Bose Einstein Statistics Derivation Pdf Download

Bose-Einstein Statistics: A Quantum Mechanical Case

Bose-Einstein statistics is a way of describing the behavior of indistinguishable particles that obey the quantum mechanical principle of symmetry. These particles are called bosons and they have integer spin values, such as 0, 1, 2, etc. Unlike fermions, which obey the Pauli exclusion principle and can only occupy one state at a time, bosons can occupy the same state with any number of particles.

In this article, we will derive the Bose-Einstein distribution function, which gives the probability of finding a certain number of bosons in a given energy state. We will also discuss some applications and consequences of this distribution, such as Bose-Einstein condensation and blackbody radiation.

The derivation is based on the lecture notes by Professor C. Yu from UC Irvine[^1^]. We will assume that the system of bosons is in thermal equilibrium with a heat reservoir at temperature T and that the energy levels of the system are discrete and non-degenerate.

To begin with, we need to find the partition function of the system, which is defined as:

$$Z=\\sum_{\\{n_i\\}}e^{-\\beta E(\\{n_i\\})}$$

where $\\beta=1/k_BT$, $E(\\{n_i\\})$ is the total energy of the system with $n_i$ particles in the i-th energy state, and the sum is over all possible configurations of $\\{n_i\\}$. Since the particles are indistinguishable, we need to divide by a factor of $N!$, where $N$ is the total number of particles, to avoid overcounting. We also need to divide by a factor of $\\prod_{i}n_i!$, where $n_i$ is the number of particles in the i-th state, to account for the symmetry of bosons. Therefore, the partition function becomes:

$$Z=\\frac{1}{N!}\\sum_{\\{n_i\\}}\\frac{e^{-\\beta E(\\{n_i\\})}}{\\prod_{i}n_i!}$$

Using the relation $E(\\{n_i\\})=\\sum_{i}\\epsilon_in_i$, where $\\epsilon_i$ is the energy of the i-th state, we can rewrite the partition function as:

$$Z=\\frac{1}{N!}\\sum_{\\{n_i\\}}\\prod_{i}\\frac{e^{-\\beta \\epsilon_in_i}}{n_i!}$$

Now we can use a trick called the grand canonical ensemble, which introduces a chemical potential $\\mu$ that controls the number of particles in the system. The idea is to multiply and divide by $e^{\\beta \\mu N}$, where $N=\\sum_{i}n_i$, and then sum over all possible values of $N$. This gives us:

$$Z=\\sum_{N=0}^{\\infty}\\frac{e^{\\beta \\mu N}}{N!}\\sum_{\\{n_i\\}}\\prod_{i}\\frac{e^{-\\beta (\\epsilon_i-\\mu)n_i}}{n_i!}$$

The inner sum can be simplified by using another trick called generating functions. We define a function $g(x)$ as:

$$g(x)=\\sum_{n=0}^{\\infty}\\frac{x^n}{n!}$$

which has the property that $g'(x)=g(x)$. Then we can write:

$$\\sum_{\\{n_i\\}}\\prod_{i}\\frac{e^{-\\beta (\\epsilon_i-\\mu)n_i}}{n_i!}=\\prod_{i}\\sum_{n_i=0}^{\\infty}\\frac{(e^{-\\beta (\\epsilon_i-\\mu)})^{n_i}}{n_i!}=\\prod_{i}g(e^{-\\beta (\\epsilon_i-\\mu)})=\\prod_{i}e^{e^{-\\beta (\\epsilon_i-\\mu)}}=e^{\\sum_{i}e^{-\\beta (\\epsilon_i-\\mu)}}$$

Substituting this back into the partition function, we get:

$$Z=\\sum_{N=0}^{\\infty}\\frac{e^{\\beta \\mu N}}{N!}e^{\\sum_{i}e^{-\\beta 061ffe29dd