PHYSICS · Multi-part · Fermi University Physics · Oscillations, Waves & Acoustics · Quantum Mechanics · Origins of quantum theory
Quantum Mechanics: From the Photon to the Hydrogen Atom, Ch Matter waves and wave-particle duality
Problem
Electrons of kinetic energy 5.0 keV, sent one at a time, pass through two narrow slits d=2.0μm apart and build up fringes on a screen L=0.60 m beyond the slits. Take the electron wavelength from λ=1.226nm/K/eV (non-relativistic), h=6.626×10−34 J s, and use small angles throughout. One classmate proposes fitting a device immediately behind the slits that records each electron's sideways position y with a resolution Δy=0.10μm. "That is twenty times finer than the slit separation," the classmate says, "so the device is very gentle, it will tell us which slit each electron used, and the fringes will still be there. If the fringes did blur, we would just buy a finer detector."
((a)) Find the spacing of neighbouring fringes on the screen, in m, with no detector present.
((b)) Using the order-of-magnitude trade-off ΔyΔpy∼h from the finite wave train, estimate the sideways momentum spread, in kg m s−1, that an electron wave localised to within Δy must contain, and hence the spread of arrival positions it produces on the screen, in m.
((c)) Is that argument right? Say what happens to your answer to (b) if the resolution is made finer still.