Finite Quantum Measure Spaces
... macroscopic world, this is not the case on a quantum scale due to the effects of annihilation and binding energy. If, for instance, x1 and x2 represent an electron and a positron respectively, then µ(x1 ) = µ(x2 ) = 9.11 × 10−31 kg whereas µ(x1 ∪ x2 ) = 0. At the heart of quantum mechanics is a phen ...
... macroscopic world, this is not the case on a quantum scale due to the effects of annihilation and binding energy. If, for instance, x1 and x2 represent an electron and a positron respectively, then µ(x1 ) = µ(x2 ) = 9.11 × 10−31 kg whereas µ(x1 ∪ x2 ) = 0. At the heart of quantum mechanics is a phen ...
The Structure of a Quantum World Jill North
... This chapter goes as follows. I first discuss the guiding principles I rely on, and the ways we use them in our scientific theorizing (section 1). I then argue that these principles support the conclusion that the wave function’s space is fundamental to a quantum world (sections 2 and 3). I end by ...
... This chapter goes as follows. I first discuss the guiding principles I rely on, and the ways we use them in our scientific theorizing (section 1). I then argue that these principles support the conclusion that the wave function’s space is fundamental to a quantum world (sections 2 and 3). I end by ...
A new look at the Milne Universe\\ and its ground state wave functions
... energy, that constitute more than 95% of the Universe [8]. According to the inflationbased ΛCDM cosmological model our Universe is composed of only 5% of normal matter (baryons), ∼25% of dark matter and ∼70% of dark energy. Dark matter is supposed to be responsible for anomaly large rotational veloc ...
... energy, that constitute more than 95% of the Universe [8]. According to the inflationbased ΛCDM cosmological model our Universe is composed of only 5% of normal matter (baryons), ∼25% of dark matter and ∼70% of dark energy. Dark matter is supposed to be responsible for anomaly large rotational veloc ...
Path integrals in quantum mechanics
... Quantum mechanics can be formulated in two equivalent ways: (i) canonical quantization, also known as operatorial quantization, which is based on linear operators acting on the Hilbert space of physical states, (ii) path integrals, based on integration over a space of functions. The former was the f ...
... Quantum mechanics can be formulated in two equivalent ways: (i) canonical quantization, also known as operatorial quantization, which is based on linear operators acting on the Hilbert space of physical states, (ii) path integrals, based on integration over a space of functions. The former was the f ...
Coupled-mode theory for general free-space resonant scattering of waves
... work, we show that the resonant scattering of freely propagating waves from resonant objects of two-dimensional 共2D兲 cylindrical or 3D spherical symmetry can also be very accurately modeled using very simple CMT analytical expressions; the resonant objects can themselves entail more than one weakly ...
... work, we show that the resonant scattering of freely propagating waves from resonant objects of two-dimensional 共2D兲 cylindrical or 3D spherical symmetry can also be very accurately modeled using very simple CMT analytical expressions; the resonant objects can themselves entail more than one weakly ...
Postulates of Quantum Mechanics
... particular physical system is, this is determined by more specific theories. Any physical system whose state space can be described by C2 can serve as an implementation of a qubit. Examples: • spin of an electron • polarization of a photon • current in a superconducting circuit ...
... particular physical system is, this is determined by more specific theories. Any physical system whose state space can be described by C2 can serve as an implementation of a qubit. Examples: • spin of an electron • polarization of a photon • current in a superconducting circuit ...
Operator Product Expansion and Conservation Laws in Non
... Experimental studies of fermions at unitarity have stimulated theoretical developments of nonrelativistic conformal field theories. The conformal extension of the Galilean algebra was found a long time ago [1, 2] and was later analyzed in the context of string theory. Mehen, Stewart, and Wise explor ...
... Experimental studies of fermions at unitarity have stimulated theoretical developments of nonrelativistic conformal field theories. The conformal extension of the Galilean algebra was found a long time ago [1, 2] and was later analyzed in the context of string theory. Mehen, Stewart, and Wise explor ...
Abstracts 报 告 摘 要 Ermakov–Ray–Reid Systems in (2+1
... which is given by the Darboux transformation from the periodic "seed". If we have enough time, the rouge wave of the variable mass Sine-Gordon is given by a nonlinear transformation from known solutions of the usual Sine-Gordon equation. 9. A new convergence acceleration algorithm related to the lat ...
... which is given by the Darboux transformation from the periodic "seed". If we have enough time, the rouge wave of the variable mass Sine-Gordon is given by a nonlinear transformation from known solutions of the usual Sine-Gordon equation. 9. A new convergence acceleration algorithm related to the lat ...
ISOBARIC-SPIN SPLITTING OF SINGLE
... case the giant resonances (always assuming that the T splitting is large enough) could still be characterized well enough b y T even though the major components of some of the contributing states could not. As discussed earlier 1) it is difficult at present to estimate the magnitude of the splitting ...
... case the giant resonances (always assuming that the T splitting is large enough) could still be characterized well enough b y T even though the major components of some of the contributing states could not. As discussed earlier 1) it is difficult at present to estimate the magnitude of the splitting ...
phyoutlinech11 notes-0 - Fort Thomas Independent Schools
... waves that pass a given point in a unit of time. • The period of a wave describes the time it takes for a complete wavelength to pass a given point. • The relationship between period and frequency in SHM holds true for waves as well; the period of a wave is inversely related to its frequency. ...
... waves that pass a given point in a unit of time. • The period of a wave describes the time it takes for a complete wavelength to pass a given point. • The relationship between period and frequency in SHM holds true for waves as well; the period of a wave is inversely related to its frequency. ...
The weak-coupling limit of large classical and quantum systems
... at distance smaller than the diameter d. The particles are moving freely up to the first contact instant, that is the first time in which two particles arrive at distance d. Then the pair interacts performing ...
... at distance smaller than the diameter d. The particles are moving freely up to the first contact instant, that is the first time in which two particles arrive at distance d. Then the pair interacts performing ...
How to get the low rank approximation T
... Nano electronic devices complicated 2D/3D geometries; 10000 ~ 10 million atoms in the active domain; many materials are used. ...
... Nano electronic devices complicated 2D/3D geometries; 10000 ~ 10 million atoms in the active domain; many materials are used. ...
III. Spin and orbital angular momentum
... Sz |sm >= mh̄|sm > . Each elementary particle has a fixed magnitude of the spin vector, given by the quantum number s. However, the projection of the spin onto one axis, typically chosen to be the z-axis, is needed in addition to the coordinates (or momenta) to fully specify the state of the partic ...
... Sz |sm >= mh̄|sm > . Each elementary particle has a fixed magnitude of the spin vector, given by the quantum number s. However, the projection of the spin onto one axis, typically chosen to be the z-axis, is needed in addition to the coordinates (or momenta) to fully specify the state of the partic ...
73 013601 (2006)
... dimensional system that obeys the KAM theorem, while the kicked harmonic oscillator is known to be a special degenerate system out of the framework of the KAM theorem 关14兴. It is very interesting to understand how both quantum mechanics and mean-field interaction affect the dynamics of such a generi ...
... dimensional system that obeys the KAM theorem, while the kicked harmonic oscillator is known to be a special degenerate system out of the framework of the KAM theorem 关14兴. It is very interesting to understand how both quantum mechanics and mean-field interaction affect the dynamics of such a generi ...
Bose-Einstein Condensation and Free DKP field
... theory BEC is commonly related to spontaneous symmetry breaking [8, 9, 10]. In quantum field theory BEC the complex scalar field was and is used for studying the thermodynamical properties of physical systems composite of bosonic particles with spin 0. However, at zero temperature, there is an alter ...
... theory BEC is commonly related to spontaneous symmetry breaking [8, 9, 10]. In quantum field theory BEC the complex scalar field was and is used for studying the thermodynamical properties of physical systems composite of bosonic particles with spin 0. However, at zero temperature, there is an alter ...
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.