DOCSLIB.ORG

Blackbody Radiation and the Quantization of Energy

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Blackbody Radiation and the Quantization of Energy Blackbody Radiation and the Quantization of Energy Energy quantization, the noncontinuous nature of energy,was discovered from observing radiant energy from blackb o dies. A blackbody absorbs electromagnetic EM energy eciently. An ideal blackb o dy absorbs all electromagnetic energy that b ears up on it. Ecient absorb ers of energy also are ecient emitters or else they would heat up without limit. Energy radiated from blackb o dies was studied b ecause the characteristics of the radiation are indep endent of the material constitution of the blackb o dy. Therefore, conclusions reached by the study of energy radiated from blackb o dies do not require quali cations related to the material studied. If you heat up a go o d absorb er emitter of EM energy, it will radiate energy with total p ower p er unit area prop ortional to the fourth p ower of temp erature in degrees Kelvin. This is the Stefan - Boltzmann law: 4 R = T ; 8 2 4 where the Stefan constant =5:6703 10 W/m K . This lawwas rst prop osed by Josef Stefan in 1879 and studied theoretically by Boltzmann a few years later, so it is named after b oth of them. The EM energy emitted from a blackb o dy actually dep ends strongly on the wavelength, , of the energy. Thus, radiantpower is a function of wavelength, R , and total p ower p er unit area is simply an integral over all wavelengths: Z R = R d: 1 The function R is called the blackbody spectrum. Recall that ! 2 c = f = ! =2f k = k The blackb o dy sp ectrum is di erent for di erent temp eratures, but has the same general shap e as the solid lines in Figure 1 show. More total p ower is emitted as the temp erature of the blackbody grows and the p eak of the sp ectrum, , shifts to shorter wavelengths. In fact, is inversely m m prop ortional to temp erature T such that: 3 T = constant =2:898 10 mK ; m which is known as Wiens' displacement law b ecause it quanti ed the displacement of the sp ectral peak as temp erature grows. Without going into any detail, it turns out that classical EM theory predicts that 4 R =2ckT Rayleigh-Jeans Law ; 2 23 5 where k =1:381 10 J/K = 8:617 10 eV/K is the Boltzmann constant. Note that at ro om temp erature the quantity kT 1=40 eV. Insp ection of this prediction, shown in Figure 1 as the dotted line, indicates that it do es not agree with observations of the blackb o dy sp ectrum. In fact, not only is there disagreement, but substitution of equation 2 into 1 results in the catastrophic result that the power per unit area radiated by a blackbody is in nite. Because the disagreement between the observed curves and the curve predicted by the Rayleigh-Jeans formula worsens as period decreases this is called the ultraviolet catastrophe. Recall that ultraviolet light is shorter wavelength than visible lightwhich is itself shorter wavelength than infrared, etc. 1 Planck’s Law 1 ) λ 1650 K 0.8 Rayleigh - Jeans Formula 0.6 1450 K 0.4 1250 K normalized spectral amplitude R( 0.2 0 0 2000 4000 6000 8000 wavelength (nm) Figure 1: Solid curves are blackb o dy sp ectra predicted by Planck's Law eqn 4 for the three temp eratures indicated, and the dotted curve is the prediction from the Rayleigh-Jeans Law eqn 2 for a temp erature of 1650 K. One of the key assumptions in the derivation of the erroneous Rayleigh-Jeans lawisthecontin- uous nature of EM energy. Planck was able to mo del the shap e of the blackbody sp ectrum much more accurately by assuming that energy is quantized, that it is not continuous: E = nhf n =0; 1; 2; 3;::: 3 n where h is Planck's constant. With this assumption, in 1900 Planck rederived the theoretical shap e of the blackb o dy sp ectrum and found the following: 2 5 2c h R = Planck's Law 4 hc=k T e 1 Fitting this functional form, which is shown for three temp eratures in Figure 1, for the unknown h, 34 15 Planck estimated Planck's constant, currentvalues of which are h =6:626 10 J-s = 4:136 10 eV-s. This is generally considered the earliest indication of the quantization of energy. Equation 3 is one that we will use throughout our study of quantum mechanics. It will turn out that energy is quantized by this formula for all forms of EM energy and for matter, as well. 2.
Load more

Recommended publications
  • Blackbody Radiation: (Vibrational Energies of Atoms in Solid Produce BB Radiation)
    Independent study in physics The Thermodynamic Interaction of Light with Matter Mirna Alhanash Project in Physics Uppsala University Contents Abstract ................................................................................................................................................................................ 3 Introduction ......................................................................................................................................................................... 3 Blackbody Radiation: (vibrational energies of atoms in solid produce BB radiation) .................................... 4 Stefan-Boltzmann .............................................................................................................................................................. 6 Wien displacement law..................................................................................................................................................... 7 Photoelectric effect ......................................................................................................................................................... 12 Frequency dependence/Atom model & electron excitation .................................................................................. 12 Why we see colours ....................................................................................................................................................... 14 Optical properties of materials: ..................................................................................................................................
    [Show full text]
  • 151545957.Pdf
    Universit´ede Montr´eal Towards a Philosophical Reconstruction of the Dialogue between Modern Physics and Advaita Ved¯anta: An Inquiry into the Concepts of ¯ak¯a´sa, Vacuum and Reality par Jonathan Duquette Facult´ede th´eologie et de sciences des religions Th`ese pr´esent´ee `ala Facult´edes ´etudes sup´erieures en vue de l’obtention du grade de Philosophiae Doctor (Ph.D.) en sciences des religions Septembre 2010 c Jonathan Duquette, 2010 Universit´ede Montr´eal Facult´edes ´etudes sup´erieures et postdoctorales Cette th`ese intitul´ee: Towards A Philosophical Reconstruction of the Dialogue between Modern Physics and Advaita Ved¯anta: An Inquiry into the Concepts of ¯ak¯a´sa, Vacuum and Reality pr´esent´ee par: Jonathan Duquette a ´et´e´evalu´ee par un jury compos´edes personnes suivantes: Patrice Brodeur, pr´esident-rapporteur Trichur S. Rukmani, directrice de recherche Normand Mousseau, codirecteur de recherche Solange Lefebvre, membre du jury Varadaraja Raman, examinateur externe Karine Bates, repr´esentante du doyen de la FESP ii Abstract Toward the end of the 19th century, the Hindu monk and reformer Swami Vivekananda claimed that modern science was inevitably converging towards Advaita Ved¯anta, an important philosophico-religious system in Hinduism. In the decades that followed, in the midst of the revolution occasioned by the emergence of Einstein’s relativity and quantum physics, a growing number of authors claimed to discover striking “par- allels” between Advaita Ved¯anta and modern physics. Such claims of convergence have continued to the present day, especially in relation to quantum physics. In this dissertation, an attempt is made to critically examine such claims by engaging a de- tailed comparative analysis of two concepts: ¯ak¯a´sa in Advaita Ved¯anta and vacuum in quantum physics.
    [Show full text]
  • Kelvin's Clouds
    Kelvin’s clouds Oliver Passon∗ July 1, 2021 Abstract In 1900 Lord Kelvin identified two problems for 19th century physics, two“clouds” as he puts it: the relative motion of the ether with respect to massive objects, and Maxwell-Boltzmann’s theorem on the equipartition of energy. These issues were eventually solved by the theory of special relativity and by quantum mechanics. In modern quotations, the content of Kelvin’s lecture is almost always distorted and misrepresented. For example, it is commonly claimed that Kelvin was concerned about the UV- catastrophe of the Rayleigh-Jeans law, while his lecture actually makes no reference at all to blackbody radiation. I rectify these mistakes and explore reasons for their occurrence. 1 Introduction In 1900, Lord Kelvin, born as William Thomson, delivered a lecture in which he identified two problems for physics at the turn of the 20th century: the question of the existence of ether and the equipartition theorem. The solution for these two clouds was eventually achieved by the theory of special relativity and by quantum mechanics [1]. Understandably, these prophetic remarks are often quoted in popular science books or in the introductory sections of textbooks on modern physics. A typical example is found in the textbook by Tipler and Llewellyn [2]: [...] as there were already vexing cracks in the foundation of what we now refer to as classical physics. Two of these were described by Lord Kelvin, in his famous Baltimore Lectures in 1900, as the “two clouds” on the horizon of twentieth-century physics: the failure of arXiv:2106.16033v1 [physics.hist-ph] 30 Jun 2021 theory to account for the radiation spectrum emitted by a blackbody and the inexplicable results of the Michelson-Morley experiment.
    [Show full text]
  • Note-A-Rific: Blackbody Radiation
    Note-A-Rific: Blackbody Radiation What is Blackbody How to Calculate Peak Failure of Classical Radiation Wavelength Physics What is Blackbody Radiation? As thermal energy is added to an object to heat it up, the object will emit radiation at various wavelengths. • All objects emit radiation at an intensity equal to its temperature (in degrees Kelvin, see Glossary) to the fourth power. 4 Intensity = T o At room temperatures, we are not aware of objects emitting radiation because it’s at such a low intensity. o At higher temperatures there is enough infrared radiation that we can feel it. o At still higher temperatures (about 1000K) we can actually see the object glowing red o At temperatures above 2000K, objects glow yellowish or white hot. No real object will absorb all radiation falling on it, nor will it re-emit all of the energy it absorbs. • Instead physicists came up with a theoretical model called a Blackbody. • A blackbody absorbs all radiation falling on it, and then releases all that energy as radiation in the form of EM waves. • As the temperature of a blackbody increases, it will emit more and more intense radiation. • At the same time, as the temperature increases, most of the radiation is released at higher and higher frequencies (lower and lower wavelengths). • The frequency at which the emitted radiation is at the highest intensity is called the peak frequency or (more typically) the peak wavelength. • This explains why an object at room temperature does not emit much radiation, and what radiation it does emit is at higher wavelengths, like infrared.
    [Show full text]
  • Blackbody Radiation and the Ultraviolet Catastrophe Introduction
    Activity – Blackbody Radiation and the Ultraviolet Catastrophe Introduction: A blackbody is a term used to describe the light given by an object that only gives off emitted light, in other words it doesn’t reflect light. Of course, in order for such an object to emit light it must get hot. In this activity, you are going to observe the nature of light given off by hot objects and determine if there is an empirical relationship between an object’s temperature and the light emitted. These are some helpful video links which provide an excellent foundation of knowledge of Blackbody Radiation. Crash Course (First Half of Video) https://www.youtube.com/watch?v=7kb1VT0J3DE&list=PL8dPuuaLjXtN0ge7yDk_UA0ldZJdhwkoV&index=44&t=438 s Professor Dave’s Summary on Blackbody Radiation and the Ultraviolet Catastrophe https://www.youtube.com/watch?v=7BXvc9W97iU PHET’s Manipulative Graph of Spectral Power Density vs. Wavelength https://phet.colorado.edu/en/simulation/blackbody-spectrum Draw a graph depicting classical blackbody radiation, (Y-axis is Spectral Power Density, X-axis is Wavelength). Complete the graph correlating the spectral power density to wavelength for the sun, as demonstrated on the Phet activity. Describe the correlation between wavelength of light emitted with the average kinetic energy of particles within a substance. Temperature in Kelvin for Red Light: Temperature in Kelvin for Yellow Light: Temperature in Kelvin for Violet Light: Questions: Dianna Cowern with PBS has an enthusiastic video explaining the origin of quantum topics. https://www.youtube.com/watch?v=FXfrncRey-4 What does the ultraviolet catastrophe refer to, and why does the curve collapse on the left-hand side of the graph? (At what approximate wavelength does the curve drop)? What is Max Planck’s explanation for the ultraviolet catastrophe, and how does it lead to the Planck equation? (Provide Planck equation and Planck’s constant).
    [Show full text]
  • Blackbody Radiation
    5. Light-matter interactions: Blackbody radiation The electromagnetic spectrum Sources of light Boltzmann's Law Blackbody radiation The cosmic microwave background The electromagnetic spectrum All of these are electromagnetic waves. The amazing diversity is a result of the fact that the interaction of radiation with matter depends on the frequency of the wave. The boundaries between regions are a bit arbitrary… Sources of light Accelerating charges emit light Linearly accelerating charge http://www.cco.caltech.edu/~phys1/java/phys1/MovingCharge/MovingCharge.html Vibrating electrons or nuclei of an atom or molecule Synchrotron radiation—light emitted by charged particles whose motion is deflected by a DC magnetic field Bremsstrahlung ("Braking radiation")—light emitted when charged particles collide with other charged particles The vast majority of light in the universe comes from molecular motions. Core (tightly bound) electrons vibrate in their motion around nuclei ~ 1018 -1020 cycles per second. Valence (not so tightly bound) electrons vibrate in their motion around nuclei ~ 1014 -1017 cycles per second. Nuclei in molecules vibrate with respect to each other ~ 1011 -1013 cycles per second. Molecules rotate ~ 109 -1010 cycles per second. One interesting source of x-rays Sticky tape emits x-rays when you unpeel it. Scotch tape generates enough x-rays to take an image of the bones in your finger. This phenomenon, known as ‘mechanoluminescence,’ was discovered in 2008. For more information and a video describing this: http://www.nature.com/nature/videoarchive/x-rays/
    [Show full text]
  • Undergraduate Course Handbook
    Undergraduate Course Handbook 2016-2017 Undergraduate Laboratory Images Images Laboratory Undergraduate The Course 1 These notes have been produced by the Department of Physics, University of Oxford. The information in this handbook is for the academic year Michaelmas Term 2016, Hilary Term 2017 and Trinity Term 2017. B A N B U Clarendon Laboratory R Y Lindemann and Martin Denys Wilkinson Building N R Wood lecture theatres Dennis Sciama lecture theatre O (via the Level 4 entrance) A D D Robert Hooke Building O A E R B L Teaching Faculty Office K E B L P A A C R K K H S Physics Teaching Laboratories S A T L (entrance down the steps) L R O GIL R D D A R D E M E U S U S ’ M gt The Department is able to make provision for students with special needs. If you think you may need any special requirements, it would be very helpful to us if you could contact the Assistant Head of Teaching (Academic) about these as soon as possible. Students in wheelchairs or with mobility needs can access the Lindemann and the Dennis Sciama Lecture Theatres by lifts from the ground floors. The Denys Wilkinson Building and the Clarendon Laboratory have toilet facilities for wheelchair users. The Martin Wood Lecture Theatre has access for wheelchairs and a reserved area within the theatre. There are induction loop systems for students with hearing difficulties in the Lindemann, Dennis Sciama and Martin Wood Lecture Theatres. Other provisions for students with special needs can be also be made.
    [Show full text]
  • Lecture 4 Introduction to Quantum Mechanical Way of Thinking
    Lecture 4 Introduction to Quantum Mechanical Way of Thinking. Today’s Program 1. Brief history of quantum mechanics (QM). 2. Wavefunctions in QM (First postulate) 3. Schrodinger’s Equation Questions you should be able to answer by the end of today’s lecture: 1. When do you observe wave-particle duality? 2. How to determine particle’s wavelength if you know its momentum? 3. What does the wavefunction represent? 4. How does wavefunction evolve in time? References for Today’s lecture: 1. Introductory Applied Quantum and Statistical Mechanics, Hagelstein, Senturia, Orlando. 2. Introduction to Quantum Mechanics, Griffiths. 1 In the end of 19th century and beginning of the 20th century, scientists were convinced they understood physics. Only a “handful” of experiments remained unexplained. These experiments became a foundation for the new kind of physics, which we now know as Quantum Mechanics. Today a lot of devices (all electronics, all optics) surrounding us are fundamentally quantum mechanical in their design and principles of operation. 1887, Hertz: Photoelectric effect. When illuminated with ultraviolet (UV) light metallic sample emitted electrons. However, the lower frequency (higher wavelength) light did not initiate electron emission even at higher incident light powers. ν I e I ν ν o Black Body Spectrum: The physicists were puzzled by the radiation of a black body, as it could not be explained by the Maxwell’s equations. Blackbody curves for various temperatures and comparison with classical theory. (This image is in the public domain. Source: Wikimedia Commons.) For short wavelength (high frequency) limit Wien’s law provided an approximation: const 1 P ,T ~ e T 5 const 3 T P,T ~ e For long wavelength (low frequency) Rayleigh-Jeans law provided another approximation: 2 T P ,T ~ Const 4 2 P,T ~ Const T Which, if correct, would have resulted in rapid increase of power at short wavelength (high frequency) labeled as “ultraviolet catastrophe”.
    [Show full text]
  • Ultraviolet Catastrophe”
    Chapter 2: The blackbody spectrum and the \ultraviolet catastrophe" Luis M. Molina Departamento de F´ısicaTe´orica,At´omicay Optica´ Quantum Physics Luis M. Molina (FTAO) Chapter 2: The blackbody spectrumQuantum and Physics the \ultraviolet 1 / 13 catastrophe" Thermal radiation What is thermal radiation? Thermal radiation is the electromagnetic radiation emitted by a body as a result of its temperature. All bodies emit such radiation to their surroundings and absorb such radiation from them. Usually, most of the radiation is emitted in frequencies outside the visible range. (for example, at the infrared at room temperature) All bodies (solids and liquids) emit a continuous spectrum of radiation. 1 Practically independent of composition 2 Strongly dependent on the temperature. Luis M. Molina (FTAO) Chapter 2: The blackbody spectrumQuantum and Physics the \ultraviolet 2 / 13 catastrophe" Blackbody Radiation When radiation impinges on a body, partly is absorbed and partly is reflected A black body is the one that absorbs all the radiation coming on it Independently of their composition, all blackbodies at the same temperature emit thermal radiation with the same spectrum. Figure 1. Incident radiation is Examples of black bodies: completely adsorbed after succesive Body painted in black (reflecting reflections. The radiation emitted by very little light) the hole will have a blackbody Cavity connected by a small hole spectrum to the outside Luis M. Molina (FTAO) Chapter 2: The blackbody spectrumQuantum and Physics the \ultraviolet 3 / 13 catastrophe" Backbody radiation: experimental results How do we measure the blackbody spectrum? We define the spectral radiancy RT (ν) such as RT (ν) dν is the energy emitted per unit time in the frequency interval [ν; ν + dν] from a unit area of the surface at temperature T.
    [Show full text]
  • 3. Particle-Like Properties of E&M Radiation
    3. Particle-like properties of E&M radiation 3.1. Maxwell’s equations... Maxwell (1831–1879) studied the following equationsa: ρ Gauss’s Law of Electricity: ~ E~ = ∇· ε0 Gauss’s Law of Magnetism: ~ B~ = 0 ∇· ∂B~ Faraday’s Law: ~ E~ = ∇× − ∂t ∂E~ Amp`ere-Maxwell Law: ~ B~ = µ0 J~ + ε0 ∇× ∂t ! aIn Chapter 1 these equations were given in cgs units. Here, the more conventional SI units are used. Gauss’s Law of Electricity describes how charges have associated with them, a divergent electric field E~ . Gauss’s Law of Magnetism describes the divergence of magnetic fields, B~ , and state that there is no magnetic “charge”. All magnetic charges are made up of magnetic dipoles Faraday’s Law shows how a time varying magnetic field can induce an electric field. The Amp`ere-Maxwell Law shows how a time varying electric field, as well as a current, J~, can induce a magnetic field. Elements of Nuclear Engineering and Radiological Sciences I NERS 311: Slide #1 The classical wave equation ... One of Maxwell’s greatest discoveries was that the simplest form (without time varying currents) of the last two equations, namely: ∂B~ 1 ∂E~ ~ E~ = ; ~ B~ = , (3.1) ∇× − ∂t ∇× c2 ∂t where c =1/√µ0ε0. (3.1) describes how an EM wave propagates. Time-varying electric fields induce time- varying magnetic fields that induce time-varying electric fields. and so on. The initiator can be a vertical antenna with vertical and descending swarms of electrons that give rise to axial magnetic fields that induce radial electric fields. Elements of Nuclear Engineering and Radiological Sciences I NERS 311: Slide #2 ..
    [Show full text]
  • Blackbody Radiation Is Totally Absorbed Within the Blackbody Blackbody = a Perfect Absorber A  1
    Modern physics Historical introduction to quantum mechanics Modern Physics, summer 2016 1 Historical introduction to quantum mechanics Gustav Kirchhoff (1824-1887) Surprisingly, the path to quantum mechanics begins with the work of German physicist Gustav Kirchhoff in 1859. Electron was discovered by J.J.Thomson in 1897 (neutron in 1932) The scientific community was reluctant to accept these new ideas. Thomson recalls such an incident: „I was told long afterwards by a distinguished physicist who had been present at my lecture that he thought I had been pulling their leg”. Modern Physics, summer 2016 2 Historical introduction to quantum mechanics Kirchhoff discovered that so called D-lines from the light emitted by the Sun came from the absorption of light from its interior by sodium atoms at the surface. Kirchhoff could not explain selective absorption. At that time Maxwell had not even begun to formulate his electromagnetic equations. Statistical mechanics did not exist and thermodynamics was in its infancy Modern Physics, summer 2016 3 Historical introduction to quantum mechanics • At that time it was known that heated solids (like tungsten W) and gases emit radiation. • Spectral radiancy Rλ is defined in such a way that Rλ dλ is the rate at which energy is radiated per unit area of surface for wavelengths lying in the interval λ to λ+d λ. • Total radiated energy R is called The spectral radiancy of radiancy and is defined as the tungsten (ribbon and cavity rate per unit surface area at radiator) at 2000 K. which energy is radiated into the forward hemisphere 2 R 23.5 W / cm R Rd 0 Modern Physics, summer 2016 4 Historical introduction to quantum mechanics Kirchhoff imagined a container – a cavity –whose walls were heated up so that they emitted radiation that was trapped in the container.
    [Show full text]
  • Lecture 30 — Atoms and the Planck Radiation Law — March 29, 1999
    Lecture 30 — Atoms and the Planck Radiation Law — March 29, 1999 Physics 221 — Lecture 30 — “Atoms and the Planck Radiation Law” March 29, 1999 Reading Meyer-Arendt Ch. 20; Hecht, Ch. 13; Young, Ch. 4. Demonstrations Planck.nb in Mathematica demonstration Problems Example 2, Ch. 20, problems 1, 2, 3, 6. Reminders See WWW for course information. 1. The end of the nineteenth century: the ultraviolet catastrophe. During the next several weeks we will be exploring quantum optics, that part of optics which is gov- erned by the laws of atoms, molecules and solids at the smallest size scales. Quan- tum electrodynamics, which underlies the phenomena of quantum optics, is perhaps the most successful theory of the twentieth century, predicting with extraordinary accuracy essentially all the properties of electrons in atoms. The origins of this the- ory lie almost a century back in time: a. Using the classical theory of the electron as an harmonic oscillator, Lorentz and others had formulated a theory of classical electrodynamics, which seemed to make perfect sense but which was seriously inconsistent with a number of experimental observations of radiation from atomic systems. b. The solution to this problem turned out to be the idea of quanta (Greek: “a little bit”) introduced by Max Planck, a professor at the Technical Univer- sity of Berlin at the end of the nineteenth century. The idea of quantization is central to our understanding of all phenomena at the atomic level. c. By the end of the nineteenth century, primitive infrared detectors were be- coming available, and physicists were starting to measure the distribution of intensity as a function of wavelength or frequency from light sources rang- ing from the sun to arc lamps (gas discharges).
    [Show full text]