Bohr Model of Hydrogen
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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. -
Franck-Hertz Experiment
IIA2 Modul Atomic/Nuclear Physics Franck-Hertz Experiment This experiment by JAMES FRANCK and GUSTAV LUDWIG HERTZ from 1914 (Nobel Prize 1926) is one of the most impressive comparisons in the search for quantum theory: it shows a very simple arrangement in the existence of discrete stationary energy states of the electrons in the atoms. ÜÔ ÖÑÒØ Á Á¾ ¹ ÜÔ ÖÑÒØ This experiment by JAMES FRANCK and GUSTAV LUDWIG HERTZ from 1914 (Nobel Prize 1926) is one of the most impressive comparisons in the search for quantum theory: it shows a very simple arrangement in the existence of discrete stationary energy states of the electrons in the atoms. c AP, Department of Physics, University of Basel, September 2016 1.1 Preliminary Questions • Explain the FRANCK-HERTZ experiment in our own words. • What is the meaning of the unit eV and how is it defined? • Which experiment can verify the 1. excitation energy as well? • Why is an anode used in the tube? Why is the current not measured directly at the grid? 1.2 Theory 1.2.1 Light emission and absorption in the atom There has always been the question of the microscopic nature of matter, which is a key object of physical research. An important experimental approach in the "world of atoms "is the study of light absorption and emission of light from matter, that the accidental investigation of the spectral distribution of light absorbed or emitted by a particular substance. The strange phenomenon was observed (first from FRAUNHOFER with the spectrum of sunlight), and was unexplained until the beginning of this century when it finally appeared: • If light is a continuous spectrum (for example, incandescent light) through a gas of a particular type of atom, and subsequently , the spectrum is observed, it is found that the light is very special, atom dependent wavelengths have been absorbed by the gas and therefore, the spectrum is absent. -
The Franck-Hertz Experiment: 100 Years Ago and Now
The Franck-Hertz experiment: 100 years ago and now A tribute to two great German scientists Zoltán Donkó1, Péter Magyar2, Ihor Korolov1 1 Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, Budapest, Hungary 2 Physics Faculty, Roland Eötvös University, Budapest, Hungary Franck-Hertz experiment anno (~1914) The Nobel Prize in Physics 1925 was awarded jointly to James Franck and Gustav Ludwig Hertz "for their discovery of the laws governing the impact of an electron upon an atom" Anode current Primary experimental result 4.9 V nobelprize.org Elv: ! # " Accelerating voltage ! " Verh. Dtsch. Phys. Ges. 16: 457–467 (1914). Franck-Hertz experiment anno (~1914) The Nobel Prize in Physics 1925 was awarded jointly to James Franck and Gustav Ludwig Hertz "for their discovery of the laws governing the impact of an electron upon an atom" nobelprize.org Verh. Dtsch. Phys. Ges. 16: 457–467 (1914). Franck-Hertz experiment anno (~1914) “The electrons in Hg vapor experience only elastic collisions up to a critical velocity” “We show a method using which the critical velocity (i.e. the accelerating voltage) can be determined to an accuracy of 0.1 V; its value is 4.9 V.” “We show that the energy of the ray with 4.9 V corresponds to the energy quantum of the resonance transition of Hg (λ = 253.6 nm)” ((( “Part of the energy goes into excitation and part goes into ionization” ))) Important experimental evidence for the quantized nature of the atomic energy levels. The Franck-Hertz experiment: 100 years ago and now Franck-Hertz experiment: published in 1914, Nobel prize in 1925 Why is it interesting today as well? “Simple” explanation (“The electrons ....”) → description based on kinetic theory (Robson, Sigeneger, ...) Modern experiments Various gases (Hg, He, Ne, Ar) Modern experiment + kinetic description (develop an experiment that can be modeled accurately ...) → P. -
Optical Properties of Bulk and Nano
Lecture 3: Optical Properties of Bulk and Nano 5 nm Course Info First H/W#1 is due Sept. 10 The Previous Lecture Origin frequency dependence of χ in real materials • Lorentz model (harmonic oscillator model) e- n(ω) n' n'' n'1= ω0 + Nucleus ω0 ω Today optical properties of materials • Insulators (Lattice absorption, color centers…) • Semiconductors (Energy bands, Urbach tail, excitons …) • Metals (Response due to bound and free electrons, plasma oscillations.. ) Optical properties of molecules, nanoparticles, and microparticles Classification Matter: Insulators, Semiconductors, Metals Bonds and bands • One atom, e.g. H. Schrödinger equation: E H+ • Two atoms: bond formation ? H+ +H Every electron contributes one state • Equilibrium distance d (after reaction) Classification Matter ~ 1 eV • Pauli principle: Only 2 electrons in the same electronic state (one spin & one spin ) Classification Matter Atoms with many electrons Empty outer orbitals Outermost electrons interact Form bands Partly filled valence orbitals Filled Energy Inner Electrons in inner shells do not interact shells Do not form bands Distance between atoms Classification Matter Insulators, semiconductors, and metals • Classification based on bandstructure Dispersion and Absorption in Insulators Electronic transitions No transitions Atomic vibrations Refractive Index Various Materials 3.4 3.0 2.0 Refractive index: n’ index: Refractive 1.0 0.1 1.0 10 λ (µm) Color Centers • Insulators with a large EGAP should not show absorption…..or ? • Ion beam irradiation or x-ray exposure result in beautiful colors! • Due to formation of color (absorption) centers….(Homework assignment) Absorption Processes in Semiconductors Absorption spectrum of a typical semiconductor E EC Phonon Photon EV ωPhonon Excitons: Electron and Hole Bound by Coulomb Analogy with H-atom • Electron orbit around a hole is similar to the electron orbit around a H-core • 1913 Niels Bohr: Electron restricted to well-defined orbits n = 1 + -13.6 eV n = 2 n = 3 -3.4 eV -1.51 eV me4 13.6 • Binding energy electron: =−=−=e EB 2 2 eV, n 1,2,3,.. -
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. -
Particle Nature of Matter
Solved Problems on the Particle Nature of Matter Charles Asman, Adam Monahan and Malcolm McMillan Department of Physics and Astronomy University of British Columbia, Vancouver, British Columbia, Canada Fall 1999; revised 2011 by Malcolm McMillan Given here are solutions to 5 problems on the particle nature of matter. The solutions were used as a learning-tool for students in the introductory undergraduate course Physics 200 Relativity and Quanta given by Malcolm McMillan at UBC during the 1998 and 1999 Winter Sessions. The solutions were prepared in collaboration with Charles Asman and Adam Monaham who were graduate students in the Department of Physics at the time. The problems are from Chapter 3 The Particle Nature of Matter of the course text Modern Physics by Raymond A. Serway, Clement J. Moses and Curt A. Moyer, Saunders College Publishing, 2nd ed., (1997). Coulomb's Constant and the Elementary Charge When solving numerical problems on the particle nature of matter it is useful to note that the product of Coulomb's constant k = 8:9876 × 109 m2= C2 (1) and the square of the elementary charge e = 1:6022 × 10−19 C (2) is ke2 = 1:4400 eV nm = 1:4400 keV pm = 1:4400 MeV fm (3) where eV = 1:6022 × 10−19 J (4) Breakdown of the Rutherford Scattering Formula: Radius of a Nucleus Problem 3.9, page 39 It is observed that α particles with kinetic energies of 13.9 MeV or higher, incident on copper foils, do not obey Rutherford's (sin φ/2)−4 scattering formula. • Use this observation to estimate the radius of the nucleus of a copper atom. -
Sterns Lebensdaten Und Chronologie Seines Wirkens
Sterns Lebensdaten und Chronologie seines Wirkens Diese Chronologie von Otto Sterns Wirken basiert auf folgenden Quellen: 1. Otto Sterns selbst verfassten Lebensläufen, 2. Sterns Briefen und Sterns Publikationen, 3. Sterns Reisepässen 4. Sterns Züricher Interview 1961 5. Dokumenten der Hochschularchive (17.2.1888 bis 17.8.1969) 1888 Geb. 17.2.1888 als Otto Stern in Sohrau/Oberschlesien In allen Lebensläufen und Dokumenten findet man immer nur den VornamenOt- to. Im polizeilichen Führungszeugnis ausgestellt am 12.7.1912 vom königlichen Polizeipräsidium Abt. IV in Breslau wird bei Stern ebenfalls nur der Vorname Otto erwähnt. Nur im Emeritierungsdokument des Carnegie Institutes of Tech- nology wird ein zweiter Vorname Otto M. Stern erwähnt. Vater: Mühlenbesitzer Oskar Stern (*1850–1919) und Mutter Eugenie Stern geb. Rosenthal (*1863–1907) Nach Angabe von Diana Templeton-Killan, der Enkeltochter von Berta Kamm und somit Großnichte von Otto Stern (E-Mail vom 3.12.2015 an Horst Schmidt- Böcking) war Ottos Großvater Abraham Stern. Abraham hatte 5 Kinder mit seiner ersten Frau Nanni Freund. Nanni starb kurz nach der Geburt des fünften Kindes. Bald danach heiratete Abraham Berta Ben- der, mit der er 6 weitere Kinder hatte. Ottos Vater Oskar war das dritte Kind von Berta. Abraham und Nannis erstes Kind war Heinrich Stern (1833–1908). Heinrich hatte 4 Kinder. Das erste Kind war Richard Stern (1865–1911), der Toni Asch © Springer-Verlag GmbH Deutschland 2018 325 H. Schmidt-Böcking, A. Templeton, W. Trageser (Hrsg.), Otto Sterns gesammelte Briefe – Band 1, https://doi.org/10.1007/978-3-662-55735-8 326 Sterns Lebensdaten und Chronologie seines Wirkens heiratete. -
Knowledge on the Web: Towards Robust and Scalable Harvesting of Entity-Relationship Facts
Knowledge on the Web: Towards Robust and Scalable Harvesting of Entity-Relationship Facts Gerhard Weikum Max Planck Institute for Informatics http://www.mpi-inf.mpg.de/~weikum/ Acknowledgements 2/38 Vision: Turn Web into Knowledge Base comprehensive DB knowledge fact of human knowledge assets extraction • everything that (Semantic (Statistical Web) Web) Wikipedia knows • machine-readable communities • capturing entities, (Social Web) classes, relationships Source: DB & IR methods for knowledge discovery. Communications of the ACM 52(4), 2009 3/38 Knowledge as Enabling Technology • entity recognition & disambiguation • understanding natural language & speech • knowledge services & reasoning for semantic apps • semantic search: precise answers to advanced queries (by scientists, students, journalists, analysts, etc.) German chancellor when Angela Merkel was born? Japanese computer science institutes? Politicians who are also scientists? Enzymes that inhibit HIV? Influenza drugs for pregnant women? ... 4/38 Knowledge Search on the Web (1) Query: sushi ingredients? Results: Nori seaweed Ginger Tuna Sashimi ... Unagi http://www.google.com/squared/5/38 Knowledge Search on the Web (1) Query:Query: JapaneseJapanese computerscomputeroOputer science science ? institutes ? http://www.google.com/squared/6/38 Knowledge Search on the Web (2) Query: politicians who are also scientists ? ?x isa politician . ?x isa scientist Results: Benjamin Franklin Zbigniew Brzezinski Alan Greenspan Angela Merkel … http://www.mpi-inf.mpg.de/yago-naga/7/38 Knowledge Search on the Web (2) Query: politicians who are married to scientists ? ?x isa politician . ?x isMarriedTo ?y . ?y isa scientist Results (3): [ Adrienne Clarkson, Stephen Clarkson ], [ Raúl Castro, Vilma Espín ], [ Jeannemarie Devolites Davis, Thomas M. Davis ] http://www.mpi-inf.mpg.de/yago-naga/8/38 Knowledge Search on the Web (3) http://www-tsujii.is.s.u-tokyo.ac.jp/medie/ 9/38 Take-Home Message If music was invented Information is not Knowledge. -
Bohr's 1913 Molecular Model Revisited
Bohr’s 1913 molecular model revisited Anatoly A. Svidzinsky*†‡, Marlan O. Scully*†§, and Dudley R. Herschbach¶ *Departments of Chemistry and Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544; †Departments of Physics and Chemical and Electrical Engineering, Texas A&M University, College Station, TX 77843-4242; §Max-Planck-Institut fu¨r Quantenoptik, D-85748 Garching, Germany; and ¶Department of Chemistry and Chemical Biology, Harvard University, Cambridge, MA 02138 Contributed by Marlan O. Scully, July 10, 2005 It is generally believed that the old quantum theory, as presented by Niels Bohr in 1913, fails when applied to few electron systems, such as the H2 molecule. Here, we find previously undescribed solutions within the Bohr theory that describe the potential energy curve for the lowest singlet and triplet states of H2 about as well as the early wave mechanical treatment of Heitler and London. We also develop an interpolation scheme that substantially improves the agreement with the exact ground-state potential curve of H2 and provides a good description of more complicated molecules such as LiH, Li2, BeH, and He2. Bohr model ͉ chemical bond ͉ molecules he Bohr model (1–3) for a one-electron atom played a major Thistorical role and still offers pedagogical appeal. However, when applied to the simple H2 molecule, the ‘‘old quantum theory’’ proved unsatisfactory (4, 5). Here we show that a simple extension of the original Bohr model describes the potential energy curves E(R) for the lowest singlet and triplet states about Fig. 1. Molecular configurations as sketched by Niels Bohr; [from an unpub- as well as the first wave mechanical treatment by Heitler and lished manuscript (7), intended as an appendix to his 1913 papers]. -
Bohr's Model and Physics of the Atom
CHAPTER 43 BOHR'S MODEL AND PHYSICS OF THE ATOM 43.1 EARLY ATOMIC MODELS Lenard's Suggestion Lenard had noted that cathode rays could pass The idea that all matter is made of very small through materials of small thickness almost indivisible particles is very old. It has taken a long undeviated. If the atoms were solid spheres, most of time, intelligent reasoning and classic experiments to the electrons in the cathode rays would hit them and cover the journey from this idea to the present day would not be able to go ahead in the forward direction. atomic models. Lenard, therefore, suggested in 1903 that the atom We can start our discussion with the mention of must have a lot of empty space in it. He proposed that English scientist Robert Boyle (1627-1691) who the atom is made of electrons and similar tiny particles studied the expansion and compression of air. The fact carrying positive charge. But then, the question was, that air can be compressed or expanded, tells that air why on heating a metal, these tiny positively charged is made of tiny particles with lot of empty space particles were not ejected ? between the particles. When air is compressed, these 1 Rutherford's Model of the Atom particles get closer to each other, reducing the empty space. We mention Robert Boyle here, because, with Thomson's model and Lenard's model, both had him atomism entered a new phase, from mere certain advantages and disadvantages. Thomson's reasoning to experimental observations. The smallest model made the positive charge immovable by unit of an element, which carries all the properties of assuming it-to be spread over the total volume of the the element is called an atom. -
Ab Initio Calculation of Energy Levels for Phosphorus Donors in Silicon
Ab initio calculation of energy levels for phosphorus donors in silicon J. S. Smith,1 A. Budi,2 M. C. Per,3 N. Vogt,1 D. W. Drumm,1, 4 L. C. L. Hollenberg,5 J. H. Cole,1 and S. P. Russo1 1Chemical and Quantum Physics, School of Science, RMIT University, Melbourne VIC 3001, Australia 2Materials Chemistry, Nano-Science Center, Department of Chemistry, University of Copenhagen, Universitetsparken 5, 2100 København Ø, Denmark 3Data 61 CSIRO, Door 34 Goods Shed, Village Street, Docklands VIC 3008, Australia 4Australian Research Council Centre of Excellence for Nanoscale BioPhotonics, School of Science, RMIT University, Melbourne, VIC 3001, Australia 5Centre for Quantum Computation and Communication Technology, School of Physics, The University of Melbourne, Parkville, 3010 Victoria, Australia The s manifold energy levels for phosphorus donors in silicon are important input parameters for the design and modelling of electronic devices on the nanoscale. In this paper we calculate these energy levels from first principles using density functional theory. The wavefunction of the donor electron's ground state is found to have a form that is similar to an atomic s orbital, with an effective Bohr radius of 1.8 nm. The corresponding binding energy of this state is found to be 41 meV, which is in good agreement with the currently accepted value of 45.59 meV. We also calculate the energies of the excited 1s(T2) and 1s(E) states, finding them to be 32 and 31 meV respectively. These results constitute the first ab initio confirmation of the s manifold energy levels for phosphorus donors in silicon. -
Series 1 Atoms and Bohr Model.Pdf
Atoms and the Bohr Model Reading: Gray: (1-1) to (1-7) OGN: (15.1) and (15.4) Outline of First Lecture I. General information about the atom II. How the theory of the atomic structure evolved A. Charge and Mass of the atomic particles 1. Faraday 2. Thomson 3. Millikan B. Rutherford’s Model of the atom Reading: Gray: (1-1) to (1-7) OGN: (15.1) and (15.4) H He Li Be B C N O F Ne Na Mg Al Si P S Cl Ar K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe Cs Ba La Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn Fr Ra Ac Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr Atoms consist of: Mass (a.m.u.) Charge (eu) 6 6 Protons ~1 +1 Neutrons ~1 0 C Electrons 1/1836 -1 12.01 10-5 Å (Nucleus) Electron Cloud 1 Å 1 Å = 10 –10 m Note: Nucleus not drawn to scale! HOWHOW DODO WEWE KNOW:KNOW: Atomic Size? Charge and Mass of an Electron? Charge and Mass of a Proton? Mass Distribution in an Atom? Reading: Gray: (1-1) to (1-7) OGN: (15.1) and (15.4) A Timeline of the Atom 400 BC .....