{"id":4767,"date":"2012-04-26T22:04:37","date_gmt":"2012-04-26T20:04:37","guid":{"rendered":"http:\/\/djalil.chafai.net\/blog\/?p=4767"},"modified":"2012-04-27T11:57:04","modified_gmt":"2012-04-27T09:57:04","slug":"bosons-and-fermions","status":"publish","type":"post","link":"https:\/\/djalil.chafai.net\/blog\/2012\/04\/26\/bosons-and-fermions\/","title":{"rendered":"Bosons and fermions"},"content":{"rendered":"

\"Bose-Einstein<\/a><\/p>\n

In quantum mechanics, the spin-statistics theorem<\/a> states that sub-atomic particles are either bosons<\/strong> or fermions<\/strong>. Photons<\/a> are bosons<\/a>, while electrons<\/a> are fermions<\/a>. Bosons have integer spin<\/a>, and the wave function of a system of two bosons is symmetric. Fermions have half integer spin, and the wave function of a system of two fermions is antisymmetric. Bosons can occupy the same quantum state, and obey to the Bose-Einstein statistics<\/a>. Fermions cannot occupy the same quantum state (Pauli exclusion principle<\/a>), and obey to the Fermi-Dirac statistics<\/a>. These statistics can be retained using some elementary combinatorics, in the spirit of the famous Maxwell-Boltzmann statistics<\/a>.<\/p>\n

Maxwell-Boltzmann statistics.<\/strong> We have \\( {m} \\) energy levels \\( {E_1,\\ldots,E_m} \\) and \\( {n} \\) distinguishable<\/strong> particles. If \\( {n_i} \\) is the number of \\( {E_i} \\) particles then \\( {n=n_1+\\cdots+n_m} \\) and the average energy of configuration \\( {(n_1,\\ldots,n_m)} \\) is \\( {E=n_1E_1+\\cdots+n_mE_m} \\). There are<\/p>\n

\\[ \\binom{n}{n_1,\\ldots,n_m}=n!\\prod_{i=1}^m\\frac{1}{n_i!} \\]<\/p>\n

ways to obtain the configuration \\( {(n_1,\\ldots,n_m)} \\) (multinomial coefficient: throw \\( {n} \\) distinguishable balls into \\( {m} \\) boxes). In order to get an additive measurement of the degree of freedom, we may take the logarithm. The additive degree of freedom per particle is then \\( {\\frac{1}{n}\\log\\binom{n}{n_1,\\ldots,n_m}} \\). Now, if \\( {n\\rightarrow\\infty} \\) with \\( {n_i\/n\\rightarrow p_i} \\) for every \\( {1\\leq i\\leq n} \\) then \\( {p_1+\\cdots+p_m=1} \\), and thanks to the Stirling formula, the asymptotic degree of freedom per particle is given by<\/p>\n

\\[ -\\sum_{i=1}^mp_i\\log(p_i). \\]<\/p>\n

This quantity is known as the Boltzmann entropy. Maximizing this entropy subject to the fixed average energy constraint \\( {E=\\sum_{i=1}^mp_iE_i} \\) gives<\/p>\n

\\[ p_i=Z^{-1}e^{-\\beta E_i} \\]<\/p>\n

where the Lagrange multiplier \\( {\\beta>0} \\) can be interpreted to be proportional to an inverse temperature, and were \\( {Z=\\sum_{i=1}^me^{-\\beta E_i}} \\) is a normalizing factor.<\/p>\n

As pointed out by Gibbs, the reasoning above is faulty. Namely, if we consider a system formed by \\( {n+n'} \\) particles, then the logarithmic degree of freedom is actually not additive, violating the second principle of thermodynamics. A way to solve this paradox is to assume that the particles are undistinguishable<\/strong>. This makes the combinatorics trivial since for every configuration \\( {(n_1,\\ldots,n_m)} \\) there is a unique way to throw our undistinguishable particles into the \\( {m} \\) boxes with \\( {n_i} \\) particles in each box for every \\( {1\\leq i\\leq m} \\). To escape from this triviality, we assume that each energy level \\( {E_i} \\) has \\( {s_i} \\) states, in other words that box \\( {i} \\) has \\( {s_i} \\) sub-boxes. The number of ways to obtain configuration \\( {(n_1,\\ldots,n_m)} \\) is then<\/p>\n

\\[ \\prod_{i=1}^m\\frac{(n_i+s_i-1)!}{n_i!(s_i-1)!}. \\]<\/p>\n

If \\( {s_1=\\cdots=s_m=1} \\) then this number is \\( {1} \\). If \\( {n_i\\ll s_i} \\) then thanks to the Stirling formula, the number of ways to obtain configuration \\( {(n_1,\\ldots,n_m)} \\) is<\/p>\n

\\[ \\approx\\prod_{i=1}^m\\frac{s_i^{n_i}}{n_i!}. \\]<\/p>\n

The variational reasoning used for the distinguishable case leads then to another distribution for the energy levels, known as the Maxwell-Boltzmann statistics:<\/p>\n

\\[ p_i\\approx Z^{-1}s_ie^{-\\beta E_i}. \\]<\/p>\n

When \\( {s_1=\\cdots=s_m} \\) we recover the same distribution.<\/p>\n

Bose-Einstein statistics.<\/strong> We follow the reasoning used for the Maxwell-Boltzmann statistics, without the assumption \\( {n_i\\ll s_i} \\). This leads to<\/p>\n

\\[ p_i\\approx Z^{-1}\\frac{s_i}{e^{\\beta E_i-\\alpha}-1}. \\]<\/p>\n

Fermi-Dirac statistics.<\/strong> Suppose this time that at most one particle can occupy a state (i.e. a sub-box). The number of ways to obtain configuration \\( {(n_1,\\ldots,n_m)} \\) is this time<\/p>\n

\\[ \\prod_{i=1}^m\\frac{s_i!}{n_i!(s_i-n_i)!}, \\]<\/p>\n

leading to<\/p>\n

\\[ p_i\\approx Z^{-1}\\frac{s_i}{e^{\\beta E_i-\\alpha}+1}. \\]<\/p>\n","protected":false},"excerpt":{"rendered":"

In quantum mechanics, the spin-statistics theorem states that sub-atomic particles are either bosons or fermions. Photons are bosons, while electrons are fermions. Bosons have integer…<\/p>\n

Continue readingBosons and fermions<\/span><\/a><\/div>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":120},"categories":[1],"tags":[],"_links":{"self":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts\/4767"}],"collection":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/comments?post=4767"}],"version-history":[{"count":25,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts\/4767\/revisions"}],"predecessor-version":[{"id":4795,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts\/4767\/revisions\/4795"}],"wp:attachment":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/media?parent=4767"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/categories?post=4767"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/tags?post=4767"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}