Check my work please. Particle states.

In summary: But I do agree that this may not be what you meant. If you meant identical particles, then the answer would be 12.
  • #1
frankR
91
0
Problem:

Consider a non-interacting system of 4 particles with each particle having single-particle states with energies equal to 0, e, 2e and 3e. Given that the total energy of the system is 6e, find the number of microstates of the system (and identify the microstates) if the particles are a) distinguishable, b) indistinguishable Bosons and c) indistinguishable Fermions.

For a) I get 3-ways to get 6e and 3*4 ways to get 6e amoung distinguishable particles.

For b) I get 3-ways to get 6e amoung the indistinguishable Bosons.

For c) I get 1-way to get 6e amoung Fermions.

Is this correct?

Also is there such thing as distinguishable Fermions. My guess is by definition of a Fermion, no!

Thanks.
 
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  • #2
I agree with your answer for (c): the only possible state in that case is where one fermion has energy 0, another e, another 2e, and the remaining one 3e.

I seem to get more than 3 ways for (b). Here are four (the ordering left-to-right on a row being immaterial since they are indistinguishable):

0 0 3 3
0 1 2 3
1 1 2 2
0 2 2 2

I think I convinced myself that your answer for (a) is an undercount. (In this case the ordering does count.)
 
  • #3
frankR said:
Also is there such thing as distinguishable Fermions. My guess is by definition of a Fermion, no!

This is probably not what you meant, but an example would be a system consisting of a neutrino, an electron, a muon, and a tauon. Four particles, very much distinguishable.
 
  • #4
Ahh I missed: 1=e, 2=e, 3=e, 4=3e

So corrected answers:

a) omega = 16 (I permute here by multiplying by 4 right?)
b) omega = 4
c) omega = 1

Thanks.
 
  • #5
Janitor said:
This is probably not what you meant, but an example would be a system consisting of a neutrino, an electron, a muon, and a tauon. Four particles, very much distinguishable.

Yeah you got me there. I should specified identical particles.

So say all are electrons.

I ask because Prof. Webb specified "indistinguishable Fermions". So that would indicate were not talking about electrons and protons, in which case the Pauli Exclusion Principle doesn't apply, Protons and Electrons can occupy the same state. Unless protons aren't Fermions. :tongue2: :yuck:
 
  • #6
frankR said:
Ahh I missed: 1=e, 2=e, 3=e, 4=3e

Oops. I missed

1 1 1 3

as well, so there are at least 5 states for (b).
 
  • #7
frankR said:
So say all are electrons.

I suppose there are conditions (say strong applied magnetic field, low temperature) where a pair of electrons would be distinguishable, because one would maintain an 'up' state of spin component, and the other a down 'state,' in some sort of metastable situation.
 
  • #8
frankR said:
a) omega = 16 (I permute here by multiplying by 4 right?)

I'm not sure. I just started writing out possible states for particles A B C D, where order counts, and I stopped when I got to 13, since that was more than your first answer of 12. There is probably some elegant way of figuring out the number. The mathematical types here will know.
 
  • #9
Okay so I also missed: 1=3e, 2=2e, 3=e, 4=e

Did we get them all this time? :cry: :rofl:
 
  • #10
frankR said:
Did we get them all this time? :cry: :rofl:

It's all my tired brain can find at the moment. I will sleep on it. :zzz:
 
  • #11
I thought about this a bit more today, and here is what I think are the answers to (a) and (b).

0 1 2 3 12
0 2 2 2 4
0 0 3 3 6
1 1 2 2 6
1 1 1 3 4

12 + 4 + 6 + 6 + 4 = 32


The five rows show the five microstates for bosons, so the answer to (b) is 5. The number of permutations of each microstate is shown in bold. The sum is 32, so that is the answer to (a). I am pretty sure that considerations of symmetrization for bosons and antisymmetrization for fermions are not an issue in this particular problem, but you might want to check up on that.
 
  • #12
Me and my fellow class mates were able to find 32 as well for a.

Nice work!

You're a janitor that does physics in his spare time? Are you like Will Hunting? :D
 
  • #13
Never saw the movie, I'm afraid.
 

1. What is the purpose of checking particle states?

The purpose of checking particle states is to ensure that the particle's properties, such as position, momentum, and energy, are accurately measured and recorded. This helps to validate the results of experiments and ensure the reliability of data.

2. What are the common methods used for checking particle states?

The most common methods for checking particle states include particle detectors, such as Geiger counters and photomultiplier tubes, and particle accelerators, which can produce and measure high-energy particles.

3. How do scientists determine the state of a particle?

Scientists determine the state of a particle by analyzing its interactions with other particles and measuring its properties, such as charge and mass. This information can then be used to identify the type and state of the particle.

4. Why is it important to accurately check particle states?

Accurately checking particle states is crucial for advancing our understanding of the fundamental building blocks of the universe and their interactions. It also helps to validate scientific theories and models, and can lead to new discoveries and breakthroughs in various fields of research.

5. Are there any challenges in checking particle states?

Yes, there are several challenges in checking particle states, including the complexity and high energy levels involved in particle interactions, the need for advanced technology and equipment, and the potential for human error. However, with careful planning and rigorous testing, scientists are able to overcome these challenges and obtain accurate results.

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