Recall that Planck introduced h to describe the possible energies of a radiation oscillator in a cavity wall. The result was ε = nhf where n = 0, 1, 2, … and f is the frequency of the oscillator.
a) What are the dimensions of h?
b) It is reasonable to expect electrons in an atom are bound due to an electrostatic force ke2/r2. Consider a Hydrogen atom. Assuming the size of the electron orbit can only depend on the electronic charge e, the electron mass m,and Planck’s constant h, use dimensional analysis to determine how the size of the orbit r must depend on e, m, and h. Use your result to estimate the size of an H atom. Is the size of such an orbit unambiguously determined by classical considerations?
c) Show that the “fine structure constant” α ≡ 2πke2/hc is dimensionless and evaluate its value.
d) It is speculated that quantum mechanical effects are important for the gravitational interaction at a distance scale called the Planck length. Using dimensional considerations determine how the Planck length depends on G, h, and c. Evaluate its value.
2. Assume the energy of a photon of frequency f to be given by ε = hf. Consider light produced by a 50W monochromatic Sodium light source of wavelength λ = 590nm.
a) Determine the energy of each photon in the Na light
b) At a distance of 40cm from the light source what is the density of photons per unit volume?
c) What is the rate at which photons enter an eye pupil of area A = 2mm2 for the location of part (b)?