Q. 7.8 - An Introduction to Thermal Physics

Question

Suppose you have a "box" in which each particle may occupy any of $10$ single-particle states.  For simplicity, assume that each of these states has energy zero.

(a)  What is the partition function of this system if the box contains only one particle?

(b)  What is the partition function of this system if the box contains two distinguishable particles?

(c)  What is the partition function if the box contains two identical bosons?

(d)  What is the partition function if the box contains two identical fermions?

(e)  What would be the partition function of this system according to equation $7.16$?

(f)  What is the probability of finding both particles in the same single particle state, for the three cases of distinguishable particles, identical bosom, and identical fermions?

Step-by-Step Solution

Answer

(a) Partition function of system with box having only one particle is $10$.

(b) Partition function of the system if the box contains two distinguishable particles is $100$.

(c) Partition function if the box contains two identical bosons is $55$.

(d) Partition function if the box contains two identical fermions is $45$.

(e) Partition function of this system according to equation 7.16 is $50$.

(f)  Probability of finding both particles in the same single particle state, for the three cases of distinguishable particles, identical bosom, and identical fermions are $10\%, 18\% and 0\%$.

Step 1. Calculation one particle

Each particle can have any $10$ states each with zero energy.

So, $E\left(r_i\right)=0 for 1\le r_i\le 10$

Formula for particle function with box having one particle:

$Z_l=\sum _i{e}^{-\beta E\left(r_i\right)}$

where,  $\beta =\frac{1}{kT}$$\beta =\frac{1}{kT}$

We substitute $E\left(r_i\right)=0$

 $$\begin{gathered} Z_l=\sum _1^{10}{e}^{\beta \left(0\right)} \\ =\sum _1^{10}1 \\ =10 \end{gathered}$$

So, Partition function of system with box having only one particle is $10$.

Step 2. Calculation two particles

For two distinguishable particles partition function is:

$Z={Z_l}^2$

Substitute $Z_l=10$

$$\begin{gathered} Z={\left(10\right)}^2 \\ =100 \end{gathered}$$

So, the partition function of the system if the box contains two distinguishable particles is $100$.

Step 3. Calculation bosons

To put two identical bosons in same single particle state the possibility is $10$.

So, to put two identical bosons in different single particle state the possibility is  calculated as:

$$\begin{gathered} {}_2{}^{10}C=\frac{10!}{2! 8!} \\ =45 \end{gathered}$$

So, total possible states are:

$$\begin{gathered} Z=10+45 \\ = 55 \end{gathered}$$

Hence, the partition function if the box contains two identical bosons is $55$.

Step 4. Calculation fermions

There is no possibility to put two identical fermions in same single particle state.

For different single particle state possibilities are calculated by:

$$\begin{gathered} {}_2{}^{10}C=\frac{10!}{2! 8!} \\ =45 \end{gathered}$$

So, the partition function if the box contains two identical fermions is $45$.

Step 5. Calculation indistinguishable particles

Partition function for $N$ indistinguishable and non interacting particles is given by,

$Z=\frac{{Z_l}^N}{N!}$

Substitute $Z_l=10$and $N=2$

$$\begin{gathered} Z=\frac{{10}^2}{2!} \\ =50 \end{gathered}$$

So, the partition function of the system when box have indistinguishable particles is $50$.

Step 6. Calculation probability

For distinguishable particles number of accessible states are $Z=100$.

For both particles in same single particle state, accessible states are $10$.

So, probability of finding both particle in same single particle state is:

$$\begin{gathered} P=\frac{10}{100} \\ =10\% \end{gathered}$$

So, probability for distinguishable particles is $10\%$.

For identical bosons number of accessible states are $Z=55$.

For both bosons in same single particle state, accessible states are $10$.

So, probability of finding both bosons in same single particle state is:

$$\begin{gathered} P=\frac{10}{55} \\ =18\% \end{gathered}$$

So, probability for identical bosons is  $18\%$.

For identical fermions number of accessible states are $Z=45$.

For both fermions in same single particle state, accessible states is $0$.

So, probability of finding both fermions in same single particle state is:

$$\begin{gathered} P=\frac{0}{45} \\ =0\% \end{gathered}$$

So, probability for identical fermions is  $0\%$.