Question

5. Consider a quantum mechanical system made of N identical particles. There are total M possible energy levels that each of these particles can occupy. (a) According to statistical thermodynamics, the probability that a particle occu- pies ith energy level with energy e; is proportional to e-Bes where B = r and T is the temperature. k is a universal constant called Boltzmann constant. What is the probability for a given particle to occupyith energy level? (b) On average, how many particles occupy įth energy level at temperature T? (c) Calculate the average energy of this system at T.

Answer 1

*If particel cire identical and indis puse Feri Diorac (FD) Statistics Toteil nwber al Particeel & enery of system=constant paulits Fermi - Dirac Distribution Low- N partical in system are identical obey po - exclusion principle murticals in syster let Ni N₂ N3 ---Nn ntical obey have states & energy El Ez &z an-En respectively and let g: is number of degenzaute quantum steile can be ocecoded by best one persbacen Particle. Since Ni particle are to be distors bute amang gi degenseite states C9;N) having the same éner E, Ni states will be filled up and Cgi-Nil Stectes semain vacargt Now gi state can be aneinged gi! passible ways. But since particles and quantum state eve iridistinguishble we have deduct Nil ways and Cgi-ni) I way, sem all possible ways to get effective numbe ongmont do total possible conseingment for ith state - Wi= [gi) i Ni ! C9; -M;)! Hence to tell number of ways for entire distribution of N puoticles in ini number of energy level la We Loo)! i N;? (g: -;)! --- Taking natural logithm on both side e que and applying stirling approximation we get

InW= &ringi - en Ni- en Cgi-N;)] = { [9ilngi – NilnW;-C81N;) loløi-;)] differntiate excation –3 to obtain most probable custoidubion eve get dCloW max = &-In Niten (gi-ni]]d Ni=0 Considering the conso er fation of total enery 6 total number of particol we can write Çd Ni =o ( constration sy total portical) cu ç Eid Ni so (conservation of energy mutiplying e o cinco by B than add in to equation ③ byd L-ln Ni + ln [gi-Ni)-2-B Ei bini=0 - Since dNi que independent of one another the expression in bracket in equ ® is zero for eaus Ni Thus which represents the probability of finding a perrtical with energy El