PHYSICS · Multi-part · A Level · Physics · Particles and radiation · Particles and radiation
Problem
A microscope is used to observe an electron. The objective of the microscope subtends a total angle 2θ at the scattering point, and the photon employed has wavelength λ.
(i) Show that, due to diffraction at the aperture, the minimum uncertainty in the measured position of the electron perpendicular to the direction of observation is approximately Δx≈2sinθλ.
(ii) Taking the transverse component of the photon’s momentum to be pγsinθ where pγ=λh, prove that the uncertainty in the electron’s transverse momentum caused by the scattering is of the order Δp≈λhsinθ.
(iii) Hence derive that the product of the two uncertainties satisfies ΔxΔp≳4πh, establishing the position–momentum uncertainty relation from this thought experiment.