In this exercise you will become familiar with planck's distribution and energy quantization. planck's distribution was important to the development of quantum mechanics and has many applications. for example the energy distribution of light from the sun follows planck's distribution. you will see how quantization of energy and the boltzmann distribution lead the correct average energy of an electromagnetic oscillator and the correct distribution of energy for black-body radiation, planck's distribution, below o(at)- where h is planck's constant and k is the boltzmann constant. this expression differs from its classical equivalent, the rayleigh-jeans law, where the average total energy of an electromagnetic oscillator which is taken to be kt. the rayleigh-jeans law is planck was able to derive the correct expression based on the revolutionary idea that electromagnetic oscillators have quantized energy, e = nhv, where n= 0, 1, 2, and the number of oscillators excited to a given quantized energy level is given by boltzmann's distribution, n = noe-nhykt with n , i the number of unexcited oscillators. make a table with the headings, energy of level = nhv, # excited (noe-hv/kt), and total energy at this level nhu noe-nm/kt for n = 0 to n = 4 so you can start to see a pattern. (3 points) write an expression for the average energy of an oscillator based on appropriate infinite sums according to the pattern you see from a. above. substitute x = e-hv/kt and write an expression for the average energy of an electromagnetic oscillator of a given wavelength (5 points) derive the correct expression for average energy of an oscillator by using the following infinite series which apply for x < iii. (6 points) a. b. c. 1+1+12 +x3 + + xn =_=> x" (-1 < x < 1) and 冶(1+1+12 +x3 + + x") = (11)2-2-0 nxn-1 dr (1-x show that for low frequency and long wavelength or high temperature the average energy can be approximated by kt (hint: use l'hopital's rule for the low frequency case and use the expansion d. σ points) ) for the high t limit including only the first order term. note that 0! = 1 ) e'- (6 o