle [9]. Thus it seems that the concept of nilpotence (or the empty set as the description of the initial mathematical state of a system as used by Conway in relation to universal models) is founda ...
... since as Jessel shows such Heaviside operators are fundamental to the formalization of Huygens’ Principle [9]. Thus it seems that the concept of nilpotence (or the empty set as the description of the initial mathematical state of a system as used by Conway in relation to universal models) is founda ...
Lie Groups and Quantum Mechanics
... direct path to this goal, attempting to make it appear “naturally”. Once we do have it, the urge to explore some of the related topology is irresistible. Next comes physics. Usually introductory quantum mechanics starts off with things like wave/particle duality, the Heisenberg uncertainty principle, ...
... direct path to this goal, attempting to make it appear “naturally”. Once we do have it, the urge to explore some of the related topology is irresistible. Next comes physics. Usually introductory quantum mechanics starts off with things like wave/particle duality, the Heisenberg uncertainty principle, ...
DOC - University of Colorado Boulder
... This trick is way out there! I would never have come up with it (?) but it's cool, and turns out to be more general than you can imagine right now. It's the basis for a similar trick to understand angular momentum in 3-D, and then spin, and moving on to quantum field theory. So it's worth learning! ...
... This trick is way out there! I would never have come up with it (?) but it's cool, and turns out to be more general than you can imagine right now. It's the basis for a similar trick to understand angular momentum in 3-D, and then spin, and moving on to quantum field theory. So it's worth learning! ...
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.