• In the two-slit experiment, we cannot say which slit the photon goes through - it goes through both as a wave • In Bragg diffraction, we could not tell which atom a photon scattered from; we will soon see this is the case for electrons as well. • If we measure a particle initially at position a and next at position b, we generally cannot tell the path it took from a to b. Thursday, March 14, 2013
Wave Particle Duality 1924: Louis-Victor de Broglie proposes that, since light acts as both a particle and a wave, "particles" also act as both particles and waves. We assume non-relativistic particles... Photons Electrons, etc. E = hf E = hf p = h/ λ = ħ k p = mv = h/ λ = ħ k p = E/c p = (2mK) 1/2 Thursday, March 14, 2013
Photons vs. "Particles" We have the same relations between momentum p and wavelength λ : p = h/ λ . We will focus on this relation, since we will study many phenomena, such as interference, that are directly connected to the wavelength. The uncertainty relations we discussed for photons - Δ E Δ t ≥ ħ /2, Δ x Δ p x ≥ ħ /2, etc. also apply to particles. But we have different relations between energy E and momentum p since the photon is massless: E = pc vs. E 2 = (pc) 2 + (mc 2 ) 2 (relativistically), or K = p 2 /2m (non- relativistically). Thursday, March 14, 2013
Experimental Confirmation The wave nature of electrons was demonstrated within a few years of its prediction by the Davisson-Germer experiment, which measured diffraction of electrons by a Ni crystal, the same phenomena as we discussed in Bragg diffraction of X-rays by a crystal. Thursday, March 14, 2013
The Data and Picture The meaning of the intensity distribution shown to the left is unclear, since we don't know what the distribution would look like for a disordered sample of Nickel - it gives a smooth falloff with angle. When the sample was heated to nearly melting, and cooled, forming a better crystal, a maximum appears for an angle near 50 o . From the figure: m λ = d sin θ . For m = 1, λ /d = 0.77. Thursday, March 14, 2013
More Data Top: diffraction pattern of 71-pm X-rays from aluminum foil. Bottom: diffraction pattern of 600 eV electrons from aluminum. The scale of the two is different - the electrons do not have 71 pm wavelength. What is the electron wavelength? K = 600 eV ➮ p = (2mK) 1/2 = (2 x 9.11x10 -31 kg x 600 eV x 1.6x10 -19 J/ eV) 1/2 = 1.32x10 -23 kg m/s. λ = h/p = 6.626x10 -34 Js / (1.32x10 -23 kg m/s) = 5.02x10 -11 m = 50.2 pm. Note: atomic separations typically a few hundred pm. Thursday, March 14, 2013
iClicker If m λ = d sin θ for the maxima, shouldn't we get maxima evenly spaced in sin θ ? (Spread further apart in angle with increasing angle towards 90 o .) Why doesn't this happen? Why are there a couple lines close together? a) They are spread out even in sin θ . b) There are two electron wavelengths, one from the kinetic energy K and the other from the total energy E. c) There are two electron wavelengths, one from λ = h/p and one from λ = v/f = v/(K/h).
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