## 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 0.289*D*).^{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*) = Δ*p*_{x}/*p*_{y }and, for small angles, tan *b* = sin(*b*). Therefore, sin(*b*) = Δ*p*_{x}/*p*_{y}._{ }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*/*p*_{y}. Combining these equations, we get

Δ*p*_{x} = *p*_{y}sin(*b*) = *p*_{y}λ/2π*D* = *h*/2π*D*.

But, the uncertainty in the *x* position of these particles is about the slit width (Δ*x* ~ *D*) so Δ*x*Δ*p*_{x} ~ *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:

Δ*x*Δ*p*_{x } ~ *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).

^{5}Development follows K. Krane, *Modern Physics*, John Wiley & Sons (1996), p. 118.

^{6}The uncertainty in *x*, Δ*x*, is given by

Δ*x* = (<*x*^{2}> − <*x*>^{2})^{1/2} (5.5)

where

<*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 +∞.

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