Non-relativistic limit in the 2+ 1 Dirac Oscillator: A Ramsey
... the Dirac oscillator in different contexts. Beyond its exact solvability, the energy spectrum presents certain peculiar degeneracies which can be related to a non-trivial symmetry Lie algebra [4]. Furthermore, its solvability can be traced back to an exact Foldy-Wouthuysen transformation [5], and it ...
... the Dirac oscillator in different contexts. Beyond its exact solvability, the energy spectrum presents certain peculiar degeneracies which can be related to a non-trivial symmetry Lie algebra [4]. Furthermore, its solvability can be traced back to an exact Foldy-Wouthuysen transformation [5], and it ...
entanglement properties of quantum many
... Our study of the entanglement properties of correlated wave functions for spin lattices was motivated by the work of Osterloh et al. [26] and more especially that of Osborne and Nielsen (ON) [27]. These authors were the rst to explore possible connections between quantum phase transitions and entan ...
... Our study of the entanglement properties of correlated wave functions for spin lattices was motivated by the work of Osterloh et al. [26] and more especially that of Osborne and Nielsen (ON) [27]. These authors were the rst to explore possible connections between quantum phase transitions and entan ...
Effective Field Theories for Topological states of Matter
... Z index, or the values ±1 which is a Z2 index. If we change the Hamiltonian, without changing its symmetry, the index can change only at points where the gap to excited states vanishes. A symmetry-protected topological phase (SPT), is a nontrivial phase which however can be connected to the trivial ...
... Z index, or the values ±1 which is a Z2 index. If we change the Hamiltonian, without changing its symmetry, the index can change only at points where the gap to excited states vanishes. A symmetry-protected topological phase (SPT), is a nontrivial phase which however can be connected to the trivial ...
Particle-wave duality - Proceedings of the Royal Society A
... can be quantified without reference to the details of any interferometer. We can, however, relate symmetry directly to interference in the following way. Taking the double slit experiment as the prototypical interferometer, we note that the positions of interference fringes on the screen can be chang ...
... can be quantified without reference to the details of any interferometer. We can, however, relate symmetry directly to interference in the following way. Taking the double slit experiment as the prototypical interferometer, we note that the positions of interference fringes on the screen can be chang ...
Conf. Ser. 724 (2016) 012029 1 - The Racah Institute of Physics
... Symmetries play an important role in the understanding of complex systems. However, complete dynamical symmetries (DS) are manifest in only a few nuclei. Generic Hamiltonians involve interaction terms with competing incompatible symmetries, which break the DS. More often one finds that the assumed s ...
... Symmetries play an important role in the understanding of complex systems. However, complete dynamical symmetries (DS) are manifest in only a few nuclei. Generic Hamiltonians involve interaction terms with competing incompatible symmetries, which break the DS. More often one finds that the assumed s ...
Multidimensional Hypergeometric Functions in Conformai Field
... where co is a differential form of the Orlik-Solomon algebra. Hypergeometric forms form a finite-dimensional complex, as the differential of a hypergeometric form is a hypergeometric form: d(lao>) = L E «(//) dfH/fH A co. The weight local system ^(a) on the complement of a configuration is the compl ...
... where co is a differential form of the Orlik-Solomon algebra. Hypergeometric forms form a finite-dimensional complex, as the differential of a hypergeometric form is a hypergeometric form: d(lao>) = L E «(//) dfH/fH A co. The weight local system ^(a) on the complement of a configuration is the compl ...
Derivation of the Lindblad Equation for Open Quantum Systems and
... Before we get there, we present the derivation of the Lindblad equation. In quantum mechanics, the time evolution of a quantum state of an isolated system is described by the Schrödinger equation. The density operator describes a quantum system in a mixed state, and the Liouville-von Neumann equati ...
... Before we get there, we present the derivation of the Lindblad equation. In quantum mechanics, the time evolution of a quantum state of an isolated system is described by the Schrödinger equation. The density operator describes a quantum system in a mixed state, and the Liouville-von Neumann equati ...
Quantum HPC Sweden
... ▪ Quantum speedup can only be realized if the evolution exp(-iAt) can be implemented using a short circuit, i.e. it does not depend on lots of data ▪ Electromagnetic wave scattering problem (Clader et al, PRL, 2013) ...
... ▪ Quantum speedup can only be realized if the evolution exp(-iAt) can be implemented using a short circuit, i.e. it does not depend on lots of data ▪ Electromagnetic wave scattering problem (Clader et al, PRL, 2013) ...
Real-time resolution of the causality paradox of time
... v共r , t兲 and 兩共t兲典 are functionals of n共r , t兲 and 兩0典 over a time interval 0 ⱕ t ⱕ T. They should be denoted by v关n , 兩0典 ; r , t兴 and 兩关n , 兩0典兴典, respectively. From now on, however, the dependence on the initial state will not be explicitly noted and we will simply write v关n ; r , t兴 and 兩关 ...
... v共r , t兲 and 兩共t兲典 are functionals of n共r , t兲 and 兩0典 over a time interval 0 ⱕ t ⱕ T. They should be denoted by v关n , 兩0典 ; r , t兴 and 兩关n , 兩0典兴典, respectively. From now on, however, the dependence on the initial state will not be explicitly noted and we will simply write v关n ; r , t兴 and 兩关 ...
On some log-cosine integrals related to (3), (4), and (6)
... [7,10,15,12]. We show that this integral may be alternatively evaluated starting from a known tabulated result [9]. The derivation yields results suitable for certain classes of logarithmic-trigonometric integrals. We then describe in Appendix A how some integrals related to 2 (3) may be further us ...
... [7,10,15,12]. We show that this integral may be alternatively evaluated starting from a known tabulated result [9]. The derivation yields results suitable for certain classes of logarithmic-trigonometric integrals. We then describe in Appendix A how some integrals related to 2 (3) may be further us ...
Lecture Notes Chapters 1-7
... quantitative calculations of the physical properties of solids has been one of the enduring agendas of condensed matter physics and the methods have acquired increasing sophistication over the years. We would like to understand the cohesion of solids – why it is, for example that mercury is a liquid ...
... quantitative calculations of the physical properties of solids has been one of the enduring agendas of condensed matter physics and the methods have acquired increasing sophistication over the years. We would like to understand the cohesion of solids – why it is, for example that mercury is a liquid ...
Non-exponential and oscillatory decays in quantum mechanics
... the calculation is known. However they are silent about the range of times during which the decay is exponential. Moreover, they define the survival probability of the initial state Ψ(0)as S(t) = |hΨ(0)|Ψ(t)i|2 . ...
... the calculation is known. However they are silent about the range of times during which the decay is exponential. Moreover, they define the survival probability of the initial state Ψ(0)as S(t) = |hΨ(0)|Ψ(t)i|2 . ...
A quantum delayed choice experiment
... with a Schrödinger cat-like BS R1 is produced by the field stored in a resonator, R2(θ) produced by a classical field. When the resonator is filled with a coherent field ( ), R1 is ...
... with a Schrödinger cat-like BS R1 is produced by the field stored in a resonator, R2(θ) produced by a classical field. When the resonator is filled with a coherent field ( ), R1 is ...
Non-abelian quantum Hall states and fractional charges in one dimension Emma Wikberg
... Chapters 1-5 in this thesis are modified versions of the corresponding sections in Non-abelian Quantum Hall states on the Thin Torus, Licentiate thesis, E. Wikberg (2009). ...
... Chapters 1-5 in this thesis are modified versions of the corresponding sections in Non-abelian Quantum Hall states on the Thin Torus, Licentiate thesis, E. Wikberg (2009). ...
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.