Understanding Electron Diffraction and Matter Waves in Quantum Mechanics

1. Matter-Waves and Electron Diffraction “The most incomprehensible thing about the world is that it is comprehensible.” – Albert Einstein Day 18, 3/31: Questions? Electron Diffraction Wave packets and uncertainty Next Week: Schrodinger equation The Wave Function Exam 2 (Thursday)

2. Exam II is next Thursday 3/31, in class. HW09 will be due Wednesday (3/30) by 5pm in my mailbox. HW09 solutions posted at 5pm on Wednesday Recently: 1. Single-photon experiments 2. Complementarity and wave-particle duality 3. Matter-Wave Interference Today: 1. Electron diffraction and matter waves 2. Wave packets and uncertainty 2

3. Double-Slit Experiment • Pass a beam of electrons through a double-slit apparatus. • Individual electrons are detected as points on the screen. • Over time, a fringe pattern of dark and light bands appears.

4. Double-Slit Experiment with Single Electrons (1989) http://www.hitachi.com/rd/research/em/doubleslit.html A. Tonomura, J. Endo, T. Matsuda, T. Kawasaki and H. Ezawa, "Demonstration of Single-Electron Buildup of an Interference Pattern,“ Amer. J. Phys. 57, 117 (1989).

6. Double-Slit Experiment with Light Recall solution for Electric field (E) from the wave equation:    cos( ) E A kx t Let’s forget about the time- dependence for now:  cos( ) E A kx   exp( ) cos( ) sin( ) ikx kx i kx Euler’s Formula says:    sin( )] Re[ exp( kx  cos( ) Re[ cos( ) A )] E A kx kx iA A ikx  exp( ) E A ikx For convenience, write:

7. Double-Slit Experiment with Electrons Before Slits   exp( ) A ikx After Slits     exp( ) exp( ) A ikx A ikx & 1 1 1 2 2 2       1 2 Total   exp( ) exp( ) A ikx A ikx 1 1 2 2

8. Double-Slit Experiment    exp( ) exp( ) A ikx A ikx 1 1 2 2 2       ( ) ( ) x  * [ ] P x | ( ) x x      ikx   ikx   exp( ) exp( ) exp( ) exp( ) A A A ikx A ikx 1 1 2 2 1 1 2 2   ikx A  ikx A    ikx A  ikx A  exp( exp( ) ) exp( exp( ) ) exp( exp( ) ) exp( exp( ) ) A A ikx ikx A A ikx ikx 1 1 1 1 2 2 2 2 1 1 2 2 2 2 1 1       )) exp(  ik x   2 2 exp( ( ( )) A A AA ik x x x 1 2 1 2 2 1 2 1       2 2 2 cos ( ) A A AA k x x 1 2 1 2 2 1

9. Double-Slit Experiment 2       ( ) ( ) x  * [ ] P x | ( ) x x   2      2 2 ( ) x 2 cos ( ) A A A A k x x 1 2 1 2 2 1        2  2      2 2 ( ) x 2 cos ( ) A A A A x x 1 2 1 2 2 1 How do we connect this with what we observe? 2   ikx A   2 ( ) x exp( ) exp( ) A ikx A 1 1 1 1 1 1 2   ikx A   2 ( ) x exp( ) exp( ) A ikx A 2 2 2 2 2 2

10. Double-Slit Experiment        2  2       2 2 [ ] P x ( ) x 2 cos ( ) A A A A x x 1 2 1 2 2 1 2 2   INTERFERENCE! 1 2 2 2 2          [ ] [ ] [ ] P x P x P x 12 1 2 1 2 1 2 • In quantum mechanics, we add the individual amplitudes and square the sum to get the total probability • In classical physics, we added the individual probabilities to get the total probability.

11. 2  1 2  2 2   1 2 2 2    1 2

12. Double-Slit Experiment

13. Determining the spacing between the slits: L 2x H 1x D m     m 0,1,2,... x x x 2 1 1      1 2  D sin    2           sin x D D m

14. L 2x H 1x D  m D         sin H L L  m L D  H Spacing between bright fringes

15. L 2x H 1x D  m L D  H Distance to first maximum for visible light: 1 m  500 nm   5 10 0.001 5 10 D   5 10 m   4 D Let , ,       7  rad  4 L     If , then 3 m L  3 mm H

16. Double-Slit Experiment 2     ( ) ( ) x     * [ ] P x ( ) x x      2 2      2 2 ( ) x 2 cos ( ) A A A A x x 1 2 1 2 2 1 2  ( ) x For a specific λ, describes where we find the interference maxima and minima of the fringe pattern. What is the wavelength of an electron?

17. Matter Waves As a doctoral student (1923), Louis de Broglie proposed that electrons might also behave like waves. • Light, often thought of as a wave, had been shown to have particle-like properties. • For photons, we know how to relate momentum to wavelength: hc   E hf From the photoelectric effect:    E p c  From special relativity:  hp   Combined:

18. Matter Waves de Broglie wavelength: hp   Confirmed experimentally by Davisson and Germer (1924).

19. Matter Waves  m L D hp    de Broglie wavelength: H In order to observe electron interference, it would be best to perform a double-slit experiment with: A) Lower energy electron beam. B) Higher energy electron beam. C) It doesn’t make any difference.

20. Matter Waves  m L D hp    de Broglie wavelength: H In order to observe electron interference, it would be best to perform a double-slit experiment with: A) Lower energy electron beam. B) Higher energy electron beam. C) It doesn’t make any difference. Lowering the energy will increase the wavelength of the electron. Can typically make electron beams with energies from 25 – 1000 eV.

21. Matter Waves  m L D hp    de Broglie wavelength: H   6.6 10   34 J s h For an electron beam of 25 eV, we expect Θ (the angle between the center and the first maximum) to be:  5 10 m   4 D Use A) Θ << 1 B) Θ < 1 C) Θ > 1 D) Θ >> 1    and remember: D 2 p  E & 2 m

22. Matter Waves  m L D hp    de Broglie wavelength: H For an electron beam of 25 eV: 2 p    2 E p mE 2 m  J s)  34 (6.6 10 x   1 2   [2 (9.1 10  kg) (25   31 19 1.6 10 x / )] x eV J eV

23. Matter Waves  m L D hp    de Broglie wavelength: H   6.6 10   34 J s h For an electron beam of 25 eV:   34 6.6 10 2.7 10  h      2.4 10  10 m p  24      4.9 10  7  5 10 m   4 D for D      (3 )(4.9 10 ) 1.5 10 m      7 6 1.5 H L m m Too small to easily see!

24. Matter Waves  m L D hp    de Broglie wavelength: H For an electron beam of 25 eV, how can we make the diffraction pattern more visible? A) Make D much smaller. B) Decrease energy of electron beam. C) Make D much bigger. D) A & B E) B & C

25. Matter Waves hp     6.6 10   34 J s h de Broglie wavelength: For an electron beam of 25 eV, how can we make the diffraction pattern more visible? A) Make D much smaller. B) Decrease energy of electron beam. C) Make D much bigger. D) A & B E) B & C 25 eV is a lower bound on the energy of a decent electron beam. Decreasing the distance between the slits will increase Θ.

26. S. Frabboni, G. C. Gazzadi, and G. Pozzi, Nanofabrication and the realization of Feynman’s two-slit experiment, Applied Physics Letters 93, 073108 (2008). Slits are 83 nm wide and spaced 420 nm apart

27. L 2x H 1x D  L  H D If we were to use protons instead of electrons, but traveling at the same speed, what happens to the distance to the first maximum, H? A) Increases B) Stays the same C) Decreases 2 p hp    E 2 m

28. Which slit did this electron go through? A) Left B) Right C) Both D) Neither E) Either left or right, we just can’t know which one.

29. Each electron passes through both slits, interferes with itself, then becomes localized when detected. • For a low-intensity beam, can have as little as one electron in the apparatus at a time. • Each electron must interfere with itself • Each electron is “aware” of both paths. • Asking which slit the electron went through disrupts the fringe pattern.

30. Light Waves High Amplitude Screen Low Amplitude Describe photon’s EM wave spread out in space. 2 probability of photon detection (amplitude of EM wave)

31. Light Waves 2 probability of photon detection (amplitude of EM wave) E-field x-section at screen x ρ(x) =probability density Photons

32. Follow same prescription for matter waves: • Describe massive particles with wave functions:    ( ) x • Interpret (wave amplitude)2as a probability density: ( ) ( ) x x    2   ( ) ( ) x  * x Electron double slit experiment. Display=Magnitude of wave function Large Magnitude (|Ψ|)= probability of detecting electron here is high Small Magnitude (|Ψ|)= probability of detecting electron here is low 32

33. What are these waves? EM Waves (light/photons) Matter Waves (electrons/etc) • Amplitude = electric field • tells you the probability of detecting a photon. • Maxwell’s Equations: 2 1 c x  • Amplitude = matter field • tells you the probability of detecting a particle. • Schrödinger Equation:   2 2  E 2 2  E      2 2 2 E E    i     2 2 2 m x t t • Solutions are sine/cosine waves: • Solutions are complex sine/cosine waves:       ( , ) E x t sin( )        t    A kx t ( , ) x t  A exp  A i kx t       cos( ) sin( ) kx i kx t ( , ) E x t cos( ) A kx t

34. Matter Waves    ( , ) x t • Describe a particle with a wave function. • Wave does not describe the path of the particle. • Wave function contains information about the probability to find a particle at x, y, z & t.   ( , x t 0) In general: ( , , , ) x y z t  ( , , , ) r    t -L L x Simplified: ( , ) x t   ( ) x & 2      ( ) ( ) x  * ( ) x ( ) x x

35. Matter Waves Wavefunction -L   ( , x t 0) L x Probability Density 2 ( , x t      0) ( , x t 0) -L L x Probability of finding particle in the interval dx is ( ) x dx      2      ( ) x dx     [ ] ( ) x 1 P x dx   Normalized wave functions

36. Matter Waves Below is a wave function for a neutron. At what value of x is the neutron most likely to be found? A) XA B) XB C) XC D) There is no one most likely place 36

37. An electron is described by the following wave function:   x ( ) x       ( ) x from to x L x L L  0 elsewhere c a b d dx x L How do the probabilities of finding the electron near (within dx) of a,b,c, and d compare? A) d > c > b > a B) a = b = c = d C) d > b > a > c D) a > d > b > c

38. An electron is described by the following wave function: ( ) x  ( ) x   x       ( ) x from to x L x L L   0 elsewhere c a b d  2 2 dx x 2     ( ) x ( ) x x L 2 L How do the probabilities of finding the electron near (within dx) of a,b,c, and d compare? A) d > c > b > a B) a = b = c = d C) d > b > a > c D) a > d > b > c

39. Plane Waves Plane waves (sines, cosines, complex exponentials) extend forever in space:           ( , ) x t exp i k x t 1 1 1           ( , ) x t exp i k x t 2 2 2           ( , ) x t exp i k x t 3 3 3           ( , ) x t exp i k x t 4 4 4 Different k’s correspond to different energies, since 1 2 2 2 2 2 p h m k m     2 E mv  2 2 2 2 m

40. Superposition 1( , ) x t   2( , ) x t   If and ( , ) x t  are solutions to a wave equation, ( , ) ( , ) x t x t  then so is 1 2 Superposition (linear combination) of two waves We can construct a “wave packet” by combining many plane waves of different energies (different k’s).

41. Superposition exp n n        i k x     ( , ) x t A t n n 41

42. Plane Waves vs. Wave Packets           ( , ) x t exp A i kx t        i k x     ( , ) x t exp A t n n n n Which one looks more like a particle?

43. Plane Waves vs. Wave Packets           ( , ) x t exp A i kx t        i k x     ( , ) x t exp A t n n n n For which type of wave are the position (x) and momentum (p) most well-defined? A) x most well-defined for plane wave, p most well-defined for wave packet. B) p most well-defined for plane wave, x most well-defined for wave packet. C) p most well-defined for plane wave, x equally well-defined for both. D) x most well-defined for wave packet, p equally well-defined for both. E) p and x are equally well-defined for both.

44. Uncertainty Principle Δx small Δp – only one wavelength Δx medium Δp – wave packet made of several waves Δx large Δp – wave packet made of lots of waves

45. Uncertainty Principle    x p In math: 2 In words: The position and momentum of a particle cannot both be determined with complete precision. The more precisely one is determined, the less precisely the other is determined.  x What do p  (uncertainty in position) and (uncertainty in momentum) mean?

46. Uncertainty Principle A Realist Interpretation: • Photons are scattered by localized particles.     • Due to the lens’ resolving power: x h     x p               h  x 2   • Due to the size of the lens: p x 2

47. Uncertainty Principle 1 N N    2     A Statistical Interpretation: x x i  1 i     x p 2 •Measurements are performed on an ensemble of similarly prepared systems. • Distributions of position and momentum values are obtained. • Uncertainties in position and momentum are defined in terms of the standard deviation.

48. Uncertainty Principle A Wave Interpretation: • Wave packets are constructed from a series of plane waves. • The more spatially localized the wave packet, the less uncertainty in position. • With less uncertainty in position comes a greater uncertainty in momentum.

49. Matter Waves (Summary) • Electrons and other particles have wave properties (interference) • When not being observed, electrons are spread out in space (delocalized waves) • When being observed, electrons are found in one place (localized particles) • Particles are described by wave functions: (probabilistic, not deterministic) • Physically, what we measure is    ( , ) x t 2 ( , ) x t    ( , ) x t (probability density for finding a particle in a particular place at a particular time) • Simultaneous measurements of x & p are constrained by the x    p Uncertainty Principle: 2