The Quantum Mirror Coating · AI Generator #GEN-4A4D45D8
Physics · quantum-mechanics
Problem
A cutting-edge astronomical telescope uses a quantum mirror coating consisting of a thin metallic film deposited on glass. Electrons in this film can be modeled as being confined to a one-dimensional infinite square well of width L=2.5nm, corresponding to the film thickness.
When the telescope observes a distant star, incident photons with wavelength λ=620nm (red light) strike the coating. For optimal reflectivity, the coating is designed so that these photons can excite electrons from the ground state (n=1) to the first excited state (n=2).
(a) What is the energy difference between these two quantum states?
(b) The incident red light has the wrong energy to cause this transition. However, due to the uncertainty principle, the electrons have a natural energy uncertainty. If an electron's position is known to within the film thickness L, what is the minimum energy uncertainty ΔE?
(c) For the quantum transition to occur, the photon energy must be within the energy uncertainty of the required transition energy. Does the red light satisfy this condition?
Use: h=6.626×10−34J⋅s, me=9.109×10−31kg, c=3.00×108m/s, ℏ=h/(2π).