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

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

2 2 Summary of 1 st lecture  classical physics explanation of black-body radiation failed  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

3 3 Outline  Introduction  cathode rays …. electrons  photoelectric effect l observation l studies l Einstein’s explanation  models of the atom  Summary

4 4Electron  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) l1869: discharge mediated by rays emitted from negative electrode (“cathode”) lrays called “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

5 5 Photoelectric effect  1887: Heinrich Hertz: lIn 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 lFurther 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

6 6 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”

7 7 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)

8 8 Hallwachs’ experiments l “photoelectric discharge” l “photoelectric excitation”

9 9 Electron l1897: three experiments measure charge/mass of cathode rays oAll measure charge/mass to similar value 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?

10 10 Further studies of 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)

11 11 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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14 14 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 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

15 15 Photoelectric effect  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

16 16 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

17 17 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.“

18 18 WHY CAN'T WE SEE ATOMS?  “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/(m v) 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

19 19 WHAT IS INSIDE AN ATOM?  THOMSON'S MODEL OF ATOM l(“RAISIN CAKE MODEL”): oatom = sphere of positive charge (diameter  10 -10 m), o with electrons embedded in it, evenly distributed (like raisins in cake)  Geiger & Marsden’s SCATTERING EXPERIMENT: 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).

20 20  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

21 21 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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23 23 Geiger & Marsden’s scattering experiment  Geiger, Marsden, 1906 - 1911 l make “beam” of particles using radioactive source l bombard foils of gold, silver, copper with beam l measure scattering angles of particles.

24 24 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

25 25 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.

26 26 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

27 27 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 1 · n 2 n = 1,2,3,…., a 1 = 0.53 x 10 -10 m = “Bohr radius” n = “principal quantum number” lallowed energy levels: E n = - E 1 /n 2, E 1 = “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.

28 28 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”; binding energy of ground state electron = E 0 = 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; electron 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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30 30 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 )

31 31  Excitation:  states of electron in hydrogen atom:

32 32 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

33 33  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;

34 34 Frequency and wavelength for light*** Relativistic relationship between a particle’s momentum and energy: For massless particles propagating at the speed of light, becomes Hence find relationship between momentum p and wavelength λ:

35 35 Momentum of a photon  Relativistic relationship between a particle’s momentum and energy: E 2 = p 2 c 2 + m 2 c 4  For massless particles propagating at the speed of light: E 2 = p 2 c 2  For photon, E = h  momentum of photon = h /c = h/

36 36 Matter waves lLouis de Broglie (1925): any moving particle has wavelength associated with it:  = h/p = h/(mv) 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 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

37 37 Atomic Model  Electrons move around nucleus only in certain stable orbits  Stable orbits are those in which an integral number of wavelengths fit into the diameter of the orbit (2  r n = n )  They emit (absorb) light only when they change from one orbital to another.  Orbits have quanta of angular momentum L = nh/2   Orbit radius increases with energy r n = n 2 r 1 (r 1 =.529 x 10 -10 m)

38 38 QUANTUM MECHANICS  = new kind of physics based on synthesis of dual nature of waves and particles; developed in 1920's and 1930's. lSchrödinger equation: (Erwin Schrödinger, 1925) ois 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! lUncertainty principle: (Werner Heisenberg, 1925) It is impossible to simultaneously know a particle's exact position and momentum (or velocity)  p   x  ħ = h/(2  ) (remember h is a very small quantity: h = 6.63 x 10 -34 J  s = 4.14 x 10 -15 eV·s) (note that here  p means “uncertainty” in our knowledge of the momentum p) lnote that there are many such uncertainty relations in quantum mechanics, for any pair of “incompatible” observables.

39 39 deBroglie’s Atom  The mystery created by Bohr’s model of the atom, “why were some orbits stable?”, was solved by deBroglie’s hypothesis that particles are also waves. lStable orbits were those for which an integral number of wavelengths fit into the diameter of the orbit (2  r n = n ) lAll other orbits, the waves destructively interfered and were not stable. lLeads naturally to quantized angular momentum (L = nh/2  )

40 40 Quantum Mechanics of the Hydrogen Atom  E n = -13.6 eV/n 2, ln = 1, 2, 3, … (principal quantum number)  Orbital quantum number l = 0, 1, 2, n-1, … oAngular Momentum, L = √ l ( l +1) (h/2  ) Magnetic quantum number - l  m  l, (there are 2 l + 1 possible values of m)  Spin quantum number: m s =  ½

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

42 42 Heisenberg Uncertainty Principle  Impossible to know both the position and the momentum of a particle precisely. lA restriction (or measurement) of one, affects the other. l  x  p  h/(2  )  Similar constraints apply to energy and time. l  E  t  h/(2  ) EXAMPLE: If an electron's position can be measured to an accuracy of 1.96×10 -8 m, how accurately can its momentum be known?  x  p  h/(2  )   p = h/(2  x)  p = 6.63x10 -34 Js /(2  1.96x10 -8 m) = 5.38 x 10 -27 N s

43 43 Periodic table  Exclusion Principle: lNo 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: llowest n llowest l llowest m l llowest m s  this determines the electronic structure of atoms

44 44 Photoelectric effect Metal plate in a vacuum, irradiated by ultraviolet light, emits charged particles (Hertz 1887), which were subsequently shown to be electrons by J.J. Thomson (1899). Electric field E of light exerts force F=-eE on electrons. As intensity of light increases, force increases, so KE of ejected electrons should increase. Electrons should be emitted whatever the frequency ν of the light, so long as E is sufficiently large For very low intensities, expect a time lag between light exposure and emission, while electrons absorb enough energy to escape from material Classical expectations HertzJ.J. Thomson I Vacuum chamber Metal plate Collecting plate Ammeter Potentiostat Light, frequency ν

45 45 Photoelectric effect (contd)*** The maximum KE of an emitted electron is then predicted to be: Maximum KE of ejected electrons is independent of intensity, but dependent on ν For ν<ν 0 (i.e. for frequencies below a cut-off frequency) no electrons are emitted There is no time lag. However, rate of ejection of electrons depends on light intensity. Einstein’s interpretation (1905): light is emitted and absorbed in packets (quanta) of energy Work function: minimum energy needed for electron to escape from metal (depends on material, but usually 2-5eV) Planck constant: universal constant of nature Einstein Millikan Verified in detail through subsequent experiments by Millikan Actual results: An electron absorbs a single quantum in order to leave the material

46 46 1.2 Compton scattering X-ray source Target Crystal (selects wavelength) Collimator (selects angle) θ Compton (1923) measured scattered intensity of X-rays (with well-defined wavelength) from solid target, as function of wavelength for different angles. Result: peak in the wavelength distribution of scattered radiation shifts to longer wavelength than source, by an amount that depends on the scattering angle θ (but not on the target material) A.H. Compton, Phys. Rev. 22 409 (1923) Detector B&M §2.7; Rae §1.2; B&J §1.3 Compton Compton experiment

47 47 Compton scattering (contd) Compton’s explanation: “billiard ball” collisions between X-ray photons and electrons in the material Conservation of energy:Conservation of momentum: θ φ p’ Classical picture: oscillating electromagnetic field would cause oscillations in positions of charged particles, re-radiation in all directions at same frequency and wavelength as incident radiation p Electron Incoming photon Before After pepe Photon

48 48 Compton scattering (contd) Assuming photon momentum related to wavelength: ‘Compton wavelength’ of electron (0.0243 Å)

49 49 Puzzle What is the origin of the component of the scattered radiation that is not wavelength-shifted?

50 50 Wave-particle duality for light*** “ There are therefore now two theories of light, both indispensable, and - as one must admit today despite twenty years of tremendous effort on the part of theoretical physicists - without any logical connection.” A. Einstein (1924) Light exhibits diffraction and interference phenomena that are only explicable in terms of wave properties Light is always detected as packets (photons); if we look, we never observe half a photon Number of photons proportional to energy density (i.e. to square of electromagnetic field strength)

51 51 1.3 Matter waves*** “As in my conversations with my brother we always arrived at the conclusion that in the case of X-rays one had both waves and corpuscles, thus suddenly -... it was certain in the course of summer 1923 - I got the idea that one had to extend this duality to material particles, especially to electrons. And I realised that, on the one hand, the Hamilton-Jacobi theory pointed somewhat in that direction, for it can be applied to particles and, in addition, it represents a geometrical optics; on the other hand, in quantum phenomena one obtains quantum numbers, which are rarely found in mechanics but occur very frequently in wave phenomena and in all problems dealing with wave motion.” L. de Broglie De Broglie Proposal: dual wave-particle nature of radiation also applies to matter. Any object having momentum p has an associated wave whose wavelength λ obeys Prediction: crystals (already used for X- ray diffraction) might also diffract particles B&M §4.1-2; Rae §1.4; B&J §1.6

52 52 Electron diffraction from crystals Davisson G.P. Thomson Davisson, C. J., "Are Electrons Waves?," Franklin Institute Journal 205, 597 (1928) The Davisson-Germer experiment (1927): scattering a beam of electrons from a Ni crystal At fixed accelerating voltage (i.e. fixed electron energy) find a pattern of pencil- sharp reflected beams from the crystal At fixed angle, find sharp peaks in intensity as a function of electron energy G.P. Thomson performed similar interference experiments with thin-film samples θiθi θrθr

53 53 Electron diffraction from crystals (contd) Modern Low Energy Electron Diffraction (LEED): this pattern of “spots” shows the beams of electrons produced by surface scattering from complex (7×7) reconstruction of a silicon surface Lawrence Bragg William Bragg (Quain Professor of Physics, UCL, 1915-1923) Interpretation used similar ideas to those pioneered for scattering of X-rays from crystals by William and Lawrence Bragg a θiθi θrθr Path difference: Constructive interference when Note difference from usual “Bragg’s Law” geometry: the identical scattering planes are oriented perpendicular to the surface Note θ i and θ r not necessarily equal Electron scattering dominated by surface layers

54 54 The double-slit interference experiment Originally performed by Young (1801) with light. Subsequently also performed with many types of matter particle (see references). D θ d Detecting screen (scintillators or particle detectors) Incoming beam of particles (or light) y Alternative method of detection: scan a detector across the plane and record arrivals at each point

55 55 Results Neutrons, A Zeilinger et al. 1988 Reviews of Modern Physics 60 1067-1073 He atoms: O Carnal and J Mlynek 1991 Physical Review Letters 66 2689-2692 C 60 molecules: M Arndt et al. 1999 Nature 401 680- 682 With multiple-slit grating Without grating Fringe visibility decreases as molecules are heated. L. Hackermüller et al. 2004 Nature 427 711-714

56 56 Double-slit experiment: interpretation Interpretation: maxima and minima arise from alternating constructive and destructive interference between the waves from the two slits Spacing between maxima: Example: He atoms at a temperature of 83K, with d=8μm and D=64cm

57 57 Double-slit experiment: bibliography

58 58 Matter waves: key points***  Interference occurs even when only a single particle (e.g. photon or electron) in apparatus, so wave is a property of a single particle lA particle can “interfere with itself”  Wavelength unconnected with internal lengthscales of object, determined by momentum  Attempt to find out which slit particle moves through causes collapse of interference pattern (see later…) Particles exhibit diffraction and interference phenomena that are only explicable in terms of wave properties Particles always detected individually; if we look, we never observe half an electron Number of particles proportional to….??? Wave-particle duality for matter particles

59 59 1.4 Heisenberg’s gamma-ray microscope and a first look at the Uncertainty Principle The combination of wave and particle pictures, and in particular the significance of the ‘wave function’ in quantum mechanics (see also §2), involves uncertainty: we only know the probability that the particle will be found near a particular location. θ/2 Light source, wavelength λ Particle Lens, having angular diameter θ Screen forming image of particle Resolving power of lens: Heisenberg B&M §4.5; Rae §1.5; B&J §2.5 (first part only)

60 60 Heisenberg’s gamma-ray microscope and the Uncertainty Principle*** Range of y-momenta of photons after scattering, if they have initial momentum p: θ/2 p p Heisenberg’s Uncertainty Principle


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