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Section 5.8: Diffraction and Uncertainty
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Electrons (or photons) incident on a single slit diffract and form the pattern shown. Restart. You can change the slit width (in reality, you cannot control a slit of this width as is necessary for diffracting electrons with such short de Broglie wavelengths in comparison with visible light). Assume the incident electrons are traveling in the y direction as they hit the slit (and have no momentum in the x direction). As they pass through the slit, they are located somewhere within the slit opening of width, D. In order for the beam to spread out, there must now be some momentum in the x direction. What happens to the diffraction pattern as you narrow down the x position of the particles (by narrowing the slit)? What does this mean about the possible values of the x component of the momentum?5
This is a manifestation of the uncertainty principle. The uncertainty in the x direction is about the same size as D (a mathematical analysis shows that this uncertainty is 0.289D).6 If we focus only on the central peak in the diffraction pattern, the component of momentum in the x direction must lie between 0 (the maximum intensity) and the amount required to spread to the edge of the central peak. Specifically, the ratio of the uncertainty in the momentum in the x direction to the momentum in the y direction is given by tan(b) = Δpx/py and, for small angles, tan b = sin(b). Therefore, sin(b) = Δpx/py. However, from our understanding of diffraction from a single slit (Section 5.4), D sin(b) = λ/2π, where D is the slit width. Furthermore, the wavelength of the incident electrons is the de Broglie wavelength (see Section 5.7), λ, given by λ = h/py. Combining these equations, we get
Δpx = pysin(b) = pyλ/2πD = h/2πD.
But, the uncertainty in the x position of these particles is about the slit width (Δx ~ D) so ΔxΔpx ~ h/2π. This is only an approximate value because the uncertainty in the x value of the momentum is arbitrary (could have been out to the secondary peaks since some small signal appears there as well). However, this is consistent with the uncertainty principle:
ΔxΔpx ~ ħ. (5.7)
Notice that this is a relationship between the x components. The y component of momentum can be very well known because the particle/wave can exist anywhere in space in the y direction (there is no slit in the y direction).
5Development follows K. Krane, Modern Physics, John Wiley & Sons (1996), p. 118.
6The uncertainty in x, Δx, is given by
Δx = (<x2> − <x>2)1/2 (5.5)
<x> = ∫ f(x) x dx (5.6)
evaluated over all space. In this case f(x) = 1/D for −D/2 < x < D/2 and zero for other values of x, thereby ensuring that ∫ f(x) dx = 1 with limits of integration from −∞ to +∞.