Firmi-Dirac Stats. - Calculate number of microstates.

In summary, the number of microstates where two particles have energy 2eV, three have 3eV, three have energy 4eV, and two have 5eV is 1,440,000. This is calculated using the Fermi-Dirac distribution equation, taking into account the indistinguishability of the particles.
  • #1
eric2921
8
0

Homework Statement



Determine the number of microstates where two particles have have energy 2eV, three have 3eV, three have energy 4eV, and two have 5eV.


10 indistinguishable Particles
1 particle allowed per state.

ε1 = 1eV ε2 = 2eV ε3 = 3eV ε4 = 4eV ε5 = 5eV ε6 = 6eV
g1 = 1 g2 = 5 g3 = 10 g4 = 10 g5 = 5 g6 = 1


Homework Equations

[itex]\textit{}[/itex]

possibly [itex]\overline{n}[/itex]i=[itex]\frac{1}{\textbf{e}^{(ε_{i}-μ)/kT}+1}[/itex]

The Attempt at a Solution



Since the particles are indistinguishable I thought that there would only be one microstate, but I don't think that this is right... can someone point me in the right direction on how to start this problem?
 
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  • #2
figured it out, was missing a very relevant equation.
[itex]
w_{\text{FD}}\left(N_1,N_2,\text{...}N_n\right)=∏ \frac{g_j!}{N_j!\left(g_j-N_j\right)!}
[/itex]
[itex]
w_{\text{FD}}\left(N_1,N_2,N_3,N_4,N_5,N_6\right)=\left(\frac{g_1!}{N_1!\left(g_1-N_1\right)!}\right)\left(\frac{g_2!}{N_2!\left(g_2-N_2\right)!}\right)\left(\frac{g_3!}{N_3!\left(g_3-N_3\right)!}\right)\left(\frac{g_4!}{N_4!\left(g_4-N_4\right)!}\right)\left(\frac{g_5!}{N_5!\left(g_5-N_5\right)!}\right)\left(\frac{g_6!}{N_6!\left(g_6-N_6\right)!}\right)
[/itex]
[itex]
w_{\text{FD}}(0,2,3,3,2,0)=\left(\frac{1!}{0!(1-0)!}\right)\left(\frac{5!}{2!(5-2)!}\right)\left(\frac{10!}{3!(10-3)!}\right)\left(\frac{10!}{3!(10-3)!}\right)\left(\frac{5!}{2!(5-2)!}\right)\left(\frac{1!}{0!(1-0)!}\right)
[/itex]
evaluates to 1,440,000
 

1. What is the difference between Fermi-Dirac statistics and Boltzmann statistics?

Fermi-Dirac statistics and Boltzmann statistics are two different methods used to describe the behavior of particles in a system. While Boltzmann statistics assume that particles can occupy any energy state, Fermi-Dirac statistics take into account the fact that particles with half-integer spin (such as electrons) cannot occupy the same energy state at the same time. This results in a different distribution of particles in energy states for Fermi-Dirac statistics compared to Boltzmann statistics.

2. How are microstates related to Fermi-Dirac statistics?

Microstates are a concept used in statistical mechanics to describe the different possible arrangements of particles in a system. In Fermi-Dirac statistics, the number of microstates is used to calculate the probability of a particle being in a particular energy state, taking into account the exclusion principle for particles with half-integer spin. This allows us to determine the distribution of particles in different energy states in a system.

3. How do you calculate the number of microstates in a system using Fermi-Dirac statistics?

The number of microstates in a system can be calculated using the formula N = (n + N - 1)! / (n!(N-1)!), where n is the number of particles in the system and N is the number of available energy states. This formula takes into account the exclusion principle for particles with half-integer spin and allows us to determine the number of ways that the particles can be arranged in the different energy states.

4. What is the significance of Fermi-Dirac statistics in quantum mechanics?

Fermi-Dirac statistics play a crucial role in quantum mechanics as they allow us to accurately describe the behavior of particles with half-integer spin, such as electrons. This is important in understanding the properties of materials and how they conduct electricity, as well as in the study of fundamental particles and their interactions.

5. Can Fermi-Dirac statistics be applied to systems with particles other than electrons?

Yes, Fermi-Dirac statistics can be applied to any system with particles that have half-integer spin. This includes not only electrons, but also other fundamental particles such as protons and neutrons, as well as composite particles like atoms and molecules. However, for systems with particles that have integer spin, such as photons, Bose-Einstein statistics should be used instead.

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