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# Disclaimer

Peeter’s lecture notes from class. May not be entirely coherent.

# Grand Canonical/Fermion-Bosons

Was mentioned that three dimensions confines us to looking at either Fermions or Bosons, and that two dimensions is a rich subject (interchange of two particles isn’t the same as one particle cycling around the other ending up in the same place — how is that different than a particle cycling around another in a two dimensional space?)

Definitions

- Fermion. Antisymmetric under exchange.
- Boson. Symmetric under exchange.

In either case our energies are

For Fermions we’ll have occupation filling of the form fig. 1.1, where there can be only one particle at any given site (an energy level for that value of momentum). For Bosonic systems as in fig. 1.2, we don’t have a restriction of only one particle for each state, and can have any given number of particles for each value of momentum.

Fig 1.1: Fermionic energy level filling for free particle in a box

Fig 1.2: Bosonic free particle in a box energy level filling

Our Hamiltonian is

where we have a number operator

such that

While the second sum is constrained, because we are summing over all , this is essentially an unconstrained sum, so we can write

**Fermions**

**Bosons**

().

Our grand partition functions are then

We can use these to compute the average number of particles

This chemical potential over temperature exponential

is called the *fugacity*. The denominator has the form

so we see that

Thus the numerator is

and

**What is the density ?**

For Fermions

Using a “particle in a box” quantization where , in a -dimensional space, we can approximate this as

This integral is actually difficult to evaluate. For (, where

This is illustrated in, where we also show the smearing that occurs as temperature increases fig. 1.3.

Fig 1.3: Occupation numbers for different energies

With

we want to ask what is the radius of the ball for which

or

so that

so that our density where is

so that

Our chemical potential at zero temperature is then

We can convince ourself that the chemical potential must have the form fig. 1.4.

Fig 1.4: Large negative chemical potential at high temperatures

Given large negative chemical potential at high temperatures our number distribution will have the form

We see that the classical Boltzmann distribution is recovered for high temperatures.

We can also calculate the chemical potential at high temperatures. We’ll find that this has the form

where this quantity is called the *Thermal de Broglie wavelength*.