Operators and Quantum Mechanics
... For Hermitian operators  and B̂ representing physical variables it is very important to know if they commute ˆ ˆ BA ˆˆ ? i.e., is AB Remember that because these linear operators obey the same algebra as matrices in general operators do not commute ...
... For Hermitian operators  and B̂ representing physical variables it is very important to know if they commute ˆ ˆ BA ˆˆ ? i.e., is AB Remember that because these linear operators obey the same algebra as matrices in general operators do not commute ...
Supersymmetric quantum mechanics and new potentials
... relation between the old system and the new one is established through the SQM. As the spectrum of one potential is the same as that of its generalized version, some papers have appeared trying to distinguish these systems through the scattering produced by them. Cooper et a115 and Kare and sukhatme ...
... relation between the old system and the new one is established through the SQM. As the spectrum of one potential is the same as that of its generalized version, some papers have appeared trying to distinguish these systems through the scattering produced by them. Cooper et a115 and Kare and sukhatme ...
The Parallel Development of Matrix and Wave Mechanics
... generally, radiation behaved as if it consisted of a finite number of independent localized energy “quanta” (Jammer, 1966, p. 30). This was a whole new kind of conceptual step which Einstein made. A step which seemed to defy the wave-like nature of light; the overall accepted conception of light. Pa ...
... generally, radiation behaved as if it consisted of a finite number of independent localized energy “quanta” (Jammer, 1966, p. 30). This was a whole new kind of conceptual step which Einstein made. A step which seemed to defy the wave-like nature of light; the overall accepted conception of light. Pa ...
Quantization of Relativistic Free Fields
... Gravitational interactions, for example, couple to zero-point energy. The infinity creates a problem when trying to construct quantum field theories in the presence of a classical gravitational field, since the vacuum energy gives rise to an infinite cosmological constant, which can be determined ex ...
... Gravitational interactions, for example, couple to zero-point energy. The infinity creates a problem when trying to construct quantum field theories in the presence of a classical gravitational field, since the vacuum energy gives rise to an infinite cosmological constant, which can be determined ex ...
K.Batrakov, Mechanisms of Terahertz Radiation Generation in
... quantum recoil. then traditional form of second-order Cherenkov resonance is realized: ...
... quantum recoil. then traditional form of second-order Cherenkov resonance is realized: ...
Are Orbitals Observable? - HYLE-
... Since what does not exist cannot be observed, it follows that in manyelectron systems orbitals cannot be observed. Ostrovsky argues that although strictly speaking, Scerri is right, approximations are ubiquitous in science, in quantum mechanics no less than in other fields (Ostrovsky 2005, p. 111). ...
... Since what does not exist cannot be observed, it follows that in manyelectron systems orbitals cannot be observed. Ostrovsky argues that although strictly speaking, Scerri is right, approximations are ubiquitous in science, in quantum mechanics no less than in other fields (Ostrovsky 2005, p. 111). ...
quantum field theory in curved spacetime
... 1. There may be no Killing vector at all, timelike or spacelike. This is the generic situation. How to deal with it is unkown, except possibly when there is an approximate Killing vector that becomes exact asymptotically. It seems most unlikely that the particle picture will prove useful here, excep ...
... 1. There may be no Killing vector at all, timelike or spacelike. This is the generic situation. How to deal with it is unkown, except possibly when there is an approximate Killing vector that becomes exact asymptotically. It seems most unlikely that the particle picture will prove useful here, excep ...
On Fractional Schrödinger Equation and Its Application
... Recently Herrmann has applied the fractional mechanical approach to several particular problems [7-12]. Some other cases of the fractional Schrödinger equation were discussed by Naber [13] and Ben Adda & Cresson [14]. ...
... Recently Herrmann has applied the fractional mechanical approach to several particular problems [7-12]. Some other cases of the fractional Schrödinger equation were discussed by Naber [13] and Ben Adda & Cresson [14]. ...
Background Material
... Chapter 1 to be related to the values of x at which the spherical Bessel functions jL(x) vanish, are not the same as in atoms, again because the radial potentials differ. However, the angular shapes of the spherical box problem are the same as in atomic structure because, in both cases, the potentia ...
... Chapter 1 to be related to the values of x at which the spherical Bessel functions jL(x) vanish, are not the same as in atoms, again because the radial potentials differ. However, the angular shapes of the spherical box problem are the same as in atomic structure because, in both cases, the potentia ...
Primitive ontology and quantum state in the GRW matter density theory
... four-dimensional space-time, at least not if one considers N-particle systems with N > 1. One then has to regard the quantum state either as a field on 3N-dimensional configuration space or as what Belot calls a “multi-field”, an object assigning properties to N-tuples of points in space. The second ...
... four-dimensional space-time, at least not if one considers N-particle systems with N > 1. One then has to regard the quantum state either as a field on 3N-dimensional configuration space or as what Belot calls a “multi-field”, an object assigning properties to N-tuples of points in space. The second ...
The Quantum World
... The source of the perplexity is more subtle. Consider what is involved in measuring the position of an electron. It requires the setting up of a chain of correlated consequences linking at one end the position of the microscopic electron and at the other end the registration of the result of that pa ...
... The source of the perplexity is more subtle. Consider what is involved in measuring the position of an electron. It requires the setting up of a chain of correlated consequences linking at one end the position of the microscopic electron and at the other end the registration of the result of that pa ...
pdf - at www.arxiv.org.
... gases could be described with the use of three conceptually different kinds of entropy. They are the thermodynamic entropy à la Clausius, the kinetic entropy that describes the gas’ relaxation to thermodynamic equilibrium, and a statistical entropy that is expected to be the logarithm of the number ...
... gases could be described with the use of three conceptually different kinds of entropy. They are the thermodynamic entropy à la Clausius, the kinetic entropy that describes the gas’ relaxation to thermodynamic equilibrium, and a statistical entropy that is expected to be the logarithm of the number ...
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.