1. Wave Properties of Particles
Serway/Jewett chapters 38.5; 40.4 – 40.7

2. Photons and Waves Revisited
Some experiments are best explained by the photon model
Some are best explained by the wave model
We must accept both models and admit that the true nature of light is not describable in terms of any single classical model
The particle and wave models complement one another

3. Dual Nature of EM radiation
To explain all experiments with EM radiation (light), one must assume that light can be described both as wave (Interference, Diffraction) and particles (Photoelectric Effect, Frank-Hertz Experiment, x-ray production, x-ray scattering from electron)
To observe wave properties must make observations using devices with dimensions comparable to the wavelength.
For instance, wave properties of X-rays were observed in diffraction from arrays of atoms in solids spaced by a few Angstroms

4. Louis de Broglie 1892 – 1987 French physicist
Originally studied history
Was awarded the Nobel Prize in 1929 for his prediction of the wave nature of electrons

5. De Broglie’s Hypothesis
Louis de Broglie postulated that the dual nature of the light must be “expanded” to ALL matter
In other words, all material particles possess wave-like properties, characterized by the wavelength, λB, related to the momentum p of the particle in the same way as for light
Planck’s Constant
Momentum of the particle
de Broglie wavelength of the particle

6. Wave Properties of Particles
Louis de Broglie postulated that because photons have both wave and particle characteristics, so too all forms of matter have both properties
For photons:
De Broglie hypothesized that particles of well defined momentum also have a wavelength, as given above, the de Broglie wavelength

7. Frequency of a Particle
In an analogy with photons, de Broglie postulated that a particle would also have a frequency associated with it
These equations present the dual nature of matter
Particle nature, p and E
Wave nature, λ and ƒ ( and k)

8. De Broglie’s Hypothesis
De Broglie’s waves are not EM waves
He called them “pilot” or “material” waves
λB depends on the momentum and not on physical size of the particle
For a non-relativistic free particle:
Momentum is p = mv, here v is the speed of the particle
For free particle total energy, E, is kinetic energy

9. Photons and Waves Revisited
Some experiments are best explained by the photon model
Some are best explained by the wave model
We must accept both models and admit that the true nature of light is not describable in terms of any single classical model
Also, the particle model and the wave model of light complement each other

10. Complementarity
The principle of complementarity states that the wave and particle models of either matter or radiation complement each other
Neither model can be used exclusively to describe matter or radiation adequately
No measurements can simultaneously reveal the particle and the wave properties of matter

11. The Principle of Complementarity and the Bohr Atom
How can we understand electron orbits in hydrogen atom from wave nature of the electron?
Remember: An electron can take only certain orbits: those for which the angular momentum, L, takes on discrete values
How does this relate to the electron’s de Broglie’s wavelength?

12. The Principle of Complementarity
Only those orbits are allowed, which can “fit” an integer (discrete) number of the electron’s de Broglie’s wavelength
Thus, one can “replace” 3rd Bohr’s postulate with the postulate demanding that the allowed orbits “fit” an integer number of the electron’s de Broglie’s wavelength
This is analogous to the standing wave condition for modes in musical instruments

13. For example we should observe
De Broglie’s Hypothesis predicts that one should see diffraction and interference of matter waves
For example we should observe
Electron diffraction
Atom or molecule diffraction

14. Estimates for De Broglie wavelength
Bullet:
m = 0.1 kg; v = 1000 m/s  λB ~ 6.63×10-36 m
Electron at 4.9 V potential:
m = 9.11×10-31 kg;
E~4.9 eV  λB ~ 5.5×10-10 m = 5.5 Å
Nitrogen Molecule at Room Temperature:
m ~ 4.2×10-26 kg;
E = (3/2)kBT  eV  λB ~2.8×10-11 m = 0.28 Å
Rubidium (87) atom at 50 nK:
λB ~ 1.2×10-6 m = 1.2 mm = 1200 Å

15. Diffraction of X-Rays by Crystals
X-rays are electromagnetic waves of relatively short wavelength (λ = 10-8 to m = 100 – 0.01 Å)
Max von Laue suggested that the regular array of atoms in a crystal (spacing in order of several Angstroms) could act as a three-dimensional diffraction grating for x-rays

16. X-ray Diffraction Pattern

17. X-Ray Diffraction
This is a two-dimensional description of the reflection (diffraction) of the x-ray beams
The condition for constructive interference is
where n = 1, 2, 3
This condition is known as Bragg’s law
This can also be used to calculate the spacing between atomic planes

18. Davisson-Germer Experiment
If particles have a wave nature, then under appropriate conditions, they should exhibit diffraction
Davisson and Germer measured the wavelength of electrons
This provided experimental confirmation of the matter waves proposed by de Broglie

19. Davisson and Germer Experiment
Electrons were directed onto nickel crystals
Accelerating voltage is used to control electron energy: E = |e|V
The scattering angle and intensity (electron current) are detected
φ is the scattering angle

20. Davisson and Germer Experiment
If electrons are “just” particles, we expect a smooth monotonic dependence of scattered intensity on angle and voltage because only elastic collisions are involved
Diffraction pattern similar to X-rays would be observed if electrons behave as waves

21. Davisson and Germer Experiment

22. Davisson and Germer Experiment
Observations:
Intensity was stronger for certain angles for specific accelerating voltages (i.e. for specific electron energies)
Electrons were reflected in almost the same way that X-rays of comparable wavelength

23. Davisson and Germer Experiment
Observations:
Current vs accelerating voltage has a maximum, i.e. the highest number of electrons is scattered in a specific direction
This can’t be explained by particle-like nature of electrons  electrons scattered on crystals behave as waves
For φ ~ 50° the maximum is at ~54V

24. Davisson and Germer Experiment
For X-ray Diffraction on Nickel

25. Davisson and Germer Experiment
(Problem 40.38) Assuming the wave nature of electrons we can use de Broglie’s approach to calculate wavelengths of a matter wave corresponding to electrons in this experiment
V = 54 V  E = 54 eV = 8.64×10-18J
This is in excellent agreement with wavelengths of X-rays diffracted from Nickel!

26. Single Electron Diffraction
In previous experiments many electrons were diffracted
Will one get the same result for a single electron?
Such experiment was performed in 1949
Intensity of the electron beam was so low that only one electron at a time “collided” with metal
Still diffraction pattern, and not diffuse scattering, was observed, confirming that
Thus individual electrons behave as waves

27. Two-slit Interference Thomas Young
The intensity is obtained by squaring the wave,
I1 ~ <h12>, I2 ~ <h22>,
I12 = <(h1 + h2)2> = <h12+h22+ 2h1h2>, where < > is average over time of the oscillating wave.
h1h2 ~ cos(2p/) and reflects the interference between waves reaching the point from the two slits.
When the waves arriving from slits 1 and 2 are in phase, p = n, and cos(2p/) = 1.
For <I1> = <I2>, I12 = 4I1.
When the waves from slits 1 and 2 are out of phase,  = n + /2, and cos(2/) = -1 and I12 = 0.

28. Electron Diffraction, Set-Up

29. Electron Diffraction, Experiment
Parallel beams of mono-energetic electrons that are incident on a double slit
The slit widths are small compared to the electron wavelength
An electron detector is positioned far from the slits at a distance much greater than the slit separation

30. Electron Diffraction, cont.
If the detector collects electrons for a long enough time, a typical wave interference pattern is produced
This is distinct evidence that electrons are interfering, a wave-like behavior
The interference pattern becomes clearer as the number of electrons reaching the screen increases

31. Active Figure 40.22
Use the active figure to observe the development of the interference pattern
Observe the destruction of the pattern when you keep track of which slit an electron goes through
PLAY
ACTIVE FIGURE

32. Electron Diffraction, Equations
A maximum occurs when
This is the same equation that was used for light
This shows the dual nature of the electron
The electrons are detected as particles at a localized spot at some instant of time
The probability of arrival at that spot is determined by calculating the amplitude squared of the sum of all waves arriving at a point

33. Electron Diffraction Explained
An electron interacts with both slits simultaneously
If an attempt is made to determine experimentally through which slit the electron goes, the act of measuring destroys the interference pattern
It is impossible to determine which slit the electron goes through
In effect, the electron goes through both slits
The wave components of the electron are present at both slits at the same time

34. Other experiments showed wave nature for neutrons, and even big molecules, which are much heavier than electrons!!
Neutrons
He atoms
C60 molecules

35. Example of Electron Diffraction
Electrons from a hot filament are incident upon a crystal at an angle φ = 30 from the normal (the line drawn perpendicular) to the crystal surface. An electron detector is place at an angle φ = 30 from the normal. Atomic layers parallel to the sample surface are spaced by d = 1.3 A.
Through what voltage V must the electron be accelerated for a maximum in the electron signal on the detector?
Will the electrons scatter at other angle?

36. Phase Speed
Phase velocity is the speed with which wave crest advances:

37. Addition of Two Waves Two sine waves traveling in the same direction:
Constructive and Destructive Interference
Two sine waves traveling in opposite directions create a standing wave
Two sine waves with different frequencies: Beats

38. Beat Notes and Group Velocity, vg
This represents a beat note with the amplitude of the beat moving at speed

39. Beats and Pulses
Two tuning forks are struck simultaneously. The vibrate at 512 and 768 Hz.
(a) What is the frequency of the separation between peaks in the beat envelope?
(b) What is the velocity of the beat envelope?

40. Beats and Pulses
Two tuning forks are struck simultaneously. The vibrate at 512 and 768 Hz.
(a) What is the separation between peaks in the beat envelope?
(b) What is the velocity of the beat envelope?
(a)
The rapidly oscillating wave is multiplied by a more slowly varying envelope
with wave vector

41. “Construction” Particles From Waves
Particles are localized in space
Waves are extended in space.
It is possible to build “localized” entities from a superposition of number of waves with different values of k-vector. For a continuum of waves, the superposition is an integral over a continuum of waves with different k-vectors.
The wave then has a non-zero amplitude only within a limited region of space
Such wave is called “wave packet”

42. Wave Packet
Mathematically a wave packet can be written as sum (integral) of many “ideal” sinusoidal waves

43. Wave Picture of Particle
Consider a wave packet made up of waves with a distribution of wave vectors k, A(k), at time t. A snapshot, of the wave in space along the x-direction is obtained by summing over waves with the full distribution of k-vectors. For a continuum this is an integral.
The spatial distribution at a time t given by:

44. Wave Picture of Particle
A(k) is spiked at a given k0, and zero elsewhere
only one wave with k = k0 (λ = λ0) contributes; thus one knows momentum exactly, and the wavefunction is a traveling wave – particle is delocalized
A(k) is the same for all k
No distinctions for momentums, so particle’s position is well defined - the wavefunction is a “spike”, representing a “very localized” particle
A(k) is shaped as a bell-curve
Gives a wave packet – “partially” localized particle

45. Wave Picture of Particle
The greater the range of wave numbers (and therefore λ‘s) in the mix, the narrower the width of the wave packet and the more localized the particle

46. Group Velocity for Particles and Waves
The group velocity in term of particle parameters is
Consider a free non-relativistic particle. The total, energy for this particle is, E = Ek = p2/2m

47. Group Velocity
The group speed of wave packet is identical to the speed of the corresponding particle,
Is this true for photon, for which u = c?
For photon total energy E = p·c

48. Group Velocity in Optical Fiber
A pulse of light is launched in an optical fiber. The amplitude A(k) of the pulses
is peaked in the telecommunications band at the wavelength in air,  = 1,500 nm.
The optical fiber is dispersive, with n = /, near  = 1,500 nm, where  is
expressed in nm. What is the group velocity?

49. Group Velocity in Optical Fiber
A pulse of light is launched in an optical fiber. The amplitude A(k) of the pulses
is peaked in the telecommunications band at the wavelength in air,  = 1,500 nm.
The optical fiber is dispersive, with n = /, near  = 1,500 nm, where  is
expressed in nm. What is the group velocity?