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... Wave particle duality of particles, de-Broglie hypothesis We have already seen the double nature of electromagnetic radiation or light. We concluded that light can behave like a wave or particle. This was called wave particle duality. In 1924 de-Broglie suggested the same duality in case of particle ...
... Wave particle duality of particles, de-Broglie hypothesis We have already seen the double nature of electromagnetic radiation or light. We concluded that light can behave like a wave or particle. This was called wave particle duality. In 1924 de-Broglie suggested the same duality in case of particle ...
Nonresonant exchange between two electrons
... Zh. Eksp. Teor. Fiz. 86, 84-93 (January 1984) An analytic expression is obtained for the matrix element of two-electron exchange in collision between an atom and an ion with different nuclear charges. A quasiclassical approximation is used that should lead to a higher accuracy (than the asymptotic e ...
... Zh. Eksp. Teor. Fiz. 86, 84-93 (January 1984) An analytic expression is obtained for the matrix element of two-electron exchange in collision between an atom and an ion with different nuclear charges. A quasiclassical approximation is used that should lead to a higher accuracy (than the asymptotic e ...
Transcript of the Philosophical Implications of Quantum Mechanics
... distance between two points, there were effectively an infinite number of paths, therefore a particle took every possible path between two points. However most of these paths mutually cancelled out the only one that didn’t being the most direct path. But oddly the particle on this direct path could ...
... distance between two points, there were effectively an infinite number of paths, therefore a particle took every possible path between two points. However most of these paths mutually cancelled out the only one that didn’t being the most direct path. But oddly the particle on this direct path could ...
Higher Order Gaussian Beams
... Can convey torque to particles Effect results from the helical phase-rotation of the field about the beam axis ...
... Can convey torque to particles Effect results from the helical phase-rotation of the field about the beam axis ...
Quantum Tunneling - Santa Rosa Junior College
... • This principle is the concept behind quantum tunneling. Equations of the • It is due to this lack of knowledge Uncertainty Principle about the particle that we call the “If the properties of the atom action of tunneling an odd and particle could be known phenomena. with certainty, then they • Inde ...
... • This principle is the concept behind quantum tunneling. Equations of the • It is due to this lack of knowledge Uncertainty Principle about the particle that we call the “If the properties of the atom action of tunneling an odd and particle could be known phenomena. with certainty, then they • Inde ...
Path integral in quantum mechanics
... we divide space and time into small segments; take a field in each segment to be constant; the differences between fields in neighboring segments become derivatives; use the trick: multiplying by is equivalent to replacing with which we often don’t write explicitly; ... eventually we can integrate o ...
... we divide space and time into small segments; take a field in each segment to be constant; the differences between fields in neighboring segments become derivatives; use the trick: multiplying by is equivalent to replacing with which we often don’t write explicitly; ... eventually we can integrate o ...
\chapter{Introduction}
... particles whatsoever or, more rigorously, a subspace $V$ of the $\mathbb{R}^3$ such that $N(V)=0$, where $N$ denotes the number of particles detected by an observer in the exterior of $V$. Intuitively this function $N:\mathbb{R}^3\rightarrow\mathbb{N}$ is an invariant under coordinate transformation ...
... particles whatsoever or, more rigorously, a subspace $V$ of the $\mathbb{R}^3$ such that $N(V)=0$, where $N$ denotes the number of particles detected by an observer in the exterior of $V$. Intuitively this function $N:\mathbb{R}^3\rightarrow\mathbb{N}$ is an invariant under coordinate transformation ...
PHYS 1443 * Section 501 Lecture #1
... Wave particle duality solution • The solution to the wave particle duality of an event is given by the following principle. • Bohr’s principle of complementarity: It is not possible to describe physical observables simultaneously in terms of both particles and waves. • Physical observables are the ...
... Wave particle duality solution • The solution to the wave particle duality of an event is given by the following principle. • Bohr’s principle of complementarity: It is not possible to describe physical observables simultaneously in terms of both particles and waves. • Physical observables are the ...
Lecture 10 Relevant sections in text: §1.7 Gaussian state Here we
... You can see that this Gaussian is peaked about the expected momentum value, as it should be, and that its width varies like 1/d, i.e., reciprocal to the position uncertainty, as expected. In summary, the Gaussian state we have defined corresponds to a particle which (on the average) is moving, and h ...
... You can see that this Gaussian is peaked about the expected momentum value, as it should be, and that its width varies like 1/d, i.e., reciprocal to the position uncertainty, as expected. In summary, the Gaussian state we have defined corresponds to a particle which (on the average) is moving, and h ...
Department of Physical Sciences (Physics)
... (iv) Orbiting satellites and spacecraft can become charged because the light from the sun ejects electrons from their outer surface and they must be designed to minimise this effect. If the skin is coated with Ni which has a work function of 4.87eV, calculate the longest wavelength of the incident s ...
... (iv) Orbiting satellites and spacecraft can become charged because the light from the sun ejects electrons from their outer surface and they must be designed to minimise this effect. If the skin is coated with Ni which has a work function of 4.87eV, calculate the longest wavelength of the incident s ...
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.