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1 Quantum physics (quantum theory, quantum mechanics) Part 2

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2 Summary of 1 st lecture classical physics explanation of black-body radiation failed (ultraviolet catastrophe) Planck’s ad-hoc assumption of “energy quanta” of energy E quantum = h, leads to a radiation spectrum which agrees with experiment. old generally accepted principle of “natura non facit saltus” violated Opens path to further developments

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3 Problems from 1 st lecture estimate Sun’s surface temperature l assume Earth and Sun are black bodies l Stefan-Boltzmann law l Earth in energetic equilibrium (i.e. rad. power absorbed = rad. power emitted), mean temperature of Earth T E = 290K lSun’s angular size Sun = 32’ show that for small frequencies, Planck’s average oscillator energy yields classical equipartition result = kT show that for standing waves on a string, number of waves in band between and + is n = (2L/ 2 )

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4 Outline Introduction cathode rays …. electrons photoelectric effect l observation l studies l Einstein’s explanation models of the atom Summary

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5 Electron Cathode rays: lDuring 2 nd half of 19 th century, many physicists do experiments with “discharge tubes”, i.e. evacuated glass tubes with “electrodes” at ends, electric field between them (HV) lJohann Hittorf (1869): discharge mediated by rays emitted from negative electrode (“cathode”) -- “Kathodenstrahlen” “cathode rays” lstudy of cathode rays by many physicists – find ocathode rays appear to be particles ocast shadow of opaque body odeflected by magnetic field onegative charge leventually realized cathode rays were particles – named them electrons

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6 Photoelectric effect 1887: Heinrich Hertz: l In experiments on e.m. waves, unexpected new observation: when receiver spark gap is shielded from light of transmitter spark, the maximum spark-length became smaller l Further investigation showed: oGlass effectively shielded the spark oQuartz did not oUse of quartz prism to break up light into wavelength components find that wavelength which makes little spark more powerful was in the UV

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7 Hertz and p.e. effect oHertz’ conclusion: “I confine myself at present to communicating the results obtained, without attempting any theory respecting the manner in which the observed phenomena are brought about”

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8 Photoelectric effect– further studies 1888: Wilhelm Hallwachs (1859-1922) (Dresden) lPerforms experiment to elucidate effect observed by Hertz: oClean circular plate of Zn mounted on insulating stand; plate connected by wire to gold leaf electroscope oElectroscope charged with negative charge – stays charged for a while; but if Zn plate illuminated with UV light, electroscope loses charge quickly o If electroscope charged with positive charge: UV light has no influence on speed of charge leakage. lBut still no explanation lCalls effect “lichtelektrische Entladung” (light- electric discharge)

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9 Hallwachs’ experiments l “photoelectric discharge” l “photoelectric excitation”

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10 Path to electron 1897: three experiments measuring e/m, all with improved vacuum: lEmil Wiechert (1861-1928) (Königsberg) oMeasures e/m – value similar to that obtained by Lorentz oAssuming value for charge = that of H ion, concludes that “charge carrying entity is about 2000 times smaller than H atom” oCathode rays part of atom? oStudy was his PhD thesis, published in obscure journal – largely ignored lWalther Kaufmann (1871-1947) (Berlin) oObtains similar value for e/m, points out discrepancy, but no explanation lJ. J. Thomson

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11 1897: Joseph John Thomson (1856-1940) (Cambridge) lConcludes that cathode rays are negatively charged “corpuscles” lThen designs other tube with electric deflection plates inside tube, for e/m measurement lResult for e/m in agreement with that obtained by Lorentz, Wiechert, Kaufmann lBold conclusion: “we have in the cathode rays matter in a new state, a state in which the subdivision of matter is carried very much further than in the ordinary gaseous state: a state in which all matter... is of one and the same kind; this matter being the substance from which all the chemical elements are built up.“

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12 Identification of particle emitted in photoelectric effect 1899: J.J. Thomson: studies of photoelectric effect: lModifies cathode ray tube: make metal surface to be exposed to light the cathode in a cathode ray tube lFinds that particles emitted due to light are the same as cathode rays (same e/m)

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13 More studies of p.e. effect 1902: Philipp Lenard lStudies of photoelectric effect oMeasured variation of energy of emitted photoelectrons with light intensity oUse retarding potential to measure energy of ejected electrons: photo-current stops when retarding potential reaches V stop oSurprises: V stop does not depend on light intensity energy of electrons does depend on color (frequency) of light

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16 Explanation of photoelectric effect 1905: Albert Einstein (1879-1955) (Bern) lGives explanation of observation relating to photoelectric effect: oAssume that incoming radiation consists of “light quanta” of energy h (h = Planck’s constant, =frequency) o electrons will leave surface of metal with energy E = h – W W = “work function” = energy necessary to get electron out of the metal o there is a minimum light frequency for a given metal (that for which quantum of energy is equal to work function), below which no electron emission happens oWhen cranking up retarding voltage until current stops, the highest energy electrons must have had energy eV stop on leaving the cathode oTherefore eV stop = h – W

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17 Verification of Einstein’s explanation 1906 – 1916: Robert Millikan (1868-1963) (Chicago) l Did not accept Einstein’s explanation l Tried to disprove it by precise measurements l Result: confirmation of Einstein’s theory, measurement of h with 0.5% precision 1923: Arthur Compton (1892-1962)(St.Louis): lObserves scattering of X-rays on electrons

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18 How to see small objects “seeing an object” l= detecting light that has been reflected off the object's surface llight = electromagnetic wave; l“visible light”= those electromagnetic waves that our eyes can detect l “wavelength” of e.m. wave (distance between two successive crests) determines “color” of light lwave hardly influenced by object if size of object is much smaller than wavelength lwavelength of visible light: between 4 10 -7 m (violet) and 7 10 -7 m (red); ldiameter of atoms: 10 -10 m generalize meaning of seeing: lseeing is to detect effect due to the presence of an object lquantum theory “particle waves”, with wavelength 1/p luse accelerated (charged) particles as probe, can “tune” wavelength by choosing mass m and changing velocity v lthis method is used in electron microscope, as well as in “scattering experiments” in nuclear and particle physics

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19 J.J. Thomson’s model: l“Plum pudding or raisin cake model” o atom = sphere of positive charge (diameter 10 -10 m), o with electrons embedded in it, evenly distributed (like raisins in cake) o i.e. electrons are part of atom, can be kicked out of it – atom no longer indivisible! Models of Atom

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20 Geiger, Marsden, Rutherford expt. l(Geiger, Marsden, 1906 - 1911) (interpreted by Rutherford, 1911) lget particles from radioactive source lmake “beam” of particles using “collimators” (lead plates with holes in them, holes aligned in straight line) lbombard foils of gold, silver, copper with beam lmeasure scattering angles of particles with scintillating screen (ZnS )

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22 Geiger Marsden experiment: result lmost particles only slightly deflected (i.e. by small angles), but some by large angles - even backward lmeasured angular distribution of scattered particles did not agree with expectations from Thomson model (only small angles expected), lbut did agree with that expected fromscattering on small, dense positively charged nucleus with diameter < 10 -14 m, surrounded by electrons at 10 -10 m

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23 Rutherford model “planetary model of atom” l positive charge concentrated in nucleus (<10 -14 m); l negative electrons in orbit around nucleus at distance 10 -10 m; lelectrons bound to nucleus by electromagnetic force.

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24 Rutherford model problem with Rutherford atom: l electron in orbit around nucleus is accelerated (centripetal acceleration to change direction of velocity); l according to theory of electromagnetism (Maxwell's equations), accelerated electron emits electromagnetic radiation (frequency = revolution frequency); l electron loses energy by radiation orbit decays l changing revolution frequency continuous emission spectrum (no line spectra), and atoms would be unstable (lifetime 10 -10 s ) we would not exist to think about this!! This problem later solved by Quantum Mechanics

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25 Bohr model of hydrogen (Niels Bohr, 1913) lBohr model is radical modification of Rutherford model; discrete line spectrum attributed to “quantum effect”; lelectron in orbit around nucleus, but not all orbits allowed; lthree basic assumptions: o1. angular momentum is quantized L = n·(h/2 ) = n ·ħ, n = 1,2,3,... electron can only be in discrete specific orbits with particular radii discrete energy levels o2. electron does not radiate when in one of the allowed levels, or “states” o3. radiation is only emitted when electron makes “transition” between states, transition also called “quantum jump” or “quantum leap” lfrom these assumptions, can calculate radii of allowed orbits and corresponding energy levels: lradii of allowed orbits: r n = a 0 · n 2 n = 1,2,3,…., a 0 = 0.53 x 10 -10 m = “Bohr radius” n = “principal quantum number” lallowed energy levels: E n = - E 0 /n 2, E 0 = “Rydberg energy” lnote: energy is negative, indicating that electron is in a “potential well”; energy is = 0 at top of well, i.e. for n = , at infinite distance from the nucleus.

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26 Energies and radii in hydrogen-like atoms Potential energy in Coulomb field lU = kq 1 q 2 /R For circular orbit, potential and kinetic energies of an electron are: l U = -kZe 2 /R K = m e v 2 /2 = kZe 2 /2R l Total energy E = U + K = -kZe 2 /2R radius for stationary orbit n lR n = n 2 ħ 2 /m e kZe 2 = n 2 a 0 /Z la o = ħ 2 /m e ke 2 = 0.53 x 10 -10 m = “Bohr radius” Energy for stationary orbit n lE n = - k 2 Z 2 m e 2 e 4 /2n 2 ħ 2 = - Z 2 E 0 /n 2 l E 0 = k 2 m e 2 e 4 /2ħ 2 = 13.6 eV (Rydberg energy or Hartree energy) values of constants l k = 1/(4πε 0 ) = 8.98· 10 9 N m 2 /c 2 l m e = 0.511 MeV/c 2 l ħ = h/2π = 1.0546 · 10 -34 J s = 6.582 · 10 -22 MeV s l e = elementary charge = 1.602 · 10 -19 C l Z = nuclear charge = 1 for hydrogen, 2 for , 79 for Au (needed for solution of problems)

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27 Ground state and excited states lground state = lowest energy state, n = 1; this is where electron is under normal circumstances; electron is “at bottom of potential well”; energy needed to get it out of the well = “binding energy”; lbinding energy of ground state electron = E 1 = energy to move electron away from the nucleus (to infinity), i.e. to “liberate” electron; this energy also called “ionization energy” lexcited states = states with n > 1 lexcitation = moving to higher state lde-excitation = moving to lower state lenergy unit eV = “electron volt” = energy acquired by an electron when it is accelerated through electric potential of 1 Volt; lelectron volt is energy unit commonly used in atomic and nuclear physics; 1 eV = 1.6 x 10 -19 J lrelation between energy and wavelength: E = h = hc/, hc = 1.24 x 10 -6 eV m

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29 Excitation and de-excitation PROCESSES FOR EXCITATION: lgain energy by collision with other atoms, molecules or stray electrons; kinetic energy of collision partners converted into internal energy of the atom; kinetic lenergy comes from heating or discharge; labsorb passing photon of appropriate energy. DE-EXCITATION: lspontaneous de-excitation with emission of photon which carries energy = difference of the two energy levels; ltypically, lifetime of excited states is 10 -8 s (compare to revolution period 10 -16 s )

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30 Excitation: states of electron in hydrogen atom:

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31 Energy levels and emission Spectra Lymann = 1 Balmern = 2 Paschenn = 3 E n = E 1 · Z 2 /n 2 Hydrogen lEn = - 13.6 eV/n 2 Balmer Series m = 2 UV Visible

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32 IONIZATION: lif energy given to electron > binding energy, the atom is ionized, i.e. electron leaves atom; surplus energy becomes kinetic energy of freed electron. lthis is what happens, e.g. in photoelectric effect lionizing effect of charged particles exploited in particle detectors (e.g. Geiger counter) laurora borealis, aurora australis: cosmic rays from sun captured in earth’s magnetic field, channeled towards poles; ionization/excitation of air caused by charged particles, followed by recombination/de-excitation;

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33 Momentum of a photon Relativistic relationship between a particle’s momentum and energy: E 2 = p 2 c 2 + m 0 2 c 4 For massless (i.e. restmass = 0) particles propagating at the speed of light: E 2 = p 2 c 2 For photon, E = h momentum of photon = h /c = h/, = h/p “(moving) mass” of a photon: E=mc 2 m = E/c 2 = h /c 2 (photon feels gravity)

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34 Matter waves lLouis de Broglie (1925): any moving particle has wavelength associated with it: = h/p lexample: oelectron in atom has 10 -10 m; ocar (1000 kg) at 60mph has 10 -38 m; owave effects manifest themselves only in interaction with things of size comparable to wavelength we do not notice wave aspect of us and our cars. lnote: Bohr's quantization condition for angular momentum is identical to requirement that integer number of electron wavelengths fit into circumference of orbit. lexperimental verification of de Broglie's matter waves: o beam of electrons scattered by crystal lattice shows diffraction pattern (crystal lattice acts like array of slits); experiment done by Davisson and Germer (1927) oElectron microscope

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35 QUANTUM MECHANICS new kind of physics based on synthesis of dual nature of waves and particles; developed in 1920's and 1930's. lSchrödinger’s “wave mechanics” (Erwin Schrödinger, 1925) oSchrödinger equation is a differential equation for matter waves; basically a formulation of energy conservation. oits solution called “wave function”, usually denoted by ; o| (x)| 2 gives the probability of finding the particle at x; oapplied to the hydrogen atom, the Schrödinger equation gives the same energy levels as those obtained from the Bohr model; othe most probable orbits are those predicted by the Bohr model; obut probability instead of Newtonian certainty!

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36 QM : Heisenberg l Heisenberg’s “matrix mechanics” (Werner Heisenberg, 1925) oMatrix mechanics consists of an array of quantities which when appropriately manipulated give the observed frequencies and intensities of spectral lines. oPhysical observables (e.g. momentum, position,..) are “operators” -- represented by matrices oThe set of eigenvalues of the matrix representing an observable is the set of all possible values that could arise as outcomes of experiments conducted on a system to measure the observable. oShown to be equivalent to wave mechanics by Erwin Schrödinger (1926)

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37 Uncertainty principle lUncertainty principle: (Werner Heisenberg, 1925) oit is impossible to simultaneously know a particle's exact position and momentum p x ħ/2 h = 6.63 x 10 -34 J s = 4.14 x 10 -15 eV·s ħ = h/(2 ) = 1.055 x 10 -34 J s = 6.582 x 10 -16 eV·s ( p means “uncertainty” in our knowledge of the momentum p) oalso corresponding relation for energy and time: E t ħ/2 (but meaning here is different) lnote that there are many such uncertainty relations in quantum mechanics, for any pair of “incompatible” (non-commuting) observables (represented by “operators”) l in general, P Q ½ [P,Q] o[P,Q] = “commutator” of P and Q, = PQ – QP o A denotes “expectation value”

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38 from The God Particle by Leon Lederman: Leaving his wife at home, Schrödinger booked a villa in the Swiss Alps for two weeks, taking with him his notebooks, two pearls, and an old Viennese girlfriend. Schrödinger's self-appointed mission was to save the patched-up, creaky quantum theory of the time. The Viennese physicist placed a pearl in each ear to screen out any distracting noises. Then he placed the girlfriend in bed for inspiration. Schrödinger had his work cut out for him. He had to create a new theory and keep the lady happy. Fortunately, he was up to the task. Heisenberg is out for a drive when he's stopped by a traffic cop. The cop says, "Do you know how fast you were going?" Heisenberg says, "No, but I know where I am."

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39 Quantum Mechanics of the Hydrogen Atom E n = -13.6 eV/n 2, l n = 1, 2, 3, … (principal quantum number) Orbital quantum number l = 0, 1, 2, n-1, … oAngular Momentum, L = (h/2 ) ·√ l ( l +1) Magnetic quantum number - l m l, (there are 2 l + 1 possible values of m) Spin quantum number: m s = ½

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40 Comparison with Bohr model Angular momentum (about any axis) assumed to be quantized in units of Planck’s constant: Electron otherwise moves according to classical mechanics and has a single well-defined orbit with radius Energy quantized and determined solely by angular momentum: Bohr model Quantum mechanics Angular momentum (about any axis) shown to be quantized in units of Planck’s constant: Energy quantized, but is determined solely by principal quantum number, not by angular momentum: Electron wavefunction spread over all radii; expectation value of the quantity 1/r satisfies

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41 Multi-electron Atoms Similar quantum numbers – but energies are different. No two electrons can have the same set of quantum numbers These two assumptions can be used to motivate (partially predict) the periodic table of the elements.

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42 Periodic table Pauli’s exclusion Principle: l No two electrons in an atom can occupy the same quantum state. When there are many electrons in an atom, the electrons fill the lowest energy states first: l lowest n lowest l lowest m l l lowest m s this determines the electronic structure of atoms

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43 Problems Calculate from classical considerations the force exerted on a perfectly reflecting mirror by a laser beam of power 1W striking the mirror perpendicular to its surface. The solar irradiation density at the earth's distance from the sun amounts to 1.3 kW/m 2 ; calculate the number of photons per m 2 per second, assuming all photons to have the wavelength at the maximum of the spectrum, i.e. ≈ max ). Assume the surface temperature of the sun to be 5800K. how close can an particle with a kinetic energy of 6 MeV approach a gold nucleus? (q = 2e, q Au = 79e) (assume that the space inside the atom is empty space)

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44 Summary electron was identified as particle emitted in photoelectric effect Einstein’s explanation of p.e. effect lends further credence to quantum idea Geiger, Marsden, Rutherford experiment disproves Thomson’s atom model Planetary model of Rutherford not stable by classical electrodynamics Bohr atom model with de Broglie waves gives some qualitative understanding of atoms, but l only semiquantitative l no explanation for missing transition lines l angular momentum in ground state = 0 (1 ) l spin?? Quantum mechanics: lobservables (position, momentum, angular momentum..) are operators which act on “state vectors”

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