Partition Functions for FermiDirac and BoseEinstein particles

In summary, the problem involves two particles in three quantum states (0, e, and 3e) at temperature T, and asks for the partition function under Fermi Dirac and Bose Einstein statistics. The solutions obtained are Z = exp[-beta * e] + exp[-2 * beta * e] + exp[-4 * beta * e] for Fermi Dirac and Z = 1 + exp[-beta * e] + 2*exp[-3 * beta * e] + exp[-4 * beta * e] + exp[-6 * beta * e] for Bose Einstein, assuming the particles are indistinguishable.
  • #1
mark.laidlaw19
21
0

Homework Statement


Consider two particles, each can be in one of three quantum states, 0, e and 3e, and are at temperature T. Find the partition function if they obey Fermi Dirac and Bose Einstein statistics.

Homework Equations

The Attempt at a Solution


I have obtained solutions to both :

FD, Z = exp[-beta * e] + exp[-2 * beta * e] + exp[-4 * beta * e]

BE,Z = 1 + exp[-beta * e] + 2*exp[-3 * beta * e] + exp[-4 * beta * e] + exp[-6 * beta * e]

I have figured these out by thinking through the possible states for the particles. I'm assuming they have to be indistinguishable. I just want to confirm that my thinking has led to the correct results.

Many thanks
 
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  • #2
What expression for Z did you not provide under 2. Homework Equations ?
 

1. What is a partition function for Fermi-Dirac and Bose-Einstein particles?

A partition function is a mathematical tool used to calculate the thermodynamic properties of a system of particles. For Fermi-Dirac particles, it takes into account the exclusion principle, which states that no two identical particles can occupy the same quantum state. For Bose-Einstein particles, it considers the tendency of particles to occupy the same quantum state, known as Bose-Einstein condensation.

2. How is the partition function different for Fermi-Dirac and Bose-Einstein particles?

The partition function for Fermi-Dirac particles includes a negative sign in its formula, reflecting the exclusion principle and resulting in a different behavior of the system compared to Bose-Einstein particles. Additionally, the energy levels for Fermi-Dirac particles are discrete, while for Bose-Einstein particles they are continuous.

3. What information can be obtained from the partition function?

The partition function allows for the calculation of various thermodynamic properties, such as the energy, entropy, and heat capacity of a system of particles. It also provides insight into the behavior of the system, such as the tendency for Bose-Einstein condensation or the Fermi energy level.

4. How is the partition function used in statistical mechanics?

In statistical mechanics, the partition function is a key element in calculating the probability of a system being in a particular state. Using the partition function, the probability distribution for the system can be derived, and from there, various thermodynamic properties can be calculated.

5. Are there any limitations to using the partition function for Fermi-Dirac and Bose-Einstein particles?

The partition function assumes that the particles in the system are in thermal equilibrium, which may not always be the case. It also does not take into account quantum effects, such as tunneling, which may be important for particles with low energy levels. Additionally, the partition function becomes more complex for systems with a large number of particles and energy levels, making it difficult to calculate accurately.

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