Partition Function of N particles in an assymetrical box

[FranciscoSili]

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

 

[TSny]

I think your work looks good.

The order of the questions indicates to me that you are to answer the first 3 dotted questions before working out the partition function. I'm guessing that in the first question, you are to write out a general expression for $P_r$ in terms of the symbols $\epsilon_r$ and $\beta$ but you are not meant to evaluate it explicitly. Likewise in the third question, express the average energy in terms of $\epsilon_r$ and $\beta$ using your expression for $P_r$ from the first question. The partition function $Z$ is defined here and you should show the identity involving the derivative of $Z$ with respect to $\beta$. Finally, in the fourth (last) question, evaluate $Z$ as you have done and then evaluate explicitly the average energy. Hope I'm not misleading you here.

 

[FranciscoSili]

I think your work looks good.

Ohh ok, that's good to hear. Thank you.

The order of the questions indicates to me that you are to answer the first 3 dotted questions before working out the partition function. I'm guessing that in the first question, you are to write out a general expression for $P_r$ in terms of the symbols $\epsilon_r$ and $\beta$ but you are not meant to evaluate it explicitly. Likewise in the third question, express the average energy in terms of $\epsilon_r$ and $\beta$ using your expression for $P_r$ from the first question. The partition function $Z$ is defined here and you should show the identity involving the derivative of $Z$ with respect to $\beta$. Finally, in the fourth (last) question, evaluate $Z$ as you have done and then evaluate explicitly the average energy. Hope I'm not misleading you here.

Yes I know that I did everything in reverse, but the most complicated calculations were the ones for the partition function, so that's what I wanted to ask, if it was correct. I should have said that in the beginning, sorry for that.

Thank you.

 

[TSny]

Yes I know that I did everything in reverse, but the most complicated calculations were the ones for the partition function, so that's what I wanted to ask, if it was correct. I should have said that in the beginning, sorry for that.

Thank you.

That's fine. Good work.