Group and phase velocity
... The evolution of a state of a system is given by the application of the Hamiltonian operator to the state wavefunction. In the Schrodinger representation, the evoultion is given by the wavefunction r , t , and not by the Hˆ ...
... The evolution of a state of a system is given by the application of the Hamiltonian operator to the state wavefunction. In the Schrodinger representation, the evoultion is given by the wavefunction r , t , and not by the Hˆ ...
Quantum Theory and Molecular Energy
... being a particular value of the coordinates. In 1 dimension: The probability of the particle described by (x) being between x and x + dx is proportional to: ...
... being a particular value of the coordinates. In 1 dimension: The probability of the particle described by (x) being between x and x + dx is proportional to: ...
REVIEW OF WAVE MECHANICS
... Hand your solutions to the following questions to Dr. Mulheran at the end of the first workshop in week 6. Some of your solutions will be marked as part of the continuous assessment of this course which contributes 20% of the overall module grade. Your solutions must be well presented; untidy work w ...
... Hand your solutions to the following questions to Dr. Mulheran at the end of the first workshop in week 6. Some of your solutions will be marked as part of the continuous assessment of this course which contributes 20% of the overall module grade. Your solutions must be well presented; untidy work w ...
Canonical quantization of scalar fields
... Recall, in classical mechanics, starting with lagrangian as a function of coordinates and their time derivatives we define conjugate ...
... Recall, in classical mechanics, starting with lagrangian as a function of coordinates and their time derivatives we define conjugate ...
Slide 1
... Measurement necessarily involves interactions between observer and the observed system Matter and radiation are the entities available to us for such measurements The relations p = h/l and E = hn are applicable to both matter and to radiation because of the intrinsic nature of wave-particle duality ...
... Measurement necessarily involves interactions between observer and the observed system Matter and radiation are the entities available to us for such measurements The relations p = h/l and E = hn are applicable to both matter and to radiation because of the intrinsic nature of wave-particle duality ...
the squared modulus of the wave function is the probability density
... This formula looks scary, right? Fortunately, however, it is not scary at all, because in the separated wavefunction each function is individually normalized, so that both integrals are equal to unity! Hence, the radial probability density takes a simple form: ...
... This formula looks scary, right? Fortunately, however, it is not scary at all, because in the separated wavefunction each function is individually normalized, so that both integrals are equal to unity! Hence, the radial probability density takes a simple form: ...
![ChemChapter_4[1]Light](http://s1.studyres.com/store/data/001894151_1-323884b777914f52c04d2bb917d4088a-300x300.png)
ChemChapter_4[1]Light
... Schrodinger • These ideas say it is impossible to know where an electron is at any point in time. Therefore we can only say where an electron is most probably located at any time. We call that region an orbital. Orbital – 3d region around a nucleus where an electron is likely to exist ...
... Schrodinger • These ideas say it is impossible to know where an electron is at any point in time. Therefore we can only say where an electron is most probably located at any time. We call that region an orbital. Orbital – 3d region around a nucleus where an electron is likely to exist ...
Quantum Mechanics
... probability of an event is equal to the square of the amplitude of the wavefunction (|ψ|²). Impossible to know all properties of a system at the same time, each must be given by probabilistic values (uncertainty principle). Matter exhibits wave-particle duality; particles may exhibit both particle a ...
... probability of an event is equal to the square of the amplitude of the wavefunction (|ψ|²). Impossible to know all properties of a system at the same time, each must be given by probabilistic values (uncertainty principle). Matter exhibits wave-particle duality; particles may exhibit both particle a ...
Waves
... Spherical Bessel functions arise in problems with spherical symmetry. de Broglie’s concept of an atom… Legendre’s equation: ...
... Spherical Bessel functions arise in problems with spherical symmetry. de Broglie’s concept of an atom… Legendre’s equation: ...
chapter 7 part 1
... high (and thick) potential walls – impenetrable – just a model that does not really correspond to reality at wave function (evanescent wave) always leaks into barrier – chapter 6 tunneling- and may be picked up at the other side, become a traveling wave again, i.e. a particle that can move so U(x,y, ...
... high (and thick) potential walls – impenetrable – just a model that does not really correspond to reality at wave function (evanescent wave) always leaks into barrier – chapter 6 tunneling- and may be picked up at the other side, become a traveling wave again, i.e. a particle that can move so U(x,y, ...
Nino Zanghì Dipartimento di Fisica dell`Università di Genova, INFN
... (ii) There is a state vector ψ in Hilbert space that evolves either unitarily or, at least, for microscopic systems very probably for a long time approximately unitarily. (iii) The state vector ψ governs the behavior of the PO by means of (possibly stochastic) ...
... (ii) There is a state vector ψ in Hilbert space that evolves either unitarily or, at least, for microscopic systems very probably for a long time approximately unitarily. (iii) The state vector ψ governs the behavior of the PO by means of (possibly stochastic) ...
Geometry, Physics, and Representation Theory Traces of intertwiners for quantum affine and
... Felder-Varchenko functions Abstract. This talk concerns two approaches for studying a family of special functions occurring in the study of the q-Knizhnik-Zamolodchikov-Bernard (q-KZB) equation. The philosophy of KZ-type equations predicts that it admits solutions via (1) traces of intertwining oper ...
... Felder-Varchenko functions Abstract. This talk concerns two approaches for studying a family of special functions occurring in the study of the q-Knizhnik-Zamolodchikov-Bernard (q-KZB) equation. The philosophy of KZ-type equations predicts that it admits solutions via (1) traces of intertwining oper ...
Waves and the Schroedinger Equation
... Thus, from the last expression, we see that stationary waves have fixed nodal points; zero amplitude versus time at fixed points). Now, we have gone about things in a reverse manner, but we can consider the following. We have written a representation of a wave-particle entity as a sinusoidal functio ...
... Thus, from the last expression, we see that stationary waves have fixed nodal points; zero amplitude versus time at fixed points). Now, we have gone about things in a reverse manner, but we can consider the following. We have written a representation of a wave-particle entity as a sinusoidal functio ...
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.