CHAPTER 5 Wave Properties of Matter and Quantum Mechanics I

Transcription

1 CHAPTER 5 Wave Properties of Matter and Quantum Mechanics I 1

2 5.1 X-Ray Scattering 5.2 De Broglie Waves 5.3 Electron Scattering 5.4 Wave Motion 5.5 Waves or Particles 5.6 Uncertainty Principle Topics 5.7 Probability, Wave Functions, and Copenhagen Interpretation 5.8 Particle in a Box 2

3 5.1 X-Ray Scattering Max von Laue suggested that if x rays were a form of electromagnetic radiation, interference effects should be observed. Crystals act as three-dimensional gratings, scattering the waves and producing observable interference effects. 3

4 X-Ray Scattering and Bragg s Law William Lawrence Bragg interpreted the x-ray scattering as the reflection of the incident x-ray beam from a unique set of planes of atoms within the crystal. 4

5 Bragg s Law There are two conditions for constructive interference of the scattered x rays: 1) The angle of incidence must equal the angle of reflection of the outgoing wave. 2) The difference in path lengths must be an integral number of wavelengths: nn λλ = 2 dd ssssss θθ nn = 1, 2, 3, 5

6 nn λλ = 2 dd ssssss θθ 6

7 DNA Rosalind Franklin produced the X-ray diffraction images of the DNS molecule that helped Watson and Crick unravel the DNA structure. 7

8 X-Ray Diffraction Nowadays 8

9 5.2 De Broglie Waves, 1920 After serving in World War I, Prince Louis V. de Broglie ( ) resumed his studies toward a doctoral degree at the University of Paris in 1924, where he reported his concept of matter waves as part of his doctoral dissertation. De Broglie spent his life in France where he enjoyed much success as an author and teacher. 9

10 For Electromagnetic Waves: According to Special Relativity, and m = 0: EE = pp cc According to Planck, Einstein, Compton, etc: EE = h ff EE = pp cc = h ff pp = h ff cc = h λλ pp = h λλ 10

11 Here comes De Broglie hypothesis: Well, maybe matter also behave like waves. And the wavelength is given by: λλ = h pp 11

12 Example 5.2 Tennis ball mm = 57 gg Electron KK = 50 eeee vv = 25 mm/ss (56 mph) λλ = h pp = h 2 mm KK λλ = h pp = JJ ss kkkk 25 mm/ss = mm = 0.17 nnnn 12 = = h cc 2 mm cc 2 KK 1240 eeee nnnn eeee 50 eeee

13 5.3 Electron Scattering, 1925 Clinton J. Davisson ( ) is shown here in 1928 (right) looking at the electronic diffraction tube held by Lester H. Germer ( ). Davisson received his undergraduate degree at the University of Chicago and his doctorate at Princeton. They performed their work at Bell Telephone Laboratory located in New York City. Davisson received the Nobel Prize in Physics in

14 Electron Scattering 14

15 15

16 Bragg Analysis: 16

17 Examples of transmission electron diffraction photographs Produced by scattering 120-keV electrons on the quasicrystal Al 80 Mn 20. Electron diffraction pattern on beryllium. The dots in (a) indicate that the sample was a crystal, whereas the rings in (b) indicate that a randomly oriented sample (or powder) was used. 17

18 Combining Bohr s Quantization of the Angular Momentum LL = nn ħ and De Broglie Waves pp = h λλ h λλ rr = pp rr = LL LL = nn ħ 2 ππ rr = nn λλ 18

19 5.4 Wave Motion 19

20 Waves: propagation of energy, not particles 20

21 Longitudinal Waves: disturbance is along the direction of wave propagation 21

22 Transverse Waves: disturbance is perpendicular to the direction of wave propagation 22

23 What is Waving?? pressure surface height transverse displacement longitudinal displacement electric and magnetic fields 23

24 Different Waveforms a pulse a wave train a continuous harmonic wave 24

25 Two Snapshots of a Wave Pulse propagating with velocity v along the x-axis tt = 0 yy xx, tt = 0 = ff xx tt 0 yy xx, tt = ff xx vv tt 25

26 Propagation towards Positive x-direction ψψ xx, tt = ff xx vv tt vv > 0 Propagation towards Negative x-direction ψψ xx, tt = ff xx + vv tt vv > 0 26

27 Fingerprint of the Wave Phenomena: xx vv tt ψψ(xx, tt) = ff xx vv tt ψψ xx, tt = 0 = AA ee xx2 σσ 2 xx vv tt ψψ xx, tt = AA ee σσ 2 2 ψψ xx, tt = 0 = AA ee xx σσ xx vv tt ψψ xx, tt = AA ee σσ ψψ xx, tt = 0 = AA cccccc kk xx ψψ xx, tt = AA cccccc kk xx vv tt 27

28 ψψ(xx, tt) = ff xx vv tt Therefore: vv = xx 2 ψψ xx 2 = 1 2 ψψ vv 2 tt 2 28

29 Wave Equation 2 ψψ xx 2 = 1 2 ψψ vv 2 tt 2 2 nd order partial differential equation Linear equation in ψψ : ψψ, αα ψψ, αα 22 ψψ, If ψψ 11 and ψψ 22 are solutions then ψψ 11 + ψψ 22 is also a solution equation. Homogeneous equation: no term involving independent variables If ψψ is a solution then aa ψψ is also a solution equation. 29

30 Principle of Superposition: If ψψ 11 and ψψ 22 are solutions to the wave equation then aa ψψ 11 + bb ψψ 22 is also a solution equation. Wave superposition is the foundation of wave phenomena such as interference and diffraction 30

31 A Harmonic Wave Solution: ψψ xx, tt = AA cccccc kk xx vv tt + εε or ψψ xx, tt = BB ssssss kk xx vv tt + εε or ψψ xx, tt = AA cccccc kk xx vv tt + εε + BB ssssss kk xx vv tt + εε or ii kk xx vv tt +εε ψψ xx, tt = AA ee 31

32 Let s consider: ψψ xx, tt = AA cccccc kk xx vv tt + εε 32

33 Wavelength in a Harmonic Wave ψψ xx, tt = AA cccccc kk xx vv tt + εε tt = ffffffffff xx kk 2 ππ λλ 33

34 Period in a Harmonic Wave ψψ xx, tt = AA cccccc kk xx vv tt + εε xx = ffffffffff ττ ττ tt ττ ττ λλ vv 34

35 Wave Speed ττ ττ vv = λλ ττ ττ 35

36 A Few Definitions: ψψ xx, tt = AA cccccc kk xx vv tt + εε vv = λλ ττ Wave Number Frequency Angular Frequency kk 2 ππ λλ ff 1 ττ ωω 2 ππ ττ = 2 ππ ff ψψ xx, tt = AA cccccc kk xx ωω tt + εε vv = ωω kk 36

37 ψψ xx, tt = AA cccccc kk xx ωω tt + εε = AA cccccc φφ xx, tt Amplitude: Phase: AA φφ xx, tt kk xx ωω tt + εε 37

38 Constant Phase and Phase Velocity φφ xx, tt kk xx ωω tt + εε ddφφ xx, tt = kk ddxx ωω ddtt ddφφ xx, tt = 0 kk dddd ωω dddd = 0 ddxx vv ddtt = ωω kk = λλ ppp = ττ 38

39 Adding Waves of the Same Frequency (or Wavelength): 39

40 (Almost Completely) In Phase: Constructive Interference 40

41 (Almost Completely) Out of Phase: Destructive Interference 41

42 Interference: 42

43 Wave Propagation Through Slits: One slit: pppppppppp aaaaaaaa = II = εε oo cc EE 2 II Two slits: 43

44 Interference of Waves: Only one slit open Both slits open 44

45 Phase difference in Young s Interferometer yy aa 2 θθ RR ss Λ aa 2 EE rr, tt = EE 1 rr, tt + EE 2 rr, tt Λ = λλ RR aa II = εε oo cc EE 2 = II 1 + II II 1 II 2 cccccc kk aa yy RR + εε 1 εε 2 45

46 Diffraction: 46

47 yy Single Slit zz dd yy YY dd θθ RR ss EE YY, tt = EE ii kk RR ωω tt 0 ee RR + dd 2 kk YY ii ee RR yy dddd dd 2 47

48 Single Slit, cont. zeros at kk YY mm dd 2 RR = mm ππ kk YY dd ssssss II YY = εε oo cc EE 2 2 RR = II 0 kk YY dd 2 RR 2 with mm = ±1, ±2, ±3 YY(mmmm) λλ RR YY mm = mm dd geometrical shadow YY mm RR = ssssss θθ mm = mm λλ dd YY 1 ssssss θθ 1 = λλ dd II YY II0 YY 1 48

49 After diffraction, the wave spreads over a range of angles: ssssss θθ θθ λλ dd 49

50 Adding Waves of Different Frequencies (or Wavelengths) 50

51 Adding Two Waves: 51

52 Two waves with slight different frequencies (and wavelengths): ψψ 1 xx, tt = ψψ 0,1 cccccc kk 1 xx ωω 1 tt + εε 1 ψψ 2 xx, tt = ψψ 0,2 cccccc kk 2 xx ωω 2 tt + εε 2 ψψ xx, tt = ψψ 1 xx, tt + ψψ 2 xx, tt 52

53 kk 1 2 kk 1 + kk 2 ωω 1 2 ωω 1 + ωω 2 εε 1 2 εε 1 + εε 2 kk 1 2 kk 1 kk 2 ωω 1 2 ωω 1 ωω 2 εε 1 2 εε 1 εε 2 ψψ 0,1 = ψψ 0,2 = ψψ 0 ψψ xx, tt = 2 ψψ 0 cccccc kk xx ωω tt + εε cccccc xx kk tt ωω + εε 53

54 kk xx ωω tt + εε = φφ phase velocity vv ppp = ωω kk xx kk tt ωω + εε = φφ mm group velocity vv gggg = ωω kk = ddωω dddd 54

55 Phase & Group Velocities vv ppp = ωω kk vv gggg = ddωω dddd = vv pph + kk ddvv pph dddd 55

56 Phase & Group Velocities 56

57 Can we use Harmonic Functions to build a Spatially Confined Function? One particular harmonic function extends all the way from to + ψψ xx = AA cccccc kk xx + εε A Harmonic Function A Spatially Confined Function 57

58 PhET 58

59 Fourier Theory: ff xx = 1 2 ππ FF kk ee ii kk xx dddd ff(tt) = 1 2 ππ FF ωω ee ii ωω tt ddωω 59

60 A Square Function: ff xx FF kk LL 2 = xx xx 2 kk kk LL/2 xx kk ππ 60

61 A Gaussian Function: ff xx FF kk xx kk

62 A Truncated Cosine Function: ff tt FF ωω 2 tt tt 2 ωω ωω tt ωω ππ 62

63 xx kk 1 2 tt ωω

64 Back to Modern Physics 64

65 Group Velocity of Wave-Particles 65

66 Group Velocity: non-relativistic case vv gg = ddωω dddd = dd ħ ωω dd ħ kk = ddee ddpp ddee nnnn ddpp = pp mm EE = h ff = ħ ωω pp = h λλ = ħ kk EE = KK + mm cc 2 EE nnnn pp2 2 mm + mm cc2 66

67 Group Velocity: general case vv gg = ddωω dddd = dd ħ ωω dd ħ kk = ddee ddpp = pp cc2 EE = ββ cc EE 2 = pp 2 cc 2 + mm 2 cc 4 = γγ 2 mm 2 cc 4 2 EE dddd = 2 pp cc 2 dddd 1 = pp2 cc 2 EE γγ γγ 2 = pp2 cc 2 EE 2 ββ 2 = pp2 cc 2 EE 2 67

68 Interference with Wave-Particles: 68

69 Young s Interference with the Electron Wave-Particle: Demonstration of electron interference using two slits similar in concept to Young s double-slit experiment for light. This result by Claus Jönsson (1961) clearly shows that electrons exhibit wave behavior. 69

70 Young s Interference with the Photon Wave-Particle: 70

71 71

72 Even a Single Particle (photon, electron, etc) has an associated wave to guide its propagation: Only one slit open Both slits open 72

73 Diffraction with Wave-Particles: 73

74 After Diffraction, the Wave spreads over a range of angles: ssssss θθ θθ λλ dd dd λλ θθ 1 dd pp = dd pp θθ = dd h λλ θθ h xx pp xx h 74

75 5.6 Heisenberg Uncertainty Principle xx kk 1 2 xx pp xx ħ 2 tt ωω 1 2 tt EE ħ 2 75

76 76

77 Appendix A 77

78 Two waves with slight different frequencies (or wavelengths): ψψ 1 xx, tt = ψψ 0,1 cccccc kk 1 xx ωω 1 tt + εε 1 ψψ 2 xx, tt = ψψ 0,2 cccccc kk 2 xx ωω 2 tt + εε 2 ψψ xx, tt = ψψ 1 xx, tt + ψψ 2 xx, tt 78

79 AA 1 2 kk 1 + kk 2 xx 1 2 ωω 1 + ωω 2 tt εε 1 + εε 2 BB 1 2 kk 1 kk 2 xx 1 2 ωω 1 ωω 2 tt εε 1 εε 2 aa 1 2 ψψ 0,1 + ψψ 0,2 bb 1 2 ψψ 0,1 ψψ 0,2 ψψ 1 xx, tt = aa + bb cccccc AA + BB ψψ 2 xx, tt = aa bb cccccc AA BB 79

80 ψψ xx, tt = ψψ 1 xx, tt + ψψ 2 xx, tt = aa + bb cccccc AA + BB + aa bb cccccc AA BB = aa cccccc AA + BB + cccccc AA BB + bb cccccc AA + BB cccccc AA BB = 2 aa cccccc AA cccccc BB 2 bb ssssss AA ssssss BB 80

81 Same Amplitude aa 1 2 ψψ 0,1 + ψψ 0,2 = ψψ 0 ψψ 0,1 = ψψ 0,2 = ψψ 0 bb 1 2 ψψ 0,1 ψψ 0,2 = 0 ψψ xx, tt = 2 aa cccccc AA cccccc BB = 2 ψψ 0 cccccc 1 2 kk 1 + kk 2 xx 1 2 ωω 1 + ωω 2 tt εε 1 + εε 2 cccccc 1 2 kk 1 kk 2 xx 1 2 ωω 1 ωω 2 tt εε 1 εε 2 81

82 kk 1 2 kk 1 + kk 2 ωω 1 2 ωω 1 + ωω 2 εε 1 2 εε 1 + εε 2 kk 1 2 kk 1 kk 2 ωω 1 2 ωω 1 ωω 2 εε 1 2 εε 1 εε 2 ψψ 0,1 = ψψ 0,2 = ψψ 0 ψψ xx, tt = 2 ψψ 0 cccccc kk xx ωω tt + εε cccccc xx kk tt ωω + εε 82