Okay, I realize that is has been a long time since last posting (over a year), but life has been busy for me and it’s hard to find the time sometimes to keep up on these things. Recently, however, I was asked a question about fermions vs. bosons from a mathematician. I relished the opportunity to explain an interesting physical topic without having to go easy on the math, but found myself doing a poor job of rigorously explaining it besides the usual spin\/occupation number\/symmetrization explanation. I knew what the two particle types were, but completely forgot how we came to derive their statistical properties (my friend is a statistician, so he was particularly interested in this topic). So I refreshed my memory, and would now like to explain it thoroughly.<\/p>\n
It would first be great if we understood why<\/i> bosons and fermions behave differently than their classical counterpart particles. What is the underlying difference? The source of the issue is indistinguishability<\/i>. Whereas in classical mechanics, we can talk about THIS mass and THAT mass separately, but in the quantum picture, we simply cannot label one electron THIS one, and one THAT one. All quantum particles are exactly the same; I can’t paint one red and one blue to tell them apart. As such, I cannot write the composite wavefunction as (ignoring spin) \\(\\psi(r_1,r_2)=\\psi_a(r_1)\\psi_b(r_2)\\), separating out each particle distinctly. However, we can create a wavefunction that doesn’t choose which is in which state:<\/p>\n
\\[\\psi_{\\pm}(r_1,r_2)=C[\\psi_a(r_1)\\psi_b(r_2)\\pm\\psi_b(r_1)\\psi_a(r_2)]\\]<\/p>\n
So, two identical particles can be written as a linear combination of how they could be distinguished. Those with the plus sign are called bosons<\/b> after the Indian physicist Satyendra Bose, while those with the minus sign are called fermions<\/b> after Italian physicist Enrico Fermi. Using relativity, one can prove the spin-statistics theorem<\/b>, which adds the additional information that bosons are integer spins, while fermions are half-integer. Also notice that the above implies the Pauli exclusion principle<\/b>, because if two fermions occupied the same state, we would have<\/p>\n
\\[\\psi_-(r_1,r_2)=C[\\psi_a(r_1)\\psi_a(r_2)-\\psi_a(r_1)\\psi_a(r_2)]=0\\]<\/p>\n
This restriction on fermions accounts for much of what we observe in nature, from the electronic structure of atoms to neutron stars. The statistics of both types of indistinguishable particles can be summed up by stating the (anti)-symmetrization requirement: \\(\\psi(r_1,r_2)=\\pm\\psi(r_2,r_1)\\).<\/p>\n
Now we can examine what happens when we have lots of these particles in a potential, say with energies \\(E_1, E_2, E_3, \\dots\\) with degeneracies \\(d_1, d_2, d_3,\\dots\\). Suppose there are \\(N\\) in all (all the same mass), and we distribute them so that there are \\(N_1\\) with energy \\(E_1\\), \\(N_2\\) with energy \\(E_2\\), and so on. The number of different ways this can be achieved, \\(\\Omega(N_1,N_2,N_3,\\dots )\\) depends on whether the particles are distinguishable, identical fermions, or identical bosons, labeled \\(\\Omega_D,\\Omega_F, \\Omega_B\\). For distinguishable particles, we first ask how many ways we can select \\(N_1\\) particles from the \\(N\\) available to place in the first “bin” \\(E_1\\). The answer is the combination<\/p>\n
\\[\\binom{N}{N_1}=\\frac{N!}{N_1!(N-N_1)!}\\]<\/p>\n
Inside of this bin, we can arrange the particles into \\(d_1\\) different choices, so there is a total of<\/p>\n
\\[\\frac{N!d_1^{N_1}}{N_1!(N-N_1)!}\\]<\/p>\n
options. The same goes for bin two, except now there are only \\((N-N_1)\\) options. Hence, repeating the procedure, we obtain<\/p>\n
\\[\\Omega_D(N_1,N_2,N_3,\\dots )=\\frac{N!d_1^{N_1}}{N_1!(N-N_1)!}\\frac{(N-N_1)!d_2^{N_2}}{N_2!(N-N_1-N_2)!}\\frac{(N-N_1-N_2)!d_3^{N_3}}{N_3!(N-N_1-N_2-N_3)!}\\cdots=N!\\prod_{i=1}^{\\infty}\\frac{d_i^{N_i}}{N_i!}\\]<\/p>\n
Next we consider the case of fermions. This one is easy: because they can’t be distinguished, it doesn’t matter which particle is in which state. There is just one \\(N\\)-particle state, and only one particle can occupy each state (Pauli exclusion principle). Since there are<\/p>\n
\\[\\binom{d_i}{N_i}\\]<\/p>\n
ways to choose the states in the n<\/i>th bin, we have<\/p>\n
\\[\\Omega_F(N_1,N_2,N_3,\\dots )=\\prod_{i=1}^{\\infty}\\frac{d_i!}{N_i!(d_i-N_i)!}\\]<\/p>\n
The hardest case is for bosons, where there is no restriction on the number of particles that can occupy each state. It really comes down to distributing \\(N_i\\) particles into \\(d_i\\) partitions, implying we should use<\/p>\n
\\[\\binom{N_i+d_i-1}{N_i}\\]<\/p>\n
Hence<\/p>\n
\\[\\Omega_B(N_1,N_2,N_3,\\dots )=\\prod_{i=1}^{\\infty}\\frac{(N_i+d_i-1)!}{N_i!(d_i-1)!}\\]<\/p>\n
To determine the most probable configuration, we want to maximize the function ( Omega ) subject to the two constraints<\/p>\n
\\[G=\\ln\\Omega+\\alpha\\left[N-\\sum_{i=1}^{\\infty}N_i\\right]+\\beta\\left[E-\\sum_{i=1}^{\\infty}N_iE_i\\right]\\]<\/p>\n
and set the \\(N_i\\) derivative to zero. To do this, we assume Stirling’s approximation,<\/p>\n
\\[\\ln(z!)\\approx z\\ln z-z\\]<\/p>\n
for \\(z\\gg 1\\). This will be essentially true as long as the occupation numbers \\(N_i\\) are very large (large enough to assume that statistics will work at all). In the identical fermion case, we must also assume that the degeneracies \\(d_i\\) are very large as well (not true in one dimension, but the degeneracies in three dimensions usually increase rapidly with the energy level. In the hydrogen atom, for example, \\(d_n=n^2\\).) Using this approximation, we get for the distinguishable case<\/p>\n
\\[G\\approx\\sum_{i=1}^{\\infty}[N_i\\ln d_i-N_i\\ln N_i+N_i-\\alpha N_i-\\beta E_iN_i]+\\ln N!+\\alpha N+\\beta E\\]<\/p>\n
so that<\/p>\n
\\[\\frac{\\partial G}{\\partial N_i}=\\ln d_i-\\ln N_i-\\alpha-\\beta E_i\\]<\/p>\n
which, when equal to zero, gives us the most probable occupation numbers of each energy level to be<\/p>\n
\\[N_i=d_i e^{-(\\alpha+\\beta E_i)}\\]<\/p>\n
This is known as the Maxwell-Boltzmann distribution, applicable only to distinguishable particles. Doing the same procedure for fermions, we obtain<\/p>\n
\\[N_i=\\frac{d_i}{e^{(\\alpha+\\beta E_i)}+1}\\]<\/p>\n
while for bosons we get<\/p>\n
\\[N_i=\\frac{d_i}{e^{(\\alpha+\\beta E_i)}-1}\\]<\/p>\n
Comparing these results with the equipartition theorem, we conclude that our Lagrange multipliers must be defined as \\(\\beta=\\frac{1}{k_BT}\\) and \\(\\alpha=-\\frac{\\mu(T)}{k_BT}\\) with \\(\\mu\\) the chemical potential of the system (essentially measuring how much the particles want to move down the concentration gradient). Hence we have the distributions for the three classes of particles<\/p>\n
\\[n(\\epsilon)=\\begin{cases} e^{-(\\epsilon-\\mu)\/k_BT} & \\text{Maxwell-Boltzmann}\\\\ \\frac{1}{e^{(\\epsilon-\\mu)\/K_BT}+1} & \\text{Fermi-Dirac}\\\\ \\frac{1}{e^{(\\epsilon-\\mu)\/K_BT}-1} & \\text{Bose-Einstein} \\end{cases}.\\]<\/p>\n
From these, lots of physics can be discussed, such as Fermi energies, thermodynamics, Bose-Einstein condensation, the blackbody spectrum, neutron stars, white dwarfs, harmonic oscillators…indeed the great majority of statistical mechanics.<\/p>\n","protected":false},"excerpt":{"rendered":"
Okay, I realize that is has been a long time since last posting (over a year), but life has been busy for me and it’s hard to find the time sometimes to keep up on these things. Recently, however, I was asked a question about fermions vs. bosons from a mathematician. I relished the opportunity […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"pagelayer_contact_templates":[],"_pagelayer_content":"","footnotes":""},"categories":[10],"tags":[],"class_list":["post-97","post","type-post","status-publish","format-standard","hentry","category-physics"],"_links":{"self":[{"href":"https:\/\/quantum-immortal.net\/blog\/wp-json\/wp\/v2\/posts\/97","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/quantum-immortal.net\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/quantum-immortal.net\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/quantum-immortal.net\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/quantum-immortal.net\/blog\/wp-json\/wp\/v2\/comments?post=97"}],"version-history":[{"count":4,"href":"https:\/\/quantum-immortal.net\/blog\/wp-json\/wp\/v2\/posts\/97\/revisions"}],"predecessor-version":[{"id":181,"href":"https:\/\/quantum-immortal.net\/blog\/wp-json\/wp\/v2\/posts\/97\/revisions\/181"}],"wp:attachment":[{"href":"https:\/\/quantum-immortal.net\/blog\/wp-json\/wp\/v2\/media?parent=97"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/quantum-immortal.net\/blog\/wp-json\/wp\/v2\/categories?post=97"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/quantum-immortal.net\/blog\/wp-json\/wp\/v2\/tags?post=97"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}