- #1

FranciscoSili

- 8

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## Homework Statement

Consider a gas sufficiently diluted containing N identical molecules of mass

*m*in a box of dimensions L

_{x}, L

_{y}, L

_{z}.

- Calculate the probability of finding the molecules in any of their quantum states.
- Calculate the energy of each quantum state ε
_{r}, as a function of the quantum numbers ##n_x##, ##n_y## and ##n_z##. - Calculate the average energy using the expression

$$\bar \varepsilon = \sum_{r} P_r \varepsilon_r$$

Demonstrate that the last expression is equivalent to $$\bar \varepsilon = -\frac {\partial ln(Z)} {\partial \beta}$$

where ## Z= \sum_{r} exp\{-\beta\varepsilon_r\} ##, it's called the partition function. - Found the partition function and calculate the average energy.

Help: the sum over ##r## can be replaced by an integral. Why?

**2. The attempt at a solution**

I have tried solving this problem this way, but I am not sure this is the correct solution and/or way.

First, I calculated the energy for a single particle in this box as $$\varepsilon = {\frac {h^2} {8m}} \left[ \left( \frac {n_x} {L_x} \right)^2 + \left( \frac {n_y} {L_y} \right)^2 + \left( \frac {n_z} {L_z} \right)^2 \right]$$

Because these particles are indistinguishable I can use the property of the partition function that reads $$Z\left(T,V,N\right) = \frac 1 {N!} Z\left(T,V,1\right)^N$$ that will let me calculate only the partition function of a single particle and then obtain it to N of them.

Now I calculate the partition function for one molecule as indicated in the problem statement, $$Z\left(T,V,1\right)=\sum_{n_x,n_y,n_z} exp\left\{\ {\frac {-h^2\beta} {8m}} \left[ \left( \frac {n_x} {L_x} \right)^2 + \left( \frac {n_y} {L_y} \right)^2 + \left( \frac {n_z} {L_z} \right)^2 \right]\right\}$$ which can be split into three different sums, multiplying one another: $$Z\left(T,V,1\right)=\sum_{n_x=1}^\infty exp\left\{\ {\frac {-\beta h^2} {8m}}\left( \frac {n_x} {L_x} \right)^2\right\} \sum_{n_y=1}^\infty exp\left\{\ {\frac {-\beta h^2} {8m}}\left( \frac {n_y} {L_y} \right)^2\right\} \sum_{n_z=1}^\infty exp\left\{\ {\frac {-\beta h^2} {8m}}\left( \frac {n_z} {L_z} \right)^2\right\}.$$ If I can change the sums for integrals which, approximately, run from 0 to infinity, the partition function becomes

\begin{equation}

Z\left(T,V,1\right)=\int_0^\infty exp\left\{\ {\frac {-\beta h^2} {8m}}\left( \frac {n_x} {L_x} \right)^2\right\} \, dn_x \int_0^\infty exp\left\{\ {\frac {-\beta h^2} {8m}}\left( \frac {n_y} {L_y} \right)^2\right\} \, dn_y \int_0^\infty exp\left\{\ {\frac {-\beta h^2} {8m}}\left( \frac {n_z} {L_z} \right)^2\right\} \, dn_z,

\end{equation}

and solving them we get

\begin{equation}

Z\left(T,V,1\right)= \left(\frac {2m\pi} {h^2\beta} \right)^{3/2} L_x L_y L_z.

\end{equation}

Then I can calculate the partition function for the hole system of N particles, then the probability ##P_r## and finally the average energy,##\bar \varepsilon##, as in the statement

Any help or comments that you might have is welcomed, since this is my second week of my Statistical Mechanics course. Thank you very much :D