Statistical Mechanics: Solving for Average Energy of System

arnesmeets
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This shouldn't be too hard but I'm struggling.

Consider N identical particles in a box of volume V. The relation between their linear momentum and kinetic energy is given by E = c\left|\overrightarrow{p}\right|, where c is the speed of light. So, the Hamiltonian of the system is

H_N\left(\overrightarrow{q_1},\,\cdots,\overrightarrow{q_N},\overrightarrow{p_1},\,\cdots,\overrightarrow{p_N}\right) = c \sum_{i = 1}^N \left|p_i\right|
Now, I'm trying to find the average energy of the system. So, I need to get Z:

Z = \int \exp\left(-\beta \cdot c \cdot \sum_i \left|p_i\right|\right) d\overrightarrow{q_1}\cdots d\overrightarrow{q_N} d\overrightarrow{p_1}\cdots d\overrightarrow{p_N} = V^{N} \cdot \left(\int \exp\left(-\beta \cdot c \cdot \left|\overrightarrow{p}\right|\right) d\overrightarrow{p}\right)^NNow, I've got two questions:

1) Is there any mistake in the calculations for Z above?
2) How to evaluate the integral?? :blushing:
 
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Does nobody have a clue?:confused:
 
You can solve the integral quite easily by transforming to spherical coordinates (r,\theta, \phi). So the integral becomes:

\int_0^{2\pi} \int_0^{\pi} \int_0^{\infty} e^{-\beta r} r^2\sin{\theta} dr d\theta d\phi = 4\pi \int_0^{\infty} r^2 e^{-\beta r}dr

I think you can now solve the rest on your own :smile:
 
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The value of H equals ## 10^{3}## in natural units, According to : https://en.wikipedia.org/wiki/Natural_units, ## t \sim 10^{-21} sec = 10^{21} Hz ##, and since ## \text{GeV} \sim 10^{24} \text{Hz } ##, ## GeV \sim 10^{24} \times 10^{-21} = 10^3 ## in natural units. So is this conversion correct? Also in the above formula, can I convert H to that natural units , since it’s a constant, while keeping k in Hz ?
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