DOCSLIB.ORG

Quantum Theory. Schrödinger Equation 4

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Quantum Theory. Schrödinger Equation 4 Answer Key Integrated Chemistry Ch 4 Section 2 Review Interference is a phenomenon in which two waves superimpose to form a resultant wave of greater or lower amplitude. Interference usually refers to the interaction of waves that are correlated or coherent with each other, either because they come from the same source or because they have the same or nearly the same frequency. Interference effects can be observed with all types of waves, for example, light, radio, acoustic (sound) and surface waves (water ripples). VOCABULARY Complete each of the following sentences using the following terms: Bohr's model, diffraction, excited state, ground state, interference, orbital, or quantum theory 1. The bending of a wave as it passes by the edge of an object is called diffraction. 2. When waves overlap each other, wave interference occurs. 3. The wave properties of electrons and other very small particles are described mathematically by quantum theory. Schrödinger equation 4. The three-dimensional region around the nucleus, an orbital, indicates the probable location of an electron. Describe the contributions of each of the following experimenters to our understanding of the atom. 1. Louis de Broglie hypothesized that electrons have wavelike properties. Other investigators verified his theory by proving that electrons can be bent or diffracted. In his 1924 PhD thesis he postulated the wave nature of electrons and suggested that all matter has wave properties. This concept is known as wave- particle duality or the de Broglie hypothesis. He won the Nobel Prize for Physics in 1929. 2. Werner Heisenberg developed uncertainty principle which is one of the fundamental building blocks of our understanding of light and matter. The principle is named after him and states that it is impossible to determine simultaneously both the position and velocity of an electron or any other particle. Heisenberg was awarded the Nobel Prize in Physics for 1932 "for the creation of quantum mechanics" 3. Erwin Schrödinger developed an equation that treats electrons like waves. His equation along with the Heisenberg uncertainty principle laid the foundation for modern quantum theory. Nobel Prize for Physics (1933) - for the formulation of the Schrödinger equation 4. Neils Bohr was a Danish physicist who made foundational contributions to understanding atomic structure and quantum theory, for which he received the Nobel Prize in Physics in 1922. Bohr was also a philosopher and a promoter of scientific research. Write true or false for each of the following statements. false 1 In its ground state, an atom has its highest possible energy. Lowest 2 An atom is in an excited state when it has a higher potential energy than it has in its ground true state. 3 In Bohr's model of the atom, the orbit is the path taken by the atom’s protons around the false nucleus. Electrons 4 Quantum theory is a scientific explanation for the fact that hydrogen atoms give off only true certain frequencies of light. Circle the letter of the best answer. 1. Which of the following statements is true? a. Diffraction experiments showed that electrons behave more like particles than like waves. b. Wave functions give only the probability of finding an electron at a given place around the nucleus. c. All atomic orbitals have the same shape and size. d. Bohr’s model of the hydrogen atom is the accepted modern theory for the shape and behavior of atoms. 2. Which of the following statements is not true? a. Interference results in an increase of energy in some areas and a decrease of energy in others. b. Photons and electrons have about the same amount of energy. c. Only waves of specific frequencies provide solutions to the Schrödinger wave equation. d. Quantum theory proves that electrons act more like particles than like waves. 1. What are quantum numbers? Quantum numbers indicate the energy, location, shape, and orientation of atomic orbitals, as well as the spins of atomic orbitals. Fill in the chart to show what each quantum number indicates. Quantum Number Symbol What Number Indicates Principal quantum n the main energy level occupied by the electron Angular momentum l the shape of the orbital Magnetic quantum m the orientation of an orbital around the nucleus Spin quantum s the two fundamental spin states of an electron in an orbital Circle the letter of the best answer. 1. Which of the following indicates the s sublevel in the third main energy level? a. s3 b. 3xyz c. s3 d. 3s 2. Which of the following statements is true? a. A single orbital holds at least two electrons. b. The total number of orbitals at a main energy level increases as n increases. c. The number of orbitals at each main energy level is twice the principal quantum number. d. As n increases, the electron’s energy increases and its average distance from the nucleus decreases. .
Load more

Recommended publications
  • 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.
    [Show full text]
  • Wave Nature of Matter: Made Easy (Lesson 3) Matter Behaving As a Wave? Ridiculous!
    Wave Nature of Matter: Made Easy (Lesson 3) Matter behaving as a wave? Ridiculous! Compiled by Dr. SuchandraChatterjee Associate Professor Department of Chemistry Surendranath College Remember? I showed you earlier how Einstein (in 1905) showed that the photoelectric effect could be understood if light were thought of as a stream of particles (photons) with energy equal to hν. I got my Nobel prize for that. Louis de Broglie (in 1923) If light can behave both as a wave and a particle, I wonder if a particle can also behave as a wave? Louis de Broglie I’ll try messing around with some of Einstein’s formulae and see what I can come up with. I can imagine a photon of light. If it had a “mass” of mp, then its momentum would be given by p = mpc where c is the speed of light. Now Einstein has a lovely formula that he discovered linking mass with energy (E = mc2) and he also used Planck’s formula E = hf. What if I put them equal to each other? mc2 = hf mc2 = hf So for my photon 2 mp = hfhf/c/c So if p = mpc = hfhf/c/c p = mpc = hf/chf/c Now using the wave equation, c = fλ (f = c/λ) So mpc = hc /λc /λc= h/λ λ = hp So you’re saying that a particle of momentum p has a wavelength equal to Planck’s constant divided by p?! Yes! λ = h/p It will be known as the de Broglie wavelength of the particle Confirmation of de Broglie’s ideas De Broglie didn’t have to wait long for his idea to be shown to be correct.
    [Show full text]
  • Bringing out the Dead Alison Abbott Reviews the Story of How a DNA Forensics Team Cracked a Grisly Puzzle
    BOOKS & ARTS COMMENT DADO RUVIC/REUTERS/CORBIS DADO A forensics specialist from the International Commission on Missing Persons examines human remains from a mass grave in Tomašica, Bosnia and Herzegovina. FORENSIC SCIENCE Bringing out the dead Alison Abbott reviews the story of how a DNA forensics team cracked a grisly puzzle. uring nine sweltering days in July Bosnia’s Million Bones tells the story of how locating, storing, pre- 1995, Bosnian Serb soldiers slaugh- innovative DNA forensic science solved the paring and analysing tered about 7,000 Muslim men and grisly conundrum of identifying each bone the million or more Dboys from Srebrenica in Bosnia. They took so that grieving families might find some bones. It was in large them to several different locations and shot closure. part possible because them, or blew them up with hand grenades. This is an important book: it illustrates the during those fate- They then scooped up the bodies with bull- unspeakable horrors of a complex war whose ful days in July 1995, dozers and heavy earth-moving equipment, causes have always been hard for outsiders to aerial reconnais- and dumped them into mass graves. comprehend. The author, a British journalist, sance missions by the Bosnia’s Million It was the single most inhuman massacre has the advantage of on-the-ground knowl- Bones: Solving the United States and the of the Bosnian war, which erupted after the edge of the war and of the International World’s Greatest North Atlantic Treaty break-up of Yugoslavia and lasted from 1992 Commission on Missing Persons (ICMP), an Forensic Puzzle Organization had to 1995, leaving some 100,000 dead.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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].
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Nobel Prize Physicists Meet at Lindau
    From 28 June to 2 July 1971 the German island town of Lindau in Nobel Prize Physicists Lake Constance close to the Austrian and Swiss borders was host to a gathering of illustrious men of meet at Lindau science when, for the 21st time, Nobel Laureates held their reunion there. The success of the first Lindau reunion (1951) of Nobel Prize win­ ners in medicine had inspired the organizers to invite the chemists and W. S. Newman the physicists in turn in subsequent years. After the first three-year cycle the United Kingdom, and an audience the dates of historical events. These it was decided to let students and of more than 500 from 8 countries deviations in the radiocarbon time young scientists also attend the daily filled the elegant Stadttheater. scale are due to changes in incident meetings so they could encounter The programme consisted of a num­ cosmic radiation (producing the these eminent men on an informal ber of lectures in the mornings, two carbon isotopes) brought about by and personal level. For the Nobel social functions, a platform dis­ variations in the geomagnetic field. Laureates too the Lindau gatherings cussion, an informal reunion between Thus chemistry may reveal man­ soon became an agreeable occasion students and Nobel Laureates and, kind’s remote past whereas its long­ for making or renewing acquain­ on the last day, the traditional term future could well be shaped by tances with their contemporaries, un­ steamer excursion on Lake Cons­ the developments mentioned by trammelled by the formalities of the tance to the island of Mainau belong­ Mössbauer, viz.
    [Show full text]
  • Niels Bohr and the Atomic Structure
    GENERAL ¨ ARTICLE Niels Bohr and the Atomic Structure M Durga Prasad Niels Bohr developed his model of the atomic structure in 1913 that succeeded in explaining the spectral features of hydrogen atom. In the process, he incorporated some non-classical features such as discrete energy levels for the bound electrons, and quantization of their angular momenta. Introduction MDurgaPrasadisa Professor of Chemistry An atom consists of two interacting subsystems. The first sub- at the University of system is the atomic nucleus, consisting of Z protons and A–Z Hyderabad. His research neutrons, where Z and A are the atomic number and atomic weight interests lie in theoretical respectively, of the concerned atom. The electrons that are part of chemistry. the second subsystem occupy (to zeroth order approximation) discrete energy levels, called the orbitals, according to the Aufbau principle with the restriction that no more than two electrons occupy a given orbital. Such paired electrons must have their spins in opposite direction (Pauli’s exclusion principle). The two subsystems interact through Coulomb attraction that binds the electrons to the nucleus. The nucleons in turn are trapped in the nucleus due to strong interactions. The two forces operate on significantly different scales of length and energy. Consequently, there is a clear-cut demarcation between the nuclear processes such as radioactivity or fission, and atomic processes involving the electrons such as optical spectroscopy or chemical reactivity. This picture of the atom was developed in the period between 1890 and 1928 (see the timeline in Box 1). One of the major figures in this development was Niels Bohr, who introduced most of the terminology that is still in use.
    [Show full text]
  • Otto Stern Annalen 4.11.11
    (To be published by Annalen der Physik in December 2011) Otto Stern (1888-1969): The founding father of experimental atomic physics J. Peter Toennies,1 Horst Schmidt-Böcking,2 Bretislav Friedrich,3 Julian C.A. Lower2 1Max-Planck-Institut für Dynamik und Selbstorganisation Bunsenstrasse 10, 37073 Göttingen 2Institut für Kernphysik, Goethe Universität Frankfurt Max-von-Laue-Strasse 1, 60438 Frankfurt 3Fritz-Haber-Institut der Max-Planck-Gesellschaft Faradayweg 4-6, 14195 Berlin Keywords History of Science, Atomic Physics, Quantum Physics, Stern- Gerlach experiment, molecular beams, space quantization, magnetic dipole moments of nucleons, diffraction of matter waves, Nobel Prizes, University of Zurich, University of Frankfurt, University of Rostock, University of Hamburg, Carnegie Institute. We review the work and life of Otto Stern who developed the molecular beam technique and with its aid laid the foundations of experimental atomic physics. Among the key results of his research are: the experimental test of the Maxwell-Boltzmann distribution of molecular velocities (1920), experimental demonstration of space quantization of angular momentum (1922), diffraction of matter waves comprised of atoms and molecules by crystals (1931) and the determination of the magnetic dipole moments of the proton and deuteron (1933). 1 Introduction Short lists of the pioneers of quantum mechanics featured in textbooks and historical accounts alike typically include the names of Max Planck, Albert Einstein, Arnold Sommerfeld, Niels Bohr, Max von Laue, Werner Heisenberg, Erwin Schrödinger, Paul Dirac, Max Born, and Wolfgang Pauli on the theory side, and of Wilhelm Conrad Röntgen, Ernest Rutherford, Arthur Compton, and James Franck on the experimental side. However, the records in the Archive of the Nobel Foundation as well as scientific correspondence, oral-history accounts and scientometric evidence suggest that at least one more name should be added to the list: that of the “experimenting theorist” Otto Stern.
    [Show full text]
  • Heisenberg's Visit to Niels Bohr in 1941 and the Bohr Letters
    Klaus Gottstein Max-Planck-Institut für Physik (Werner-Heisenberg-Institut) Föhringer Ring 6 D-80805 Munich, Germany 26 February, 2002 New insights? Heisenberg’s visit to Niels Bohr in 1941 and the Bohr letters1 The documents recently released by the Niels Bohr Archive do not, in an unambiguous way, solve the enigma of what happened during the critical brief discussion between Bohr and Heisenberg in 1941 which so upset Bohr and made Heisenberg so desperate. But they are interesting, they show what Bohr remembered 15 years later. What Heisenberg remembered was already described by him in his memoirs “Der Teil und das Ganze”. The two descriptions are complementary, they are not incompatible. The two famous physicists, as Hans Bethe called it recently, just talked past each other, starting from different assumptions. They did not finish their conversation. Bohr broke it off before Heisenberg had a chance to complete his intended mission. Heisenberg and Bohr had not seen each other since the beginning of the war in 1939. In the meantime, Heisenberg and some other German physicists had been drafted by Army Ordnance to explore the feasibility of a nuclear bomb which, after the discovery of fission and of the chain reaction, could not be ruled out. How real was this theoretical possibility? By 1941 Heisenberg, after two years of intense theoretical and experimental investigations by the drafted group known as the “Uranium Club”, had reached the conclusion that the construction of a nuclear bomb would be feasible in principle, but technically and economically very difficult. He knew in principle how it could be done, by Uranium isotope separation or by Plutonium production in reactors, but both ways would take many years and would be beyond the means of Germany in time of war, and probably also beyond the means of Germany’s adversaries.
    [Show full text]