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

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

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

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

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

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

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)

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

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

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

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?

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

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

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  0.0375 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

Diffraction of X-Rays by Crystals X-rays are electromagnetic waves of relatively short wavelength (λ = 10-8 to 10-12 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

X-ray Diffraction Pattern

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

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

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

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

Davisson and Germer Experiment

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

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

Davisson and Germer Experiment For X-ray Diffraction on Nickel

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!

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

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.

Electron Diffraction, Set-Up

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

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

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

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

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

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

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?

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

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

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

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?

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

“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”

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

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:

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

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

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

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

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 = 1.50 + 102/, near  = 1,500 nm, where  is expressed in nm. What is the group velocity?

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 = 1.50 + 102/, near  = 1,500 nm, where  is expressed in nm. What is the group velocity?