Teaching the Atomic Theory: A Visual
... You, that I now mean by Elements, as those Chymists that speak plainest do by their Principles, certain Primitive and Simple, or perfectly unmingled bodies; which not being made of any other bodies, or of one another, are the Ingredients of which all those call’d perfectly mixt Bodies are immediatel ...
... You, that I now mean by Elements, as those Chymists that speak plainest do by their Principles, certain Primitive and Simple, or perfectly unmingled bodies; which not being made of any other bodies, or of one another, are the Ingredients of which all those call’d perfectly mixt Bodies are immediatel ...
The Quantum Mechanics of MRI
... Well defined wavelength and momentum but no preferential position. "Uncertainty on position": x "Uncertainty on momentum": p 0 ...
... Well defined wavelength and momentum but no preferential position. "Uncertainty on position": x "Uncertainty on momentum": p 0 ...
Do we need the Concept of Particle?
... Heisenberg and Bohr against the possibility of making exclusive use of wave mechanics, can in fact be dealt with by this theory, provided it is applied to a system large enough to include a relevant part of the measurement device. One must only note that the wavemechanical model has no ambition othe ...
... Heisenberg and Bohr against the possibility of making exclusive use of wave mechanics, can in fact be dealt with by this theory, provided it is applied to a system large enough to include a relevant part of the measurement device. One must only note that the wavemechanical model has no ambition othe ...
I. Wave Mechanics
... if the wave function is to represent a physically realistic particle. That is, a wave function must be normalizable. This boils down to the requirement that x, t 0 as x . Some wave functions satisfy the wave equation but are not normalizable. c. Expectation values Imagine a large number of ...
... if the wave function is to represent a physically realistic particle. That is, a wave function must be normalizable. This boils down to the requirement that x, t 0 as x . Some wave functions satisfy the wave equation but are not normalizable. c. Expectation values Imagine a large number of ...
shp_05 - Columbia University
... Observation tells us that physical quantities are not continuous down to the smallest scales, but tend to be discrete. O.K., we can live with that. But QM has another surprise: if you look small enough, matter – that is, “particles” –start to exhibit wavelike behavior. We have already seen hints of ...
... Observation tells us that physical quantities are not continuous down to the smallest scales, but tend to be discrete. O.K., we can live with that. But QM has another surprise: if you look small enough, matter – that is, “particles” –start to exhibit wavelike behavior. We have already seen hints of ...
AS_Unit1_Quantum_06_Wave_Particle_Duality
... 1.Find the wavelength of an electron of mass 9.00 × 10-31 kg moving at 3.00 × 107 m s-1 2. Find the wavelength of a cricket ball of mass 0.15 kg moving at 30 m s-1. 3. It is also desirable to be able to calculate the wavelength associated with an electron when the accelerating voltage is known. Ther ...
... 1.Find the wavelength of an electron of mass 9.00 × 10-31 kg moving at 3.00 × 107 m s-1 2. Find the wavelength of a cricket ball of mass 0.15 kg moving at 30 m s-1. 3. It is also desirable to be able to calculate the wavelength associated with an electron when the accelerating voltage is known. Ther ...
Exact reduced dynamics and
... evolution, the concurrence of the reduced density matrix of the two coupled spins is also obtained exactly. It is shown that the dynamics of the entanglement depends on the initial state of the system and the coupling strength between the two coupled spins, the thermal temperature of the spin enviro ...
... evolution, the concurrence of the reduced density matrix of the two coupled spins is also obtained exactly. It is shown that the dynamics of the entanglement depends on the initial state of the system and the coupling strength between the two coupled spins, the thermal temperature of the spin enviro ...
particle physics - Columbia University
... Observation tells us that physical quantities are not continuous down to the smallest scales, but tend to be discrete. O.K., we can live with that. But QM has another surprise: if you look small enough, matter – that is, “particles” –start to exhibit wavelike behavior. We have already seen hints of ...
... Observation tells us that physical quantities are not continuous down to the smallest scales, but tend to be discrete. O.K., we can live with that. But QM has another surprise: if you look small enough, matter – that is, “particles” –start to exhibit wavelike behavior. We have already seen hints of ...
Teaching Modern Physics - IMSA Digital Commons
... What implications does a probabilistic universe have for free will? Is consciousness just a series of random quantum measurements that give the semblance of purpose? Is it easier or harder to reconcile quantum mechanics with an intervening God? ...
... What implications does a probabilistic universe have for free will? Is consciousness just a series of random quantum measurements that give the semblance of purpose? Is it easier or harder to reconcile quantum mechanics with an intervening God? ...
Document
... Even with a qualitative treatment it is possible to see how the Feynman diagram picture agrees with observed reactions. Aim: compare rates of e e g g and e e g g g Start e e g g and consider possible contributing diagrams. Use the simplest possible diagrams (leading or ...
... Even with a qualitative treatment it is possible to see how the Feynman diagram picture agrees with observed reactions. Aim: compare rates of e e g g and e e g g g Start e e g g and consider possible contributing diagrams. Use the simplest possible diagrams (leading or ...
Document
... The wave function is often complex-valued The absolute square |ψ|2 = ψ*ψ is always real and positive ψ* is the complete conjugate of ψ It is proportional to the probability per unit volume of finding a particle at a given point at some instant The wave function contains within it all the informa ...
... The wave function is often complex-valued The absolute square |ψ|2 = ψ*ψ is always real and positive ψ* is the complete conjugate of ψ It is proportional to the probability per unit volume of finding a particle at a given point at some instant The wave function contains within it all the informa ...
An Ontological Interpretation of the Wave Function - Philsci
... and (x1 , y1 , z1 ). This means that the picture of discontinuous motion also exists for one-body systems. Since quantum mechanics does not provide further information about the positions of physical entities at each instant, the discontinuous motion described by the theory is also essentially rando ...
... and (x1 , y1 , z1 ). This means that the picture of discontinuous motion also exists for one-body systems. Since quantum mechanics does not provide further information about the positions of physical entities at each instant, the discontinuous motion described by the theory is also essentially rando ...
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.