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In the book

The final result is given as:

[tex] \frac{\overline {n_j}}{g_j} = e^{-\beta \epsilon_j}e^{-\alpha}[/tex]

where ##\overline {n_j}## is the occupation number and ##g_j## is the number of states of j

After solving for ##e^{-\alpha} = \frac{N}{V}\left(\frac{h^2}{2\pi mkT}\right)^{\frac{3}{2}}## and integrating density of states to find ##g_j = \frac{V}{h^3} 4\pi p^2 dp##:

We obtain the maxwell-boltzmann distribution:

[tex]\overline {n_j} = \overline {n_{(p)}} dp = N 4\pi \left(\frac{\beta}{2\pi m}\right)^{\frac{3}{2}} p^2 e^{\frac{-p^2}{2mkT}} dp[/tex]

I obtain the correct speed distribution ##\propto p^2 e^{\frac{-p^2}{2mkT}}##, but

In Blundell's Book, a shorter approach is taken using gibbs' expression for entropy to find the boltzmann probability:

Here's the earlier reference to equation (4.13):

*'Macroscopic and Statistical Thermodynamics'*they derived the Maxwell-Boltzmann distribution by maximizing entropy using lagrangian multipliers with constants ##\alpha## and ##\beta##.The final result is given as:

[tex] \frac{\overline {n_j}}{g_j} = e^{-\beta \epsilon_j}e^{-\alpha}[/tex]

where ##\overline {n_j}## is the occupation number and ##g_j## is the number of states of j

^{th}energy level.After solving for ##e^{-\alpha} = \frac{N}{V}\left(\frac{h^2}{2\pi mkT}\right)^{\frac{3}{2}}## and integrating density of states to find ##g_j = \frac{V}{h^3} 4\pi p^2 dp##:

We obtain the maxwell-boltzmann distribution:

[tex]\overline {n_j} = \overline {n_{(p)}} dp = N 4\pi \left(\frac{\beta}{2\pi m}\right)^{\frac{3}{2}} p^2 e^{\frac{-p^2}{2mkT}} dp[/tex]

I obtain the correct speed distribution ##\propto p^2 e^{\frac{-p^2}{2mkT}}##, but

**What is the probability/fraction of finding a particle with momentum p?**In Blundell's Book, a shorter approach is taken using gibbs' expression for entropy to find the boltzmann probability:

Here's the earlier reference to equation (4.13):

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