Chapter 2 Wave Mechanics and the Schrödinger equation
... (zeros). The normalizable eigenfunctions un are the wave functions of bound states with a discrete spectrum of energy levels En . It is clear that bound states should exist only for Vmin < E < Vmax . The lower bound follows because otherwise the wave function is convex, and hence cannot be normaliza ...
... (zeros). The normalizable eigenfunctions un are the wave functions of bound states with a discrete spectrum of energy levels En . It is clear that bound states should exist only for Vmin < E < Vmax . The lower bound follows because otherwise the wave function is convex, and hence cannot be normaliza ...
wave - UniMAP Portal
... Standing Waves On a rope with one fixed end, it is possible to set up waves that do not travel; they simply vibrate in place. These are called standing waves. ...
... Standing Waves On a rope with one fixed end, it is possible to set up waves that do not travel; they simply vibrate in place. These are called standing waves. ...
3D quantum mechanics, hydrogen atom
... What is the momentum of an electron in the 1st excited state of an infinite square well potential? The infinite square well energy levels are En = n2ħ2π2/2mL2. The first excited state is n = 2. The only way we know how to get the momentum is from the energy. The kinetic energy in this state is E2 = ...
... What is the momentum of an electron in the 1st excited state of an infinite square well potential? The infinite square well energy levels are En = n2ħ2π2/2mL2. The first excited state is n = 2. The only way we know how to get the momentum is from the energy. The kinetic energy in this state is E2 = ...
Lecture 2
... localized Gaussian type of wavefunction that in the case of the former the position is completely uncertain whereas the momentum is definite and in the later case position has much more uncertainty , but the momentum is no more definite. ...
... localized Gaussian type of wavefunction that in the case of the former the position is completely uncertain whereas the momentum is definite and in the later case position has much more uncertainty , but the momentum is no more definite. ...
Abstract 1. Introduction It is well known that intense sound waves
... It is well known that intense sound waves can induce quasi-stationary movements of fluid and gas media. In nonlinear acoustics such movements are called acoustic streams or sound wind. In classical hydrodynamics various types of acoustic flows have been studied (see e.g. [1, 2]). An intense sound wa ...
... It is well known that intense sound waves can induce quasi-stationary movements of fluid and gas media. In nonlinear acoustics such movements are called acoustic streams or sound wind. In classical hydrodynamics various types of acoustic flows have been studied (see e.g. [1, 2]). An intense sound wa ...
Correlation Effects in Quantum Dot Wave Function Imaging
... semiconductor quantum dots1–3 (QDs) provide spectacular images of QD wave functions.4–9 The measured intensities are generally identified with the density of carrier states at the resonant tunneling (Fermi) energy, resolved in either real4–6 or reciprocal7–9 space. However, Coulomb blockade phenomen ...
... semiconductor quantum dots1–3 (QDs) provide spectacular images of QD wave functions.4–9 The measured intensities are generally identified with the density of carrier states at the resonant tunneling (Fermi) energy, resolved in either real4–6 or reciprocal7–9 space. However, Coulomb blockade phenomen ...
Chapter 19 - eLisa UGM
... De Broglie’s Hypothesis • Louis de Broglie postulated that the dual nature of the light must be “expanded” to ALL matter – In other words, all material particles possess wave-like properties, characterized by the wavelength, λB, related to the momentum p of the particle in the same way as for light ...
... De Broglie’s Hypothesis • Louis de Broglie postulated that the dual nature of the light must be “expanded” to ALL matter – In other words, all material particles possess wave-like properties, characterized by the wavelength, λB, related to the momentum p of the particle in the same way as for light ...
waves.
... the motion of the wave source and/or by the motion of the observer. (More about the Doppler effect later in the course!) ...
... the motion of the wave source and/or by the motion of the observer. (More about the Doppler effect later in the course!) ...
ΟΝ THE WAVE FUNCTION OF THE PHOTON
... expressing the fact that the wave function describes only a single photon. This condition must be handled with some care, since stationary states in infinite space are described by non-normalizable wave functions. Therefore, Eq. (42) strictly speaking can hold only in a finite volume. In the formula ...
... expressing the fact that the wave function describes only a single photon. This condition must be handled with some care, since stationary states in infinite space are described by non-normalizable wave functions. Therefore, Eq. (42) strictly speaking can hold only in a finite volume. In the formula ...
Interpretation of quantum mechanics by the double solution theory
... separated by a physical process which destroys the initial superposition, the superposition is the physical reality. Just after submitting my thesis, I therefore had to generalize the guiding ideas by considering, on one hand, a wave which would not be plane monochromatic, and on the other hand, by ...
... separated by a physical process which destroys the initial superposition, the superposition is the physical reality. Just after submitting my thesis, I therefore had to generalize the guiding ideas by considering, on one hand, a wave which would not be plane monochromatic, and on the other hand, by ...
CHEM-UA 127: Advanced General Chemistry I
... Finally, suppose we start with a state Ψ(x, 0) = (1/ 2)[ψ1 (x) + ψ2 (x)], and we let this state evolve in time. At any point in time, the state Ψ(x, t) will be some mixture of ψ1 (x) and ψ2 (x), and this mixture changes with time. Now, at some specific instance in time t, we measure the energy and o ...
... Finally, suppose we start with a state Ψ(x, 0) = (1/ 2)[ψ1 (x) + ψ2 (x)], and we let this state evolve in time. At any point in time, the state Ψ(x, t) will be some mixture of ψ1 (x) and ψ2 (x), and this mixture changes with time. Now, at some specific instance in time t, we measure the energy and o ...
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.