# Derivation of FD/BE-distribution using single-particle state

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1. Feb 18, 2017

### lampCable

1. The problem statement, all variables and given/known data
I'm trying to understand a derivation of the Fermi-Dirac and Bose-Einstein distributions. In my textbook Thermal Physics by D. V. Schroeder it says: "The idea is to first consider a "system" consisting of one single-particle state, rather than a particle itself. Thus the system will consist of a particular spatial wavefunction (and, for particles with spin, a particular spin orientation). This idea seems strange at first, because we normally work with wavefunctions of definite energy, and each of these wavefunctions shares its space with all the other wavefunctions." Then he says that it does not matter so much. Then: "So let's concentrate on just one single-particle state of a system (say, a particle in a box), whose energy when occupied by a single particle is $\epsilon$. When the state is unoccupied its energy is 0; if it can be occupied by $n$ particles, then the energy will be $n\epsilon$."

Now, what I do not understand is why the energy of a state would be dependent on how many particles that can be in it. I mean for bosons there can be any amount of particles in any state, so that would mean that the energy of every state is infinite?

Or does he really mean that if a state is occupied by $n$ particles, then the energy is $n\epsilon$? But that doesn't make sense to me either because I thought that if a particle is in a particular state then the particle has a particular energy, whereas in the textbook it seems like if a particle is in a particular state, its energy will be different depending on how many particles is in that state.

2. Relevant equations

3. The attempt at a solution

2. Feb 20, 2017

### lampCable

Okay, so I think I have found the answer, and I thought that I'd share it in case anyone else has the same question. Maybe the question was a bit unclear, but what I didn't understand was essentially how we used the "single-particle state" to define the system.

So we're considering a system that consists of a single particle state $\psi$. The system, assuming that the particles in the system are noninteracting, is described by the wavefunction $\psi_{system}(x_1,x_2,...,x_n) = \psi(x_1)\times\psi(x_2)\times...\times\psi(x_n)$, i.e. the product of all the $n$ particles in the single particle state. Now, if the energy of the $\psi$ state is $\epsilon$, the energy of $\psi_{system}$ becomes $n\epsilon$.

So from this we can derive the probability of the state containing $n$ particles $$P(n) = \frac{1}{Z_G}e^{-n(\epsilon-\mu)/kT},$$ where $Z_G$ is the grand partition function. Using this, we can calculate the average number of particles in the state using $$\bar{n} = \sum_n nP(n).$$ And this is just the occupancy of a particular state with energy $\epsilon$. So using the properties of fermions and bosons, we can find the Fermi-Dirac and Bose-Einstein distributions.

Last edited: Feb 20, 2017