Quantum Mechanics. An essential theory to understand properties of matter and light. Chemical Electronic Magnetic Thermal Optical Etc.

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1 Quantum Mechanics An essential theory to understand properties of matter and light. Chemical Electronic Magnetic Thermal Optical Etc. Fall 2018 Prof. Sergio B. Mendes 1

2 CHAPTER 3 Experimental Basis of Quantum Physics Fall 2018 Prof. Sergio B. Mendes 2

3 Topics Discovery of the X Ray and the Electron Determination of Electron Charge Line Spectra Quantization Blackbody Radiation Photoelectric Effect X-Ray Production Compton Effect Pair Production and Annihilation Fall 2018 Prof. Sergio B. Mendes 3

4 3.1 Discovery of the X Rays X rays were discovered by Wilhelm Röntgen in Observed x rays were emitted by cathode rays bombarding glass Fall 2018 Prof. Sergio B. Mendes 4

5 Cathode Ray Experiments In the 1890s scientists and engineers were familiar with cathode rays. These rays were generated from one of the metal plates in an evacuated tube across which a large electric potential had been established. It was considered that cathode rays had something to do with atoms. It was known that cathode rays could penetrate matter and their properties were under intense investigation during the 1890s. Fall 2018 Prof. Sergio B. Mendes 5

6 Observation of X Rays Wilhelm Röntgen studied the effects of cathode rays passing through various materials. He noticed that a phosphorescent screen near the tube glowed during some of these experiments. He called them x rays and deduced that they were produced by the cathode rays bombarding the glass walls of his vacuum tube. These rays were unaffected by magnetic fields and penetrated materials more than cathode rays. Fall 2018 Prof. Sergio B. Mendes 6

7 Röntgen s X Ray Tube Röntgen constructed an x-ray tube by allowing cathode rays to impact the glass wall of the tube to produce x rays. He used x rays to image the bones of a hand on a phosphorescent screen. Fall 2018 Prof. Sergio B. Mendes 7

8 First X Ray Diagnostic!! Fall 2018 Prof. Sergio B. Mendes 8

9 Discovery of the Electron Electrons were discovered by J. J. Thomson, Observed that cathode rays were charged particles Fall 2018 Prof. Sergio B. Mendes 9

negatively charged particles (electrons) by deflecting

10 Thomson s Experiment Thomson used an evacuated cathode-ray tube to show that the cathode rays were negatively charged particles (electrons) by deflecting them in electric and magnetic fields. Fall 2018 Prof. Sergio B. Mendes 10

11 Analysis of Thomson s Experiment BB = 0 FF yy = qq EE = mm aa yy ttaaaa θθ = vv yy vv xx = qq EE mm ll vv xx 2 vv yy = aa yy tt = qq EE mm ll vv xx qq mm = vv xx 2 tttttt θθ EE ll Fall 2018 Prof. Sergio B. Mendes 11

Magnetic Field turned ON and adjust for: qq EE + qq vv BB = 0 vv xx = EE BB 1.

12 Magnetic Field turned ON and adjust for: qq EE + qq vv BB = 0 vv xx = EE BB CC kkkk = qq mm = vv xx 2 tttttt θθ EE ll = EE tttttt θθ BB 2 ll Fall 2018 Prof. Sergio B. Mendes 12

13 3.2 Millikan s Experiment and Determination of Electron Charge Fall 2018 Prof. Sergio B. Mendes 13

14 Major Steps: mmmm + bb vv = FF = mm dddd dddd = 0 mmmm + bb vv dd = 0 mmmm + qq EE bb vv = FF = 0 = mm dddd dddd mmmm + qq EE bb vv uu = 0 qq EE bb vv uu bb vv dd = 0 Fall 2018 Prof. Sergio B. Mendes 14

15 Charges were multiples of unit e qq = nn ee = bb vv dd + vv uu EE ee = CC mm = kkkk Fall 2018 Prof. Sergio B. Mendes 15

16 3.3 Line Spectra Fall 2018 Prof. Sergio B. Mendes 16

$Color Separation by a Diffraction$ grating equation mm = 1 mm = 0 1

17 Color Separation by a Diffraction Grating dd ssssss θθ mm = mm λλ grating equation mm = 1 mm = Fall 2018 Prof. Sergio B. Mendes 17

18 Emission and Absorption Experiments Fall 2018 Prof. Sergio B. Mendes 18

19 Emission Spectra of a Few Atoms Identification of Chemical Elements Composition of Materials Fall 2018 Prof. Sergio B. Mendes 19

Balmer s series for the hydrogen atom, 1885 λλ = 364.

20 Balmer s series for the hydrogen atom, 1885 λλ = nnnn kk2 kk 2 4 kk = 3, 4, 5, Fall 2018 Prof. Sergio B. Mendes 20

21 A simple rearragenment: λλ = nnnn kk2 kk 2 4 kk = 3, 4, 5, RR HH nnnn = mm 1 λλ = RR HH kk 2 Fall 2018 Prof. Sergio B. Mendes 21

Rydberg equation, 1890 1 λλ = RR HH 1 nn 2 1

22 Rydberg equation, λλ = RR HH 1 nn 2 1 kk 2 kk > nn Fall 2018 Prof. Sergio B. Mendes 22

23 3.4 Quantization Discrete quantities!! Electric charge Atomic mass Discrete spectral lines?? (many more to come) Fall 2018 Prof. Sergio B. Mendes 23

24 3.5 Blackbody Radiation Fall 2018 Prof. Sergio B. Mendes 24

25 Matter & Electromagnetic Radiation in Thermal Equilibrium Constant temperature Absorption and emission are balanced Continuous spectral distribution TT Fall 2018 Prof. Sergio B. Mendes 25

26 Experimental Spectral Distribution Fall 2018 Prof. Sergio B. Mendes 26

27 Wien s Law: Empirical Laws: λλ mmmmmm TT mm KK Stefan-Boltzmann Law: RR TT = ΙΙ λλ, TT ddλλ = εε σσ TT 4 σσ = W/(m 2 K 4 ) 0 εε = 0, 1 Fall 2018 Prof. Sergio B. Mendes 27

28 Classical Calculation of Blackbody Radiation Power per-unit-area per-unit-frequency: ΙΙ ff, TT = 1 4 cc uu ff Appendix A.1 cc tt cc tt uu ff cc tt AA uu ff = NN ff EE Fall 2018 Prof. Sergio B. Mendes 28

29 Density of Modes (standing waves) of Electromagnetic Radiation Number of modes per-unit-volume per-unit-frequency NN ff =?? NN ff = 8 ππ ff2 cc 3 Fall 2018 Prof. Sergio B. Mendes 29

30 1D Standing Waves ff xx ff xx = AA sin(kk xx) xx kk LL = mm ππ mm = kk LL ππ mm = 1, 2, 3, Fall 2018 Prof. Sergio B. Mendes 30

31 3D Standing Waves ff xx = AA sin(kk xx xx) sin kk yy yy sin(kk zz zz) kk xx LL xx = mm xx ππ kk yy LL yy = mm yy ππ kk zz LL zz = mm zz ππ mm xx = kk xx LL xx ππ mm yy = kk yy LL yy ππ mm zz = kk zz LL zz ππ mm xx mm yy mm zz = kk xx kk yy kk zz VV ππ 3 MM = ππ 3 kk3 VV ππ 3 dddd VV = 1 2 ππ 2 kk2 dddd = 4 ππ ff2 cc 3 dddd kk = 2 ππ ff cc Fall 2018 Prof. Sergio B. Mendes 31

32 dddd VV = 4 ππ ff2 cc 3 dddd NN ff = 2 dddd VV dddd = 2 4 ππ ff2 cc 3 ff2 = 8 ππ cc 3 Fall 2018 Prof. Sergio B. Mendes 32

33 Average Energy for Each Mode of the Electromagnetic Radiation EE =?? EE = 0 EE pp EE dddd 0 pp EE dddd = 0 EE ee kkkkdddd EE 0 ee EE kkkk dddd = kk TT Appendix A.2 pp EE = ee EE kkkk EE = kk TT Fall 2018 Prof. Sergio B. Mendes 33

34 Bringing All the Pieces Together: ΙΙ ff, TT = 1 cc uu ff 4 ff2 = 2 ππ 2 kk TT cc uu ff = NN ff EE = 8 ππ ff2 kk TT cc3 NN ff = 8 ππ ff2 cc 3 EE = kk TT Fall 2018 Prof. Sergio B. Mendes 34

35 Rayleigh-Jeans Formula ΙΙ ff, TT = 2 ππ ff2 cc2 kk TT ΙΙ λλ, TT dddd = ΙΙ ff, TT ddff λλ ff = cc ff ddλλ + λλ ddff = 0 ddff dddd = ff λλ = cc λλ 2 ΙΙ λλ, TT = 2 ππ cc λλ4 kk TT Fall 2018 Prof. Sergio B. Mendes 35

36 Theory and Experiment Rayleigh-Jeans ΙΙ λλ, TT = 2 ππ cc λλ4 kk TT Fall 2018 Prof. Sergio B. Mendes 36

37 Planck s Assumption EE =?? pp EE = ee EE kkkk EE = 0 EE pp EE dddd 0 pp EE dddd EE nn = nn h ff EE = nn=0 nn=0 EE nn pp EE nn pp EE nn EE = h ff ee h ff kkkk 1 Appendix A.3 Fall 2018 Prof. Sergio B. Mendes 37

38 Planck Theory of Blackbody Radiation ΙΙ ff, TT = 1 4 cc uu ff = cc 2 2 ππ h ff3 ee h ff kkkk 1 uu ff = NN ff EE = 8 ππ ff2 cc 3 h ff ee h ff kkkk 1 NN ff = 8 ππ ff2 cc 3 EE = h ff ee h ff kkkk 1 Fall 2018 Prof. Sergio B. Mendes 38

39 Comparison: Rayleigh-Jeans ΙΙ ff, TT = 2 ππ ff2 cc2 kk TT Planck ΙΙ ff, TT = cc 2 2 ππ h ff3 ee h ff kkkk 1 h = JJ ss Fall 2018 Prof. Sergio B. Mendes 39

40 Frequency and Wavelength Descriptions: ΙΙ ff, TT = cc 2 2 ππ h ff3 ee h ff kkkk 1 ΙΙ λλ, TT dddd = ΙΙ ff, TT ddff dddd dddd = ff λλ = cc λλ 2 λλ ff = cc ΙΙ λλ, TT = 2 ππ h cc2 λλ 5 ee h cc λλλλλλ 1 Fall 2018 Prof. Sergio B. Mendes 40

41 Wien s Law from Planck s Theory ΙΙ λλ, TT = 2 ππ h cc2 λλ 5 ee h cc λλλλλλ 1 ddιι λλ, TT dddd = 0 h cc λλ mmmmmm kk TT λλ mmmmmm kk TT h cc kk = mm KK Fall 2018 Prof. Sergio B. Mendes 41

42 Stefan-Boltzmann Law from Planck s Theory ΙΙ λλ, TT = 2 ππ h cc2 λλ 5 ee h cc λλλλλλ 1 RR TT = ΙΙ λλ, TT ddλλ = 2 ππ5 kk h 3 cc 2 TT4 σσ = 2 ππ5 kk 4 15 h 3 cc 2 = W/(m 2 K 4 ) Fall 2018 Prof. Sergio B. Mendes 42

43 3.6 Methods of Electron Emission: Thermionic emission: Application of heat allows electrons to gain enough energy to escape. Secondary emission: The electron gains enough energy by transfer from another high-speed particle that strikes the material from outside. Field emission: A strong external electric field pulls the electron out of the material. Photoelectric effect: Incident light (electromagnetic radiation) shining on the material transfers energy to the electrons, allowing them to escape. Fall 2018 Prof. Sergio B. Mendes 43

44 Photoelectric Effect Electromagnetic radiation interacts with electrons within the metal and can increase the electrons kinetic energy. Eventually electrons get enough extra kinetic energy that allow them to escape from the metal. We call the ejected electrons photoelectrons. Fall 2018 Prof. Sergio B. Mendes 44

45 Same frequency, different light intensities For a given frequency, the retarding potential energy is the same regardless of the light intensity Fall 2018 Prof. Sergio B. Mendes 45

46 Same light intensity, different frequencies For a given light intensity, the retarding potential energy required to stop the current increases with the light frequency. Fall 2018 Prof. Sergio B. Mendes 46

47 Retarding potential energy versus frequency The retarding potential energy required to stop the current increases linearly with the light frequency. Fall 2018 Prof. Sergio B. Mendes 47

48 Some Conclusions from Experiments: The kinetic energies of the photoelectrons are independent of the light intensity. The maximum kinetic energy of the photoelectrons, for a given emitting material, depends only on the frequency of the light. The smaller the work function of the emitter material, the smaller is the threshold frequency of the light that can eject photoelectrons. When the photoelectrons are produced, however, their number is proportional to the intensity of light. The photoelectrons are emitted almost instantly following illumination of the photocathode, independent of the intensity of the light. Fall 2018 Prof. Sergio B. Mendes 48

49 A Few Puzzles for Classical Physics: Why the maximum kinetic energy of the photoelectrons does not depend on the light intensity? Why the maximum kinetic energy depends on the light frequency? Why there is a threshold frequency for the photocurrent? Why the photoelectrons are ejected almost immediately after illumination, regardless of the light intensity? Fall 2018 Prof. Sergio B. Mendes 49

50 Einstein s Approach: Einstein suggested that electromagnetic radiation is quantized into particles called photons. Each photon has the energy quantum EE = h ff where ff is the frequency of the light and h is Planck s constant. The photon travels at the speed of light cc in a vacuum, and its wavelength is given by λλ = cc ff Fall 2018 Prof. Sergio B. Mendes 50

51 From those assumptions: Conservation of energy yields: h ff φφ = 1 2 mm vv2 where φφ is the work function of the metal. The retarding potentials measured in the photoelectric effect are the opposing potentials needed to stop the most energetic electrons. ee VV oo = 1 2 mm vv mmmmmm 2 Fall 2018 Prof. Sergio B. Mendes 51

52 The kinetic energy of the electron does not depend on the light intensity at all, but only on the light frequency and the work function of the material. 1 2 mm vv2 = h ff φφ The stopping potential is linearly proportional to the light frequency, with a slope h, the same constant found by Planck. ee VV oo = 1 2 mm vv mmmmmm 2 = h ff φφ = h ff ff oo Fall 2018 Prof. Sergio B. Mendes 52

53 Work Function of Materials Fall 2018 Prof. Sergio B. Mendes 53

Millikan on Photoelectric Effect: Millikan published data in 1916 for the photoelectric effect in which he shone light of varying frequency on a sodium electrode and measured the maximum kinetic

54 Millikan on Photoelectric Effect: Millikan published data in 1916 for the photoelectric effect in which he shone light of varying frequency on a sodium electrode and measured the maximum kinetic energies of the photoelectrons. He found that no photoelectrons were emitted below a frequency of 4.39 X Hz (or longer than a wavelength of 683 nm). The results were independent of light intensity, and the slope of a straight line drawn through the data produced a value of Planck s constant in excellent agreement with Planck s theory. Even though Millikan admitted his own data were sufficient proof of Einstein s photoelectric effect equation, Millikan was not convinced of the photon concept for light and its role in quantum theory. Fall 2018 Prof. Sergio B. Mendes 54

55 Photoelectric effect can be described by γγ + eee ee. Can the reverse process ee eee + γγ happen? The answer is yes. Fall 2018 Prof. Sergio B. Mendes 55

56 3.7 X-Ray Production Bremsstrahlung, from the German word for braking radiation EE ff + h ff = EE ii ee eee + γγ 0, ee VV oo = EE ff EE ff + h cc λλ = EE ii ee VV oo = EE ii Fall 2018 Prof. Sergio B. Mendes 56

57 Three Different Materials The relative intensity of x rays produced in an x-ray tube is shown for an accelerating voltage of 35 kv. Notice that λλ mmmmmm is the same for all three targets. EE ii = ee VV oo EE ff + h cc λλ = EE ii EE ff = 0, ee VV oo λλmmmmmm = nnnn λλ = h cc EE ii EE ff λλ mmmmmm = h cc ee VV oo Fall 2018 Prof. Sergio B. Mendes 57

58 3.8 Compton Effect Fall 2018 Prof. Sergio B. Mendes 58

59 (Classical) Light-Matter Interaction and Scattered Light Electric field in the light wave drives the oscillation of electrons present in matter. Frequency of oscillation coincides with the frequency of the incident light. Re-emitted (scattered) light has the same frequency as the incident light. Thomson radiation Fall 2018 Prof. Sergio B. Mendes 59

Compton Effect, 1923 Instead of the collective interaction between light wave & electrons, Compton considered photon-electron interaction Described the phenomenon as

60 Compton Effect, 1923 Instead of the collective interaction between light wave & electrons, Compton considered photon-electron interaction Described the phenomenon as single-photon & single-electron collision γγ + ee γγ + ee Conservation of electric charge Conservation of energy Conservation of linear momentum Fall 2018 Prof. Sergio B. Mendes 60

61 Linear Momentum for Photons: EE 2 pp 2 cc 2 = mm 2 cc 4 mm = 0 EE = pp cc EE = h ff = pp cc pp = h ff cc = h λλ pp = h λλ Fall 2018 Prof. Sergio B. Mendes 61

62 Fall 2018 Prof. Sergio B. Mendes 62

63 Conservation of Energy: h ff + EE ii = h fff + EE ff h ff = h cc λλ EE ii = mm cc 2 h ff = h cc λλ EE ff 2 = mm cc pp ee 2 cc 2 h cc λλ + mm cc2 = h cc λλλ + mm cc2 2 + pp ee 2 cc 2 Fall 2018 Prof. Sergio B. Mendes 63

64 Conservation of Linear Momentum: xx: yy: h λλ = h λλ cccccc θθ + pp ee cccccc φφ 0 = h λλ ssssss θθ pp ee ssssss φφ Fall 2018 Prof. Sergio B. Mendes 64

65 From linear momentum equations: h λλ = h λλ cccccc θθ + pp ee cccccc φφ 0 = h λλ ssssss θθ pp ee ssssss φφ h λλ 2 + h λλλ 2 2 h λλ h λλλ cccccc θθ = pp ee 2 Fall 2018 Prof. Sergio B. Mendes 65

66 From energy equation: h cc λλ + mm cc2 = h cc λλλ + mm cc2 2 + pp ee 2 cc 2 h cc λλ h cc λλλ h cc λλ h cc λλλ mm cc 2 = pp ee 2 cc 2 h λλ h λλλ h λλ h λλλ mm cc = pp ee 2 Fall 2018 Prof. Sergio B. Mendes 66

67 Combining the two equations: h λλ 2 + h λλλ 2 2 h λλ h λλλ cccccc θθ = pp ee 2 h λλ h λλλ h λλ h λλλ mm cc = pp ee 2 2 h λλ h λλλ cccccc θθ = 2 h λλ h λλλ + 2 h λλ h λλλ mm cc h λλ h λλλ 1 cccccc θθ = h λλ h λλλ mm cc Fall 2018 Prof. Sergio B. Mendes 67

68 Compton Equation: h λλ h λλλ 1 cccccc θθ = h λλ h λλλ mm cc h mm cc 1 cccccc θθ = λλλ λλ Compton equation Independent of wavelength Compton wavelength: h mm ee cc = nnnn Usually observed with x rays Fall 2018 Prof. Sergio B. Mendes 68

69 Experimental Results by Compton: Compton s original data showing (a) the primary x-ray beam from Mo unscattered and (b) the scattered spectrum from carbon at 135 showing both the modified and unmodified wave. Fall 2018 Prof. Sergio B. Mendes 69

70 3.9 Pair Production γγ ee + ee MMMMMM h ff > 2 mm ee cc 2 Conservation of electric charge Cannot satisfy simultaneously conservation of energy and linear momentum Conservation of electric charge Conservation of energy Conservation of linear momentum Fall 2018 Prof. Sergio B. Mendes 70

71 Positron-Electron Annihilation ee + ee + γγ + γγ 2 mm ee cc 2 h ff 1 + h ff 2 0 h ff 1 cc h ff 2 cc Conservation of electric charge Conservation of energy Conservation of linear momentum h ff = mm ee cc MMMMMM Annihilation of a positronium atom (consisting of an electron and positron) producing two photons. Fall 2018 Prof. Sergio B. Mendes 71

Positron Emission Tomography (PET) A useful medical diagnostic tool to

physiological processes in the region to be examined.

72 Positron Emission Tomography (PET) A useful medical diagnostic tool to study the path and location of a positronemitting radiopharmaceutical in the human body. Appropriate radiopharmaceuticals are chosen to concentrate by physiological processes in the region to be examined. The positron travels only a few millimeters before annihilation, which produces two photons that can be detected to give the positron position. Fall

73 Appendix A.1 cccccc θθ = 2 ππ ππ/2 0 ddφφ 0 dddd ssssss θθ cccccc θθ 2 ππ ππ/2 0 dddd 0 dddd ssssss θθ ππ/2 = dddd 0 ssssss 2 θθ 2 = cccccc 2 θθ 4 ππ/2 0 = 1 2 Fall 2018 Prof. Sergio B. Mendes 73

74 Appendix A.2 0 EE ee EE kk TT dddd 0 ee EE kk TT dddd = EE kk TT ee EE EE= kk TT EE=0 0 kk TT ee EE kk TT dddd 0 ee EE kk TT dddd = kk TT Fall 2018 Prof. Sergio B. Mendes 74

75 EE = nn=0 EE nn pp EE nn = nn=0 pp EE nn nn=0 Appendix A.3 nn=0 nn h ff nn h ffee kkkk nn h ff ee kkkk = kkkk nn=0 nnnn ee nnαα ee nnαα = kkkk αα dd llll nn=0 ddαα nn=0 ee nnnn αα h ff kkkk = kkkk αα dd llll 1 1 ee αα ddαα = kkkk αα dd llll 1 ee αα dddd = kkkk αα ee αα 1 ee αα = h ff ee h ff kkkk 1 Fall 2018 Prof. Sergio B. Mendes 75

76 Appendix A.4 ΙΙ λλ, TT = 2 ππ h cc2 λλ 5 ee h cc λλλλλλ 1 ddιι λλ, TT dddd = 0 λλ 6 5 ee h cc λλλλλλ 1 h cc kk TT 1 λλ 2 λλ 5 h cc ee λλ kk TT ee h cc λλ kk TT 1 = h cc kk TT ee h cc λλ kk TT λλ ee h cc λλ kk TT 1 = 0 Fall 2018 Prof. Sergio B. Mendes 76

CHAPTER 3 The Experimental Basis of Quantum Theory

CHAPTER 3 The Experimental Basis of Quantum Theory 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 Discovery of the X Ray and the Electron Determination of Electron Charge Line Spectra Quantization As far as I can