HW8

1. A sphere of 1 kg mass is confined in an otherwise empty box. Assuming the sphere is in its ground state and to be represented by a standing wave, based on the uncertainty principle what will be the sphere’s velocities in the two lateral directions of a box of 30 cm by 40 cm widths? (4 points). Based on the quantum mechanical model of a particle in an infinitely deep well/box, what will be the sphere’s velocities in the two lateral directions of the box of 30 cm by 40 cm widths? (4 points). Will these velocities by observable during the time span of a typical physicist’s professional career, assumed to be 45 years? (2 points if you provide a justification of your answer, otherwise 1 point if answered with just one word)

2. Assuming that the lifetime of the 3rd excited state of the lone electron in a He⁺ ion is 2 10^-7 seconds. (a) What is the energy width of the spectral line that corresponds to the transition to the ground state? (5 points) (b) What is the corresponding relative spread in wavelength expressed in percent? (5 points)

3. Assume Plank’s constant would be 35 orders of magnitude larger than it actually is (and all of the other physical constants retain their normal values). Would that have implications with respect to the required widths of standard roads (dimension y) with two lanes that are supposed to allow for car travel in two opposite directions (+- dimension x)? (4 points if you provide a good justification of your answer that includes calculations and reasonable estimations for a typical passenger’s car mass, widths, and a maximal speed that is allowed by state laws (on a straight stretch of road). How does the order of magnitude of the position uncertainty of such a moving car (perpendicular to its velocity vector) compares to the diameter of a typical corona virus? (1 point). Hint you may use a particle in a 2D “very long and reasonably narrow box model” or the uncertainty principle in Werner Heisenberg’s formulation, or both approaches if you like.