The Structure of Hydrogen
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Group Problems #22 - Solutions
Group Problems #22 - Solutions Friday, October 21 Problem 1 Ritz Combination Principle Show that the longest wavelength of the Balmer series and the longest two wavelengths of the Lyman series satisfy the Ritz Combination Principle. The wavelengths corresponding to transitions between two energy levels in hydrogen is given by: n2 λ = λlimit 2 2 ; (1) n − n0 where n0 = 1 for the Lyman series, n0 = 2 for the Balmer series, and λlimit = 91:1 nm for the Lyman series and λlimit = 364:5 nm for the Balmer series. Inspection of this equation shows that the longest wavelength corresponds to the transition between n = n0 and n = n0 + 1. Thus, for the Lyman series, the longest two wavelengths are λ12 and λ13. For the Balmer series, the longest wavelength is λ23. The Ritz combination principle states that certain pairs of observed frequencies from the hydrogen spectrum add together to give other observed frequencies. For this particular problem, we then have: ν12 + ν23 = ν13 =) h(ν12 + ν23) = hν13 (2) 1 1 1 =) hc + = hc (3) λ12 λ23 λ13 1 1 1 =) + = : (4) λ12 λ23 λ13 Inverting Eqn. 1 above gives: 2 2 2 1 1 n − n0 1 n0 = 2 = 1 − 2 : (5) λ λlimit n λlimit n To calculate 1/λ12 and 1/λ13, we use the Lyman series numbers and find 1/λ12 = −1 −1 3=(4 ∗ 91:1) = 0:00823 nm and 1/λ13 = 8=(9 ∗ 91:1) = 0:00976 nm . To compute 1/λ23, we use the Balmer series numbers and find 1/λ23 = 5=(9 ∗ 364:5) = 0:00152 −1 nm . -
Observations of the Lyman Series Forest Towards the Redshift 7.1 Quasar ULAS J1120+0641 R
A&A 601, A16 (2017) Astronomy DOI: 10.1051/0004-6361/201630258 & c ESO 2017 Astrophysics Observations of the Lyman series forest towards the redshift 7.1 quasar ULAS J1120+0641 R. Barnett1, S. J. Warren1, G. D. Becker2, D. J. Mortlock1; 3; 4, P. C. Hewett5, R. G. McMahon5; 6, C. Simpson7, and B. P. Venemans8 1 Astrophysics Group, Blackett Laboratory, Imperial College London, London SW7 2AZ, UK e-mail: [email protected] 2 Department of Physics & Astronomy, University of California, Riverside, 900 University Avenue, Riverside, CA 92521, USA 3 Department of Mathematics, Imperial College London, London SW7 2AZ, UK 4 Department of Astronomy, Stockholm University, Albanova, 10691 Stockholm, Sweden 5 Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK 6 Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK 7 Gemini Observatory, Northern Operations Center, N. A’ohoku Place, Hilo, HI 96720, USA 8 Max-Planck Institut für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany Received 15 December 2016 / Accepted 9 February 2017 ABSTRACT We present a 30 h integration Very Large Telescope X-shooter spectrum of the Lyman series forest towards the z = 7:084 quasar ULAS J1120+0641. The only detected transmission at S=N > 5 is confined to seven narrow spikes in the Lyα forest, over the redshift range 5:858 < z < 6:122, just longward of the wavelength of the onset of the Lyβ forest. There is also a possible detection of one further unresolved spike in the Lyβ forest at z = 6:854, with S=N = 4:5. -
Unique Astrophysics in the Lyman Ultraviolet
Unique Astrophysics in the Lyman Ultraviolet Jason Tumlinson1, Alessandra Aloisi1, Gerard Kriss1, Kevin France2, Stephan McCandliss3, Ken Sembach1, Andrew Fox1, Todd Tripp4, Edward Jenkins5, Matthew Beasley2, Charles Danforth2, Michael Shull2, John Stocke2, Nicolas Lehner6, Christopher Howk6, Cynthia Froning2, James Green2, Cristina Oliveira1, Alex Fullerton1, Bill Blair3, Jeff Kruk7, George Sonneborn7, Steven Penton1, Bart Wakker8, Xavier Prochaska9, John Vallerga10, Paul Scowen11 Summary: There is unique and groundbreaking science to be done with a new generation of UV spectrographs that cover wavelengths in the “Lyman Ultraviolet” (LUV; 912 - 1216 Å). There is no astrophysical basis for truncating spectroscopic wavelength coverage anywhere between the atmospheric cutoff (3100 Å) and the Lyman limit (912 Å); the usual reasons this happens are all technical. The unique science available in the LUV includes critical problems in astrophysics ranging from the habitability of exoplanets to the reionization of the IGM. Crucially, the local Universe (z ≲ 0.1) is entirely closed to many key physical diagnostics without access to the LUV. These compelling scientific problems require overcoming these technical barriers so that future UV spectrographs can extend coverage to the Lyman limit at 912 Å. The bifurcated history of the Space UV: Much the course of space astrophysics can be traced to the optical properties of ozone (O3) and magnesium fluoride (MgF2). The first causes the space UV, and the second divides the “HST UV” (1150 - 3100 Å) and the “FUSE UV” (900 - 1200 Å). This short paper argues that some critical problems in astrophysics are best solved by a future generation of high-resolution UV spectrographs that observe all wavelengths between the Lyman limit and the atmospheric cutoff. -
Fundamental Nature of the Fine-Structure Constant Michael Sherbon
Fundamental Nature of the Fine-Structure Constant Michael Sherbon To cite this version: Michael Sherbon. Fundamental Nature of the Fine-Structure Constant. International Journal of Physical Research , Science Publishing Corporation 2014, 2 (1), pp.1-9. 10.14419/ijpr.v2i1.1817. hal-01304522 HAL Id: hal-01304522 https://hal.archives-ouvertes.fr/hal-01304522 Submitted on 19 Apr 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution| 4.0 International License Fundamental Nature of the Fine-Structure Constant Michael A. Sherbon Case Western Reserve University Alumnus E-mail: michael:sherbon@case:edu January 17, 2014 Abstract Arnold Sommerfeld introduced the fine-structure constant that determines the strength of the electromagnetic interaction. Following Sommerfeld, Wolfgang Pauli left several clues to calculating the fine-structure constant with his research on Johannes Kepler’s view of nature and Pythagorean geometry. The Laplace limit of Kepler’s equation in classical mechanics, the Bohr-Sommerfeld model of the hydrogen atom and Julian Schwinger’s research enable a calculation of the electron magnetic moment anomaly. Considerations of fundamental lengths such as the charge radius of the proton and mass ratios suggest some further foundational interpretations of quantum electrodynamics. -
Dirac's Fine-Structure Formula for an “Abnormally High”
Dirac’s fine-structure formula for an “abnormally high” nuclear charge The solution of an old riddle A. LOINGER Dipartimento di Fisica dell’Università di Milano Via Celoria, 16 − 20133 Milano, Italy Summary. − In Dirac’s fine-structure formula for the hydrogenlike atoms a critical role is played by the square root of the following expression: the unity minus the square of the product of the atomic number by the fine-structure constant (which is approximately equal to 1/137). Indeed, for a fictitious, ideal nucleus for which the above product is greater than or equal to one, the mentioned square root becomes imaginary, or zero. I prove in this Note that the origin of such theoretical “breaking down” is quite simple and is inherent in the special theory of relativity of classical physics. PACS 11.10 − Relativistic wave equations; 03.65 – Semiclassical theories and applications. 1. − The problem As is well known, Dirac’s fine-structure formula for the energy of an electron in the Coulomb field generated by a fixed nuclear charge Ze is as follows: −1 2 2 2 E Z α (1.1) = 1+ , 2 2 m0c 2 2 2 1 2 []nr + (κ − Z α ) 2 where: E = W + m0c is the relativistic total energy; m0 is the rest-mass of the 2 electron; α := e (hc) is the fine-structure constant; nr = 0,1,2,3,... is the radial quantum number; κ = ±1,±2,±3,... is the auxiliary quantum number (according to a Weylian terminology); when nr = 0 , the quantum κ takes only the positive values [1]. -
Arxiv:1704.03416V1 [Astro-Ph.GA] 11 Apr 2017 2
Physics of Lyα Radiative Transfer Mark Dijkstra (Institute of Theoretical Astrophysics, University of Oslo) (Dated: April 7, 2017) Lecture notes for the 46th Saas-Fee winterschool, 13-19 March 2016, Les Diablerets, Switzerland. See http://obswww.unige.ch/Courses/saas-fee-2016/ arXiv:1704.03416v1 [astro-ph.GA] 11 Apr 2017 2 Contents 1. Preface 3 2. Introduction 3 3. The Hydrogen Atom and Introduction to Lyα Emission Mechanisms 4 3.1. Hydrogen in our Universe 4 3.2. The Hydrogen Atom: The Classical & Quantum Picture 6 3.3. Radiative Transitions in the Hydrogen Atom: Lyman, Balmer, ..., Pfund, .... Series 8 3.4. Lyα Emission Mechanisms 9 4. A Closer Look at Lyα Emission Mechanisms & Sources 10 4.1. Collisions 10 4.2. Recombination 13 5. Astrophysical Lyα Sources 14 5.1. Interstellar HII Regions 14 5.2. The Circumgalactic/Intergalactic Medium (CGM/IGM) 16 6. Step 1 Towards Understanding Lyα Radiative Transfer: Lyα Scattering Cross-section 22 6.1. Interaction of a Free Electron with Radiation: Thomson Scattering 22 6.2. Interaction of a Bound Electron with Radiation: Lorentzian Cross-section 25 6.3. Interaction of a Bound Electron with Radiation: Relation to Lyα Cross-Section 26 6.4. Voigt Profile of Lyα Cross-Section 27 7. Step 2 Towards Understanding Lyα Radiative Transfer: The Radiative Transfer Equation 29 7.1. I: Absorption Term: Lyα Cross Section 30 7.2. II: Volume Emission Term 30 7.3. III: Scattering Term 30 7.4. IV: ‘Destruction’ Term 36 7.5. Lyα Propagation through HI: Scattering as Double Diffusion Process 38 8. -
The Fine Structure Constant
GENERAL ⎜ ARTICLE The Fine Structure Constant Biman Nath The article discusses the importance of the fine structure constant in quantum mechanics, along with the brief history of how it emerged. Al- though Sommerfelds idea of elliptical orbits has been replaced by wave mechanics, the fine struc- ture constant he introduced has remained as an Biman Nath is an important parameter in the field of atomic struc- astrophysicist at the ture. Raman Research Institute, Bengaluru. The values of the constants of Nature, such as Newton’s gravitational constant ‘G’, determine the nature of our Universe. Among these constants, there are a few which are pure numbers and have no units. For example, there is the ‘fine structure constant’, denoted by α, which has a value roughly 1/137. The value of α is related to the electromagnetic force between subatomic charged parti- cles, and essentially determines how an atom holds to- gether its electrons. It is however not obvious why this constant has this par- ticular value. Why 1/137 and not some other value? One might think the question is meaningless, but it is not. If the value of this constant had been slightly smaller or larger, even by as little as 4%, then stars would not have been able to sustain the nuclear reac- tions in their core that produce carbon. As a result, there would not have been any carbon-based lifeforms in our Universe. Therefore, the question why α ≈ 1/137 is not completely irrelevant. Scientists have even wondered if its value remains a con- stant over time. -
Balmer and Rydberg Equations for Hydrogen Spectra Revisited
1 Balmer and Rydberg Equations for Hydrogen Spectra Revisited Raji Heyrovska Institute of Biophysics, Academy of Sciences of the Czech Republic, 135 Kralovopolska, 612 65 Brno, Czech Republic. Email: [email protected] Abstract Balmer equation for the atomic spectral lines was generalized by Rydberg. Here it is shown that 1) while Bohr's theory explains the Rydberg constant in terms of the ground state energy of the hydrogen atom, quantizing the angular momentum does not explain the Rydberg equation, 2) on reformulating Rydberg's equation, the principal quantum numbers are found to correspond to integral numbers of de Broglie waves and 3) the ground state energy of hydrogen is electromagnetic like that of photons and the frequency of the emitted or absorbed light is the difference in the frequencies of the electromagnetic energy levels. Subject terms: Chemical physics, Atomic physics, Physical chemistry, Astrophysics 2 An introduction to the development of the theory of atomic spectra can be found in [1, 2]. This article is divided into several sections and starts with section 1 on Balmer's equation for the observed spectral lines of hydrogen. 1. Balmer's equation Balmer's equation [1, 2] for the observed emitted (or absorbed) wavelengths (λ) of the spectral lines of hydrogen is given by, 2 2 2 λ = h B,n1[n2 /(n2 - n 1 )] (1) 2 where n1 and n2 (> n1) are integers with n1 having a constant value, hB,n1 = n1 hB is a constant for each series and hB is a constant. For the Lyman, Balmer, Paschen, Bracket and Pfund series, n1 = 1, 2, 3, 4 and 5 respectively. -
Spectra of Photoionized Regions 1. Introduction
Spectra of Photoionized Regions Friday, January 21, 2011 CONTENTS: 1. Introduction 2. Recombination Processes A. Overview B. Level networks C. Free-bound continuum D. Two-photon continuum E. Collisions 3. Additional Processes A. Lyman-α emission B. Helium processes C. Resonance fluorescence 4. Diagnostics A. Temperature B. Density C. Composition D. Star formation rates 1. Introduction Our next task is to consider the emitted spectrum from an H II region. This is one of the most important subjects observationally since spectroscopy provides our principal set of diagnostics on the physics occurring in the region. This material is (mostly) covered in Chapters 4 and 5 of Osterbrock & Ferland (on which most of these notes are based). The sources of emission we will consider fall into two categories: Cooling: Infrared fine structure lines Optical forbidden lines Free-free continuum Recombination: Free-bound continuum Recombination lines Two-photon continuua 1 Generally, the forbidden lines (optical or IR) come from metals, whereas the recombination features are proportional to abundance and arise mainly from H and He. Since the cooling radiation has already been discussed, we will focus on recombination and then proceed to diagnostics. 2. Recombination Processes A. OVERVIEW We first consider the case of recombination of hydrogen at very low densities where the collision rates are small, i.e. any hydrogen atom that forms can decay radiatively before any collision. Later on we will introduce collisions between particles (electrons, protons) with excited atoms. The recombination process at low density begins with a radiative recombination, H+ + e− → H0(nl) + γ , which populates an excited state of the hydrogen atom. -
The Hydrogen Atom and the Lyman Series Block: ______Answer Each Question in the Space Required
Chemistry Worksheet NAME: _____________________________ The Hydrogen Atom and the Lyman series Block: _________ Answer each question in the space required. Show all work. Pay attention to significant figures and units. 1. Hydrogen is the simplest atom in the universe, thus it is the most commonly used example for delving into the quantum world. The energy levels of hydrogen can be calculated using the 2 principle quantum number, n, in the equation En = −13.6 eV (1/n ), for n = 1, 2, 3, 4 … ∞. Recall that eV stands for the energy unit called the electron volt, which relates to joules (J) by the following conversion: 1 eV = 1.602 × 10−19 J. a. What is the value of principal quantum number, n, for the ground state of hydrogen? b. Calculate the energy (in eV and in joules) for the ground state of hydrogen. c. Calculate the energy, En, for the first four excited states of the hydrogen atom. d. Is the third excited state higher or lower in energy than the ground state? Justify your answer in one or two sentences. e. The ionization energy of hydrogen is defined as the minimum energy required to excite an electron from the ground state to a state where the electron is separated from the nucleus (the proton). From your answers above, what is the ionization energy for hydrogen? Justify your answer in one or two sentences. f. How much energy would be required to ionize an entire mole of hydrogen atoms? Copyright© 2007, Alan D. Crosby, Newton South High School, Newton, MA 02459 Chemistry Worksheet NAME: _____________________________ The Hydrogen Atom and the Lyman series Block: _________ 2. -
2.5. the Fine Structure of a Hydrogen Atom
Phys460.nb 33 0 0 0= ψ a A H'-H'A ψb = (2.281) 0 0 0 0 0 0 0 0 0 0 ψa A H' ψb - ψ a H'A ψb =A a ψa H' ψb -A b ψa H' ψb =(A a -A b) ψa H' ψb 0 0 If Aa ≠A b, this equation means that ψa H' ψb = 0. 0 0 For this situation, although we have a degeneracy, one can just do non-degenerate perturbation for ψa and ψb (separately) and there will be no singularities at all. 2.5. the fine structure of a hydrogen atom 2.5.1. Relativistic correction In QMI, we solved an ideal model for a hydrogen atom (i.e. a particle in 1/r potential). In a real hydrogen atom, that model missed some of the physics, and one of them is relativistic effects. Q: what is the energy of a particle, if the particle is moving at speed v and the rest mass m. m E=Mc 2 = c2 2 (2.282) 1- v c2 Q: what is the momentum of a particle, if the particle is moving at speed v and the rest mass m. m p =Mv= v 2 (2.283) 1- v c2 As a result, E= p2 c2 +m 2 c4 (2.284) To prove this relation, we start from the r.h.s., p2 c2 +m 2 c4 = v2 2 2 2 2 2 m2 c41- 2 2 2 2 4 2 2 2 2 4 m v m v c c2 m v c +m c -m c v m c m (2.285) c2 +m 2 c4 = + = = = c2 =E v2 v2 v2 v2 v2 1- 2 1- 2 1- 2 1- 2 1- 2 2 c c c c c 1- v c2 This relation between E and p is an very important relation for relativistic physics! Q: what is kinetic energy? A: First, find the energy of a particle when it is not moving p = 0. -
Fine and Hyperfine Structure of the Hydrogen Atom
UNIT 4 Fine and hyperfine structure of the hydrogen atom Notes by N. Sirica and R. Van Wesep Previously, we solved the time-independent Sch¨odingerequation for the Hydro- gen atom, described by the Hamiltonian p2 e2 H = − (1) 2m r Where e is the electron’s charge in your favorite units. However, this is not really the Hamiltonian for the Hydrogen atom. It is non-relativistic and it does not contain spin. In order to completely describe the Hydrogen we would need to use the Dirac equation. We will not introduce that equation here, but we will say a few words about the most important energy level of the relativistic Hydrogen atom, namely the rest mass energy. E = mc2 ≈ 0.5MeV (2) We can compare this to the ground state (ionization) energy we found for the Hamiltonian in 1. 2 E = −Eion ≈ 13.6eV ¿ mc (3) Even though the rest mass energy is so much larger, it appears constant in the non-relativistic regime. Since differences in energy are important, we could ignore it before. The rest mass energy may be large, but it does not enter the world of everyday experience. Regardless, it is fruitful to investigate the relative size of the ionization energy to the rest mass energy. E me4 1 e4 α2 ion = = = (4) mc2 2~2 mc2 2c2~2 2 86 UNIT 4: Fine and hyperfine structure of the hydrogen atom e2 1 Where α = ~c = 137 is a fundamental constant of nature. No one understands it, but it is important that it is small.