In this video, we will discuss thermal distribution of system of particles. We can extend our discussion of system of identical particles to systems with more than two particles, many particle system. The total wavefunction is constructed with the products of individual one particle wavefunctions as before. For a system of N bosons, the total wavefunction must be symmetric under exchange of any two particles. This total wavefunction, symmetric total wavefunction, can be constructed by simply multiplying all of these individual single wavefunctions and sum it over all possible permutations, this P represents the permutation. There will be a total of N factorial terms because that's the number of permutations that are possible for N particle system. They are symmetric because we are simply adding up all these different permutations. Swapping two particle simply means that you're changing the order of summation in this big summation, which obviously has no effect. This construction will give you a symmetric total wavefunction automatically. Also there is no restrictions that two or more particles can occupy the same single-particle state. That is, in this notation here, 1, 2, these are indexing the bosons in the system and a, b and these alphabets represents the single-particle quantum state and a could be equal to b, that's okay. Because the fact that two or more single-particle quantum states are the same, have no impact because we're adding them all up, there is no cancellation among terms. In contrast, fermions, we need to construct an anti-symmetric combination under exchange of any two particles. That's a little trickier. To do that, we construct the total wave function looking like this. The total wave function is, once again, is some summation of these product states. Once again, these number 1, 2, 3, 4 to N, indexes the particles in the system, and a, b, those guys indicate the single-particle quantum state that each of these particle occupies. Now, there is this plus and minus sign that we added in front of the permutation operator P, so we're still adding up or summing up all different possible permutations in this N particle system. Therefore, there are still a total of N factorial terms in this permutation, but we are changing sign depending on the number of pairwise permutation. You start out with one sequence. If you exchange any two of them, this term is added to the summation with a minus sign, and then you permute another pair. Then this times that term is added with a plus sign and then permute another pair. That term is added with a minus sign and so on and so forth. Even number of pairwise permutation, the terms corresponding to even number of pairwise permutation is added. The terms corresponding to the odd number of pair-wise permutations is subtracted; is added with a minus sign. This ensures that the total wavefunction is antisymmetric, and any two particles occupying the same state, if any of these alphabets, a, b, c, d to n, if any of them are the same, what that means is that there will be two terms with opposite sign that is subtracted to each other, so that results in a cancellation, and that makes your wavefunction identically zero. This total construction of total wavefunction satisfies Pauli's Exclusion Principle as it shift. A convenient way of constructing the antisymmetric wavefunction is to use so-called the Slater determinant, and you basically arrange these single-particle states in this way, in a matrix form, and calculate the determinant of this matrix. That is another way of expressing this antisymmetric combination that we showed in the previous slide. Now, it should be noted that the total wavefunction of N-particle system is expressed as a linear combination of product of single-particle wavefunctions. We didn't make any comment about that, but it should be noted that this is valid only when the interaction among particles are weak. If there is a strong interaction among these particles, then you can't really use your single-particle state and the product of those as your wavefunction. You need to solve that entire system and find the eigenstate of this many particle system, including all of the interparticle interaction. In general however, this total wavefunction constructed by using this product of single-particle wavefunctions can be used as the basis function to construct the actual accurate and particle system wavefunction. Now, without delving too deeply into the statistical physics, I will introduce the probability function that gives you the energy distribution of particles. In classical physics, all particles are non-identical or distinguishable. In those cases, we can use classical Maxwell-Boltzmann distribution function. What that means is that the average number of particles at energy E is given by this simple exponential factor. E is the energy, K_b is the Boltzmann constant, t is the temperature, and Mu here is the chemical potential of the system. For fermions, the distribution functions look like this. This is called the Fermi-Dirac distribution function. For bosons, the distribution function is given here. This is called the Bose-Einstein distribution function, and you can see that they're quite similar. If you ignore this plus 1 or minus 1 in the quantum mechanical distribution function, they simply become the Maxwell-Boltzmann classical distribution function. You can plot these probability functions as shown here. You can see that the Bose-Einstein distribution function generally gives you higher occupation number. Because multiple Bosons can occupy the same state, whereas the Fermions, the mean occupation number is limited to one. Maximum one always on average, smaller than that according to the Pauli's Exclusion Principle, the classical Maxwell-Boltzmann distribution function gives you a value somewhere in between and in the limit of very high-energy as shown here, they all converge into the same and that should be obvious from the explicit functional forms given in the previous slide when the energy is large, then the exponential factor in the denominator of the quantum mechanical distribution function becomes large compared to one. You can ignore that plus or minus one and they become identical to the Maxwell-Boltzmann distribution function. I want to finish this lecture by giving you two examples where these distribution functions are used. Here shows the carrier distributions in a semiconductor. Energy zero represents the bottom of the conduction band and this function here, g_c, is the density of states of the conduction band. The density of state represents the number of available states electrons can occupy. That density of state is given by this curve here. The probability function is given here by the Fermi-Dirac distribution function. If you multiply the number of available states with the probability function, that will give you the actual number of electrons at any given energy. The electrons inside a conduction band of the semiconductor has a distribution looking like this in conduction band. What this shows is that the electrons are congregated near the bottom of the conduction band. If you go higher up into the conduction band, the number of electrons that you will find will decrease exponentially according to the Fermi direct distribution function, the lower energy end is limited by the number of available states. As you approach the bottom of the conduction band, number of available states become smaller and smaller eventually goes to zero at the conduction band edge, and therefore the number of electrons there also goes to zero. This is an example of how Fermi-Dirac distribution function is used to describe electronic properties of a semiconductor material, in this case. An example of Bose-Einstein distribution function is shown here. In this case, we use the density of state, number of available states for photons in free space. That's given here. You multiply the Bose-Einstein distribution function because photons, the light quantum is a boson, so we used the Bose-Einstein distribution function. That gives you the famous black body radiation law by Planck. This solves these deficiency of the classical theory, which predicts the intensity of light, will diverge as the energies increase. This phenomenon used to be called the ultraviolet catastrophe. By using the Bose-Einstein probability function, you avoid the ultraviolet catastrophe and have the correct distribution of photons as a function of energy at any given temperature t. This equation correctly describes the light emission from a hot body, ranging from the sun and other stars to hot stove.
