# Partition function of 2 bosons in two energy level 0 and E

• Apashanka
In summary, this conversation is about how to find the microstates for 2 identical particles and then find the single particle partition function. The first approach works if the particles are indistinguishable, but the second approach works if the particles can never occupy the same state together.
Apashanka
Homework Statement
Partition function of 2 bosons in two energy level 0 and E
Relevant Equations
Partition function of 2 bosons in two energy level 0 and E
For 2 bosons each of which can occupy any of the energy levels 0 and E the microstates will be 3
0 E
a a
aa -
- aa
the partition function is therefore $$z=1+e^{-\beta E}+e^{-2\beta E}...(1)$$
Another approach to do..
The single particle partition function is
$$z=1+e^{-\beta E}$$ and therefore for 2 identical particle the partition function becomes $$z^2/2!..…(2)$$ and as such (1) and (2) doesn't match……
Am I wrong somewhere??
Thanks,
Apashanka

The second approach only works if you can neglect the cases where more than one particle is in the same state.

mfb said:
The second approach only works if you can neglect the cases where more than one particle is in the same state.
Therefore here the first one is the correct isn't it??

@mfb But for distinguishable particle the second approach works ...

Yes, because they can never occupy the exact same state together. Just states at the same energy.
If the problem statement says "2 bosons" you can assume that they are indistinguishable, otherwise it would be pointless to say that they are bosons.

mfb said:
Yes, because they can never occupy the exact same state together. Just states at the same energy.
If the problem statement says "2 bosons" you can assume that they are indistinguishable, otherwise it would be pointless to say that they are bosons.
What I mean to say is if a and b be two distinguishable particle
Then the microstates will be
0 E
a b
ab -
- ab
b a
In this $$z=1+2e^{-\beta E}+e^{-2\beta E}$$
And the single particle partition function is $$z_{sing}=1+e^{-\beta E}$$
Therefore $$z=z^2_{single}$$ matches beside two particle occupies the same energy state...

Yes, that scenario is different from your original problem.

Actually I got stuck doing this problem

What I am here getting is the if considering single particle partition function z and then for 2 identical particles the partition function becomes ##z^2/2! =(1+e^{-\beta E})^2/2!## and then taking the mean energy as ##-\frac{\partial}{\partial \beta}ln z## then none of the answer matches
Otherwise if finding the microstates and then the partition function of these 2 bosons ##z=1+e^{-\beta E}+e^{-2\beta E}## then also none of the answer matches...
Can anyone suggest where is the mistake??

Well, the approach with ##z^2/2## can't work, it needs distinguishable particles and/or temperatures so high that the difference between bosons and fermions doesn't matter because states are unlikely to have more than one particle.

None of the answers make sense. (d) is the only one where the numerator is the derivative of the denominator, but the 2 in the denominator makes no sense. I guess either (a) or (d) were supposed to be the answer and the 2 shouldn't be there in (a) or the 2 should be a 1 in (d).

mfb said:
Well, the approach with ##z^2/2## can't work, it needs distinguishable particles and/or temperatures so high that the difference between bosons and fermions doesn't matter because states are unlikely to have more than one particle.

None of the answers make sense. (d) is the only one where the numerator is the derivative of the denominator, but the 2 in the denominator makes no sense. I guess either (a) or (d) were supposed to be the answer and the 2 shouldn't be there in (a) or the 2 should be a 1 in (d).
Yes sir thanks a lot

## 1. What is the Partition Function of 2 Bosons in Two Energy Levels 0 and E?

The partition function of 2 bosons in two energy levels 0 and E is a mathematical concept used in statistical mechanics to describe the distribution of energy among a system of particles. It represents the sum of all possible energy states that the particles can occupy, taking into account their quantum properties.

## 2. How is the Partition Function Calculated for 2 Bosons in Two Energy Levels 0 and E?

The partition function for 2 bosons in two energy levels 0 and E is calculated by summing over all possible combinations of particles in each energy level. This includes the ground state with 0 particles, the first excited state with 1 particle, and the second excited state with 2 particles. The formula for the partition function is Z = 1 + e^(-E/kT) + e^(-2E/kT), where k is the Boltzmann constant and T is the temperature.

## 3. What is the Significance of the Partition Function in Statistical Mechanics?

The partition function is a fundamental concept in statistical mechanics as it allows us to calculate important thermodynamic quantities such as the internal energy, entropy, and free energy of a system. It also provides a way to analyze the behavior of a system at different temperatures and understand phase transitions.

## 4. Can the Partition Function be Applied to Other Systems Besides Bosons in Two Energy Levels?

Yes, the partition function can be applied to a wide range of physical systems, including fermions, molecules, and solids. It is a versatile tool in statistical mechanics and can be adapted to different systems by modifying the energy levels and the number of particles in each state.

## 5. How Does the Partition Function of 2 Bosons in Two Energy Levels 0 and E Change with Temperature?

The partition function of 2 bosons in two energy levels 0 and E decreases as temperature increases. This is because at higher temperatures, more particles are able to occupy higher energy states, leading to a decrease in the number of particles in the lower energy states. This is reflected in the exponential term in the partition function formula, where the probability of occupying higher energy states increases with temperature.

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