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University of San Francisco Modern Physics for Frommies II The Universe of Schr ödinger’s Cat

University of San Francisco Modern Physics for Frommies II The Universe of Schr ödinger’s Cat Lecture 3. Agenda (1). Administrative matters More on Wave-Particle Duality Young’s Double-Slit Experiment Wave Nature of Matter De Broglie’s Hypothesis Early Models of the Atom

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University of San Francisco Modern Physics for Frommies II The Universe of Schr ödinger’s Cat

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  1. University of San Francisco Modern Physics for Frommies II The Universe of Schrödinger’s Cat Lecture 3 Modern Physics II Lecture 3

  2. Agenda (1) • Administrative matters • More on Wave-Particle Duality • Young’s Double-Slit Experiment • Wave Nature of Matter • De Broglie’s Hypothesis • Early Models of the Atom • Plum Puddings and Blackberry Jam • Rutherford scattering and the nucleus • Another UV Catastrophe • Troubles with Atomic Spectra Modern Physics II Lecture 3

  3. Agenda (2) • The Atom of Niels Bohr • de Broglie’s Hypothesis and the Bohr Atom • The Atom of Bohr Kneels1 1 “The Atom of Niels Bohr” and “The Atom of Bohr Kneels” are chapter tiles taken from a book that I read as a child and whose title I cannot remember. I think the author was George Gamow to whom I am indebted. Modern Physics II Lecture 3

  4. Administrative Matters Correction to previous course wiki URL: http://modphysifrommi1.wiki.usfca.edu/ Thanks to Bud Bronstein, one of us, and to Ginny Wallace of USF’s ITS Correction to CLEA URL: http://www.gettysburg.edu/~marschal/clea/CLEAhome.html Thanks to Jonathan Marsh Modern Physics II Lecture 3 First Physics and Astronomy colloquium

  5. First Physics and Astronomy Colloquium: Wednesday, 10 February 2010 at 4 PM Professor Richard Muller, Department of Physics, UC Berkeley Topic TBA Refreshments at 3:30 PM Harney Science Center Room 127 Modern Physics II Lecture 3

  6. Waves and Interference: Superposition principle: Linear combinations of solutions are also solutions. Because of phase, addition is not simply the addition of intensities. It is almost “vectorial” in nature. Simplest cases: Constructive interference: waves are in phase or out of phase by n • (360º or 2p rad.) where n = 0, 1, 2, 3,… An integral number of wavelengths or cycles. Destructive interference: waves are out of phase by (n +1/2) •(360º or 2p rad.). An odd ½ integral number of wavelengths Modern Physics II Lecture 3

  7. Constructive Waves add y = y1 + y2 at every point in space and/or in time. 2 = 1 Modern Physics II Lecture 3

  8. Destructive Waves subtract y = y1 - y2 at every point in space and/or in time. 2 = 1 -180º Modern Physics II Lecture 3

  9. Young’s Double Slit Experiment: Thomas Young (1773-1829), a British Physician, performed this experiment in 1804. Modern Physics II Lecture 3

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  12. Effect of changing l or d : Increase l, pattern spreads out Decrease d, pattern spreads Difference between sound and light diffracting around doorway Use of diffraction pattern to analyze wavelengths Modern Physics II Lecture 3

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  14. Now let’s turn down the power so only one photon passes through the apparatus at a time. HeNe laser: l= 632.991 nm, 10,000 photons/sec = 3.13 fW of power Replace screen with photographic film or a moveable photomultiplier tube. Modern Physics II Lecture 3

  15. Count photons for a fixed time at various positions across the slits and for one slit blocked and both slits open. What is each photon interfering with if it is alone in the apparatus? In a sense, each photon passes through both slits and interferes with itself!! Modern Physics II Lecture 3

  16. de Broglie’s Hypothesis Major symmetry fan: Waves sometimes act like particles (given) Particles sometimes act like waves (hypothesis) Sometimes called the de Broglie wavelength of a particle Louis de Broglie (1892-1987) Sounds nuts, but remember h is very small ( 10-34 J·sec) A couple of examples may restore your gullibility. Modern Physics II Lecture 3

  17. Wavelength of a ball: 0.20 kg moving with speed of 15 m/sec Very small, something like 20 Planck lengths l of any ordinary object is much to small to be detected. Interference and diffraction are significant only when the sizes of objects or slits are not much larger than l Modern Physics II Lecture 3

  18. Wavelength of an electron: Accelerated thru a potential difference of 100 V so we do not need to use relativistic mechanics. Scale of atomic size Modern Physics II Lecture 3

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  20. Electron Diffraction and Interference 100 eV e- dsinq Path length difference d sinq q q           d Crystal lattice, atomic separation d      Suppose q =24º is smallest angle for diffraction maximum, what is d? Modern Physics II Lecture 3

  21. For diffraction maximum, path difference is integral multiple of the de Broglie wavelength. For smallest q d sinq = l Modern Physics II Lecture 3

  22. Neutrons with v = 2 km/sec This wavelength is that of near infrared radiation  Easy to make double slits that will give interference if neutrons act like waves. Such patterns are observed In fact , this result has been extended to include whole atoms, molecules and exotic concoctions like buckyballs. Modern Physics II Lecture 3

  23. Electron Microscopy Rayleigh criterion Decrease l  improved resolution Optics becomes more difficult as l gets smaller Glass cutoff  230 nm Quartz cutoff  180 nm NaF cutoff  130 nm Modern Physics II Lecture 3

  24. Use electrons and focus with magnetic “lenses”. E- accelerated through 100kV has l 0.004 nm In practice, aberrations in magnetic lenses currently limit this resolution to about 0.1 to 0.5 nm at best. 1000 times better than a light microscope This is M = 106. Hard to do, common Ms are 104 - 105 Modern Physics II Lecture 3

  25. Early Atomic Models • Early atomic models • Thomson and the electron, the Lorentz model • Rutherford scattering and the nuclear atom • Lines rather than continua • UV catastrophe II type instability • Lines rather than continua • The Atom of Niels Bohr • The hydrogen spectrum and the Rydberg formula • Quantization of angular momentum and the Bohr model • Electron waves and the Bohr model Modern Physics II Lecture 3

  26. electrons ca. 1890 Rutherford scattering (1911)  (+) charge confined to nucleus with r = 10-15-10-14 m Modern Physics II Lecture 3

  27. Rutherford Scattering and the Nuclear Atom 1911 Ernest Rutherford (1871-1937) et al. concluded a study of a particle scattering from metal foils. An excess of events at large scattering angles lead him to the conclusion that the positve charge in atoms is concentrated in a small, massive charged core, the nucleus. Atomic size ~ 10-10 m = 1 Å Nuclear size ~ 10-15 – 10-14 m = 1 fm Modern Physics II Lecture 3

  28. Atomic Instability - AnotherUV Catastrophe Classical Rutherford atom (1911) e- are accelerated and should radiate. e- lose energy and spiral into nucleus Atoms should be unstable and the universe should only have lasted a small fraction (10-9) of a second Oh, and there are other problems. Modern Physics II Lecture 3

  29. Line Spectra Emission from rarified gasses Modern Physics II Lecture 3

  30. Absorption in rarified gasses Modern Physics II Lecture 3

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  32. Line spectra cannot be explained by a classical model Modern Physics II Lecture 3

  33. The Hydrogen Spectrum Modern Physics II Lecture 3

  34. !885 J. J. Balmer (1825-1898) showed that the 4 visible lines ( 410, 434, 486 and 656 nm) fit the formula below. Later measurements extended this Balmer series of lines into the UV. The lines become closer together with decreasing l and become indistinguishable near 365 nm. 365 nm corresponds to n =  Modern Physics II Lecture 3

  35. Later experiments found similar series in the UV (Lyman series) and the IR (Paschen series). Balmer’s formula can be generalized as the Rydberg formula. These are the experimental data which Niels Bohr attempted to explain with his modification of the Rutherford model Modern Physics II Lecture 3

  36. Neils Bohr (1885-1962) Bohr’s Postulates Electrons in atoms cannot lose energy continuously, but must do so in quantum “jumps”. Electrons move about the nucleus in circular orbits, but only certain orbits are allowed. Modern Physics II Lecture 3

  37. An electron in an orbit has a definite energy and moves in the orbit without radiating energy. The allowed orbits are referred to as stationary states. Emission and absorption of radiation can only occur in conjunction with a transition between 2 stationary states. This results in an emitted or absorbed photon of frequency such that hf = E1 - E2 What makes an orbit allowed? Maybe energy is not the only quantized quantity. Modern Physics II Lecture 3

  38. This quantization of L had no firm theoretical foundation, Bohr tried various “quantum conditions”. This one worked. Imposing quantization of angular momentum and his other postulates on classical electrodynamics and classical orbital mechanics allowed Bohr to derive the Rydberg forrnula. Stability of atoms is insured. Ground state is the lowest state. There is no lower state it can attain by the emission of more energy Modern Physics II Lecture 3

  39. Binding energy is the energy one must supply to an electron in a state, n, to remove that electron from the atom. Modern Physics II Lecture 3

  40. de Broglie Waves and Bohr’s Quantization Bohr’s model was largely ad hoc. Assumptions were made so that theory would agree with experiment. No reason why orbits should be quantized. No reason why ground state should be stable De Broglie proposed that an electron in a stable orbit is actually a circular standing matter wave. Modern Physics II Lecture 3

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  42. The circumference of the wave must contain an integral number of wavelengths Bohr published his model in 1913. de Broglie did not propose matter waves until 1923. Bohr tried many quantization conditions in attempting to explain the experimental data. Quantization of angular momentum worked. Modern Physics II Lecture 3

  43. The Atom of Bohr Kneels Successes: Predicts the correct Rydberg constant for alkali atoms and things like He+ and Li++. Replace e with Zeffein the Coulomb force Failures: Is successful only for single electron atoms Fails to explain the “fine structure” of spectral lines, even for alkali atoms The ground state of Hydrogen has L=0 not 1 Cannot explain the bonding of atoms into molecules. Modern Physics II Lecture 3

  44. The Bohr model is an ad hoc theory which fits the hydrogen spectrum. It is a semiclassical theory. We now know that it does not correctly describe atoms. This description requires a true quantum mechanical theory. Important 1st step from a purely classical theory to a quantum mechanical one. “appeared to me like a miracle and appears as a miracle even today.” - A. Einstein, ca. 1940 Modern Physics II Lecture 3

  45. Erwin Schrödinger 1887 - 1961 Werner Heisenberg 1901 - 1967 Modern Physics II Lecture 3

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