15.4 Bohr Model and Angular Momentum and Magnetic Motion
Total Page:16
File Type:pdf, Size:1020Kb

Load more
Recommended publications
-
The Experimental Determination of the Moment of Inertia of a Model Airplane Michael Koken [email protected]
The University of Akron IdeaExchange@UAkron The Dr. Gary B. and Pamela S. Williams Honors Honors Research Projects College Fall 2017 The Experimental Determination of the Moment of Inertia of a Model Airplane Michael Koken [email protected] Please take a moment to share how this work helps you through this survey. Your feedback will be important as we plan further development of our repository. Follow this and additional works at: http://ideaexchange.uakron.edu/honors_research_projects Part of the Aerospace Engineering Commons, Aviation Commons, Civil and Environmental Engineering Commons, Mechanical Engineering Commons, and the Physics Commons Recommended Citation Koken, Michael, "The Experimental Determination of the Moment of Inertia of a Model Airplane" (2017). Honors Research Projects. 585. http://ideaexchange.uakron.edu/honors_research_projects/585 This Honors Research Project is brought to you for free and open access by The Dr. Gary B. and Pamela S. Williams Honors College at IdeaExchange@UAkron, the institutional repository of The nivU ersity of Akron in Akron, Ohio, USA. It has been accepted for inclusion in Honors Research Projects by an authorized administrator of IdeaExchange@UAkron. For more information, please contact [email protected], [email protected]. 2017 THE EXPERIMENTAL DETERMINATION OF A MODEL AIRPLANE KOKEN, MICHAEL THE UNIVERSITY OF AKRON Honors Project TABLE OF CONTENTS List of Tables ................................................................................................................................................ -
Rotational Motion (The Dynamics of a Rigid Body)
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Robert Katz Publications Research Papers in Physics and Astronomy 1-1958 Physics, Chapter 11: Rotational Motion (The Dynamics of a Rigid Body) Henry Semat City College of New York Robert Katz University of Nebraska-Lincoln, [email protected] Follow this and additional works at: https://digitalcommons.unl.edu/physicskatz Part of the Physics Commons Semat, Henry and Katz, Robert, "Physics, Chapter 11: Rotational Motion (The Dynamics of a Rigid Body)" (1958). Robert Katz Publications. 141. https://digitalcommons.unl.edu/physicskatz/141 This Article is brought to you for free and open access by the Research Papers in Physics and Astronomy at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Robert Katz Publications by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. 11 Rotational Motion (The Dynamics of a Rigid Body) 11-1 Motion about a Fixed Axis The motion of the flywheel of an engine and of a pulley on its axle are examples of an important type of motion of a rigid body, that of the motion of rotation about a fixed axis. Consider the motion of a uniform disk rotat ing about a fixed axis passing through its center of gravity C perpendicular to the face of the disk, as shown in Figure 11-1. The motion of this disk may be de scribed in terms of the motions of each of its individual particles, but a better way to describe the motion is in terms of the angle through which the disk rotates. -
Schrödinger Equation)
Lecture 37 (Schrödinger Equation) Physics 2310-01 Spring 2020 Douglas Fields Reduced Mass • OK, so the Bohr model of the atom gives energy levels: • But, this has one problem – it was developed assuming the acceleration of the electron was given as an object revolving around a fixed point. • In fact, the proton is also free to move. • The acceleration of the electron must then take this into account. • Since we know from Newton’s third law that: • If we want to relate the real acceleration of the electron to the force on the electron, we have to take into account the motion of the proton too. Reduced Mass • So, the relative acceleration of the electron to the proton is just: • Then, the force relation becomes: • And the energy levels become: Reduced Mass • The reduced mass is close to the electron mass, but the 0.0054% difference is measurable in hydrogen and important in the energy levels of muonium (a hydrogen atom with a muon instead of an electron) since the muon mass is 200 times heavier than the electron. • Or, in general: Hydrogen-like atoms • For single electron atoms with more than one proton in the nucleus, we can use the Bohr energy levels with a minor change: e4 → Z2e4. • For instance, for He+ , Uncertainty Revisited • Let’s go back to the wave function for a travelling plane wave: • Notice that we derived an uncertainty relationship between k and x that ended being an uncertainty relation between p and x (since p=ћk): Uncertainty Revisited • Well it turns out that the same relation holds for ω and t, and therefore for E and t: • We see this playing an important role in the lifetime of excited states. -
Magnetism, Magnetic Properties, Magnetochemistry
Magnetism, Magnetic Properties, Magnetochemistry 1 Magnetism All matter is electronic Positive/negative charges - bound by Coulombic forces Result of electric field E between charges, electric dipole Electric and magnetic fields = the electromagnetic interaction (Oersted, Maxwell) Electric field = electric +/ charges, electric dipole Magnetic field ??No source?? No magnetic charges, N-S No magnetic monopole Magnetic field = motion of electric charges (electric current, atomic motions) Magnetic dipole – magnetic moment = i A [A m2] 2 Electromagnetic Fields 3 Magnetism Magnetic field = motion of electric charges • Macro - electric current • Micro - spin + orbital momentum Ampère 1822 Poisson model Magnetic dipole – magnetic (dipole) moment [A m2] i A 4 Ampere model Magnetism Microscopic explanation of source of magnetism = Fundamental quantum magnets Unpaired electrons = spins (Bohr 1913) Atomic building blocks (protons, neutrons and electrons = fermions) possess an intrinsic magnetic moment Relativistic quantum theory (P. Dirac 1928) SPIN (quantum property ~ rotation of charged particles) Spin (½ for all fermions) gives rise to a magnetic moment 5 Atomic Motions of Electric Charges The origins for the magnetic moment of a free atom Motions of Electric Charges: 1) The spins of the electrons S. Unpaired spins give a paramagnetic contribution. Paired spins give a diamagnetic contribution. 2) The orbital angular momentum L of the electrons about the nucleus, degenerate orbitals, paramagnetic contribution. The change in the orbital moment -
Pierre Simon Laplace - Biography Paper
Pierre Simon Laplace - Biography Paper MATH 4010 Melissa R. Moore University of Colorado- Denver April 1, 2008 2 Many people contributed to the scientific fields of mathematics, physics, chemistry and astronomy. According to Gillispie (1997), Pierre Simon Laplace was the most influential scientist in all history, (p.vii). Laplace helped form the modern scientific disciplines. His techniques are used diligently by engineers, mathematicians and physicists. His diverse collection of work ranged in all fields but began in mathematics. Laplace was born in Normandy in 1749. His father, Pierre, was a syndic of a parish and his mother, Marie-Anne, was from a family of farmers. Many accounts refer to Laplace as a peasant. While it was not exactly known the profession of his Uncle Louis, priest, mathematician or teacher, speculations implied he was an educated man. In 1756, Laplace enrolled at the Beaumont-en-Auge, a secondary school run the Benedictine order. He studied there until the age of sixteen. The nest step of education led to either the army or the church. His father intended him for ecclesiastical vocation, according to Gillispie (1997 p.3). In 1766, Laplace went to the University of Caen to begin preparation for a career in the church, according to Katz (1998 p.609). Cristophe Gadbled and Pierre Le Canu taught Laplace mathematics, which in turn showed him his talents. In 1768, Laplace left for Paris to pursue mathematics further. Le Canu gave Laplace a letter of recommendation to d’Alembert, according to Gillispie (1997 p.3). Allegedly d’Alembert gave Laplace a problem which he solved immediately. -
Magnetism, Angular Momentum, and Spin
Chapter 19 Magnetism, Angular Momentum, and Spin P. J. Grandinetti Chem. 4300 P. J. Grandinetti Chapter 19: Magnetism, Angular Momentum, and Spin In 1820 Hans Christian Ørsted discovered that electric current produces a magnetic field that deflects compass needle from magnetic north, establishing first direct connection between fields of electricity and magnetism. P. J. Grandinetti Chapter 19: Magnetism, Angular Momentum, and Spin Biot-Savart Law Jean-Baptiste Biot and Félix Savart worked out that magnetic field, B⃗, produced at distance r away from section of wire of length dl carrying steady current I is 휇 I d⃗l × ⃗r dB⃗ = 0 Biot-Savart law 4휋 r3 Direction of magnetic field vector is given by “right-hand” rule: if you point thumb of your right hand along direction of current then your fingers will curl in direction of magnetic field. current P. J. Grandinetti Chapter 19: Magnetism, Angular Momentum, and Spin Microscopic Origins of Magnetism Shortly after Biot and Savart, Ampére suggested that magnetism in matter arises from a multitude of ring currents circulating at atomic and molecular scale. André-Marie Ampére 1775 - 1836 P. J. Grandinetti Chapter 19: Magnetism, Angular Momentum, and Spin Magnetic dipole moment from current loop Current flowing in flat loop of wire with area A will generate magnetic field magnetic which, at distance much larger than radius, r, appears identical to field dipole produced by point magnetic dipole with strength of radius 휇 = | ⃗휇| = I ⋅ A current Example What is magnetic dipole moment induced by e* in circular orbit of radius r with linear velocity v? * 휋 Solution: For e with linear velocity of v the time for one orbit is torbit = 2 r_v. -
Lecture #4, Matter/Energy Interactions, Emissions Spectra, Quantum Numbers
Welcome to 3.091 Lecture 4 September 16, 2009 Matter/Energy Interactions: Atomic Spectra 3.091 Periodic Table Quiz 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 Name Grade /10 Image by MIT OpenCourseWare. Rutherford-Geiger-Marsden experiment Image by MIT OpenCourseWare. Bohr Postulates for the Hydrogen Atom 1. Rutherford atom is correct 2. Classical EM theory not applicable to orbiting e- 3. Newtonian mechanics applicable to orbiting e- 4. Eelectron = Ekinetic + Epotential 5. e- energy quantized through its angular momentum: L = mvr = nh/2π, n = 1, 2, 3,… 6. Planck-Einstein relation applies to e- transitions: ΔE = Ef - Ei = hν = hc/λ c = νλ _ _ 24 1 18 Bohr magneton µΒ = eh/2me 9.274 015 4(31) X 10 J T 0.34 _ _ 27 1 19 Nuclear magneton µΝ = eh/2mp 5.050 786 6(17) X 10 J T 0.34 _ 2 3 20 Fine structure constant α = µ0ce /2h 7.297 353 08(33) X 10 0.045 21 Inverse fine structure constant 1/α 137.035 989 5(61) 0.045 _ 2 1 22 Rydberg constant R¥ = mecα /2h 10 973 731.534(13) m 0.0012 23 Rydberg constant in eV R¥ hc/{e} 13.605 698 1(40) eV 0.30 _ 10 24 Bohr radius a0 = a/4πR¥ 0.529 177 249(24) X 10 m 0.045 _ _ 4 2 1 25 Quantum of circulation h/2me 3.636 948 07(33) X 10 m s 0.089 _ 11 1 26 Electron specific charge -e/me -1.758 819 62(53) X 10 C kg 0.30 _ 12 27 Electron Compton wavelength λC = h/mec 2.426 310 58(22) X 10 m 0.089 _ 2 15 28 Electron classical radius re = α a0 2.817 940 92(38) X 10 m 0.13 _ _ 26 1 29 Electron magnetic moment` µe 928.477 01(31) X 10 J T 0.34 _ _ 3 30 Electron mag. -
Modern Physics: Problem Set #5
Physics 320 Fall (12) 2017 Modern Physics: Problem Set #5 The Bohr Model ("This is all nonsense", von Laue on Bohr's model; "It is one of the greatest discoveries", Einstein on the same). 2 4 mek e 1 ⎛ 1 ⎞ hc L = mvr = n! ; E n = – 2 2 ≈ –13.6eV⎜ 2 ⎟ ; λn →m = 2! n ⎝ n ⎠ | E m – E n | Notes: Exam 1 is next Friday (9/29). This exam will cover material in chapters 2 through 4. The exam€ is closed book, closed notes but you may bring in one-half of an 8.5x11" sheet of paper with handwritten notes€ and equations. € Due: Friday Sept. 29 by 6 pm Reading assignment: for Monday, 5.1-5.4 (Wave function for matter & 1D-Schrodinger equation) for Wednesday, 5.5, 5.8 (Particle in a box & Expectation values) Problem assignment: Chapter 4 Problems: 54. Bohr model for hydrogen spectrum (identify the region of the spectrum for each λ) 57. Electron speed in the Bohr model for hydrogen [Result: ke 2 /n!] A1. Rotational Spectra: The figure shows a diatomic molecule with bond- length d rotating about its center of mass. According to quantum theory, the d m ! ! ! € angular momentum ( ) of this system is restricted to a discrete set L = r × p m of values (including zero). Using the Bohr quantization condition (L=n!), determine the following: a) The allowed€ rotational energies of the molecule. [Result: n 2!2 /md 2 ] b) The wavelength of the photon needed to excite the molecule from the n to the n+1 rotational state. -
Bohr Model of Hydrogen
Chapter 3 Bohr model of hydrogen Figure 3.1: Democritus The atomic theory of matter has a long history, in some ways all the way back to the ancient Greeks (Democritus - ca. 400 BCE - suggested that all things are composed of indivisible \atoms"). From what we can observe, atoms have certain properties and behaviors, which can be summarized as follows: Atoms are small, with diameters on the order of 0:1 nm. Atoms are stable, they do not spontaneously break apart into smaller pieces or collapse. Atoms contain negatively charged electrons, but are electrically neutral. Atoms emit and absorb electromagnetic radiation. Any successful model of atoms must be capable of describing these observed properties. 1 (a) Isaac Newton (b) Joseph von Fraunhofer (c) Gustav Robert Kirch- hoff 3.1 Atomic spectra Even though the spectral nature of light is present in a rainbow, it was not until 1666 that Isaac Newton showed that white light from the sun is com- posed of a continuum of colors (frequencies). Newton introduced the term \spectrum" to describe this phenomenon. His method to measure the spec- trum of light consisted of a small aperture to define a point source of light, a lens to collimate this into a beam of light, a glass spectrum to disperse the colors and a screen on which to observe the resulting spectrum. This is indeed quite close to a modern spectrometer! Newton's analysis was the beginning of the science of spectroscopy (the study of the frequency distri- bution of light from different sources). The first observation of the discrete nature of emission and absorption from atomic systems was made by Joseph Fraunhofer in 1814. -
1 Introduction 2 Spins As Quantized Magnetic Moments
Spin 1 Introduction For the past few weeks you have been learning about the mathematical and algorithmic background of quantum computation. Over the course of the next couple of lectures, we'll discuss the physics of making measurements and performing qubit operations in an actual system. In particular we consider nuclear magnetic resonance (NMR). Before we get there, though, we'll discuss a very famous experiment by Stern and Gerlach. 2 Spins as quantized magnetic moments The quantum two level system is, in many ways, the simplest quantum system that displays interesting behavior (this is a very subjective statement!). Before diving into the physics of two-level systems, a little on the history of the canonical example: electron spin. In 1921, Stern proposed an experiment to distinguish between Larmor's classical theory of the atom and Sommerfeld's quantum theory. Each theory predicted that the atom should have a magnetic moment (i.e., it should act like a small bar magnet). However, Larmor predicted that this magnetic moment could be oriented along any direction in space, while Sommerfeld (with help from Bohr) predicted that the orientation could only be in one of two directions (in this case, aligned or anti-aligned with a magnetic field). Stern's idea was to use the fact that magnetic moments experience a linear force when placed in a magnetic field gradient. To see this, not that the potential energy of a magnetic dipole in a magnetic field is given by: U = −~µ · B~ Here, ~µ is the vector indicating the magnitude and direction of the magnetic moment. -
Spin the Evidence of Intrinsic Angular Momentum Or Spin and Its
Spin The evidence of intrinsic angular momentum or spin and its associated magnetic moment came through experiments by Stern and Gerlach and works of Goudsmit and Uhlenbeck. The spin is called intrinsic since, unlike orbital angular momentum which is extrinsic, it is carried by point particle in addition to its orbital angular momentum and has nothing to do with motion in space. The magnetic moment ~µ of silver atom was measured in 1922 experiment by Stern and Gerlach and its projection µz in the direction of magnetic fieldz ^ B (through which the silver beam was passed) was found to be just −|~µj and +j~µj instead of continuously varying between these two as limits. Classically, magnetic moment is proportional to the angular momentum (12) and, assuming this proportionality to survive in quantum mechanics, quan- tization of magnetic moment leads to quantization of corresponding angular momentum S, which we called spin, q q µ = g S ) µ = g ~ m (57) z 2m z z 2m s as is the case with orbital angular momentum, where the ratio q=2m is called Bohr mag- neton, µb and g is known as Lande-g factor or gyromagnetic factor. The g-factor is 1 for orbital angular momentum and hence corresponding magnetic moment is l µb (where l is integer orbital angular momentum quantum number). A surprising feature here is that spin magnetic moment is also µb instead of µb=2 as one would naively expect. So it turns out g = 2 for spin { an effect that can only be understood if linearization of Schr¨odinger equation is attempted i.e. -
Finite Element Model Calibration of an Instrumented Thirteen-Story Steel Moment Frame Building in South San Fernando Valley, California
Finite Element Model Calibration of An Instrumented Thirteen-story Steel Moment Frame Building in South San Fernando Valley, California By Erol Kalkan Disclamer The finite element model incuding its executable file are provided by the copyright holder "as is" and any express or implied warranties, including, but not limited to, the implied warranties of merchantability and fitness for a particular purpose are disclaimed. In no event shall the copyright owner be liable for any direct, indirect, incidental, special, exemplary, or consequential damages (including, but not limited to, procurement of substitute goods or services; loss of use, data, or profits; or business interruption) however caused and on any theory of liability, whether in contract, strict liability, or tort (including negligence or otherwise) arising in any way out of the use of this software, even if advised of the possibility of such damage. Acknowledgments Ground motions were recorded at a station owned and maintained by the California Geological Survey (CGS). Data can be downloaded from CESMD Virtual Data Center at: http://www.strongmotioncenter.org/cgi-bin/CESMD/StaEvent.pl?stacode=CE24567. Contents Introduction ..................................................................................................................................................... 1 OpenSEES Model ........................................................................................................................................... 3 Calibration of OpenSEES Model to Observed Response