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$h$ factor The factor of $1/h^{3N}$ is a total hack. The integral over phase space has dimensions, whereas $Z$ only makes sense if it's dimensionless. The $h$ factors are there to make $Z$ dimensionless. Suppose you have a system with only one particle in one dimension. Then the integral in phase space goes over one position variable, $dq$ and one momentum ...


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Now, in my book there is not this factor $1/h^{3N} N!$. Where this factor comes from? Why do we need to include it there? $\mathbf{1/h^{3N}}~$: Some people like the value of the partition integral to be independent of units, so they add the denominator $h^{3N}$ to the formula where $h$ has the same units as $pq$. This makes the expression dimensionless ...


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When calculating expectation values, you need to know a few things: What is my random variable? What is my distribution function? What is my desired quantity in terms of the random variable? The general form in one dimension would look like this $$ \langle G(x) \rangle = \frac{\int_{x_{min}}^{x_{max}} f(x) G(x) dx}{\int_{x_{min}}^{x_{max}} f(x) dx} \, ...


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For the $h$ factor: (the $N!$ is the Gibbs correction as already explained by others ) Step 1: Semi-classical setting The $h$ factor can actually be understood in a semiclassical way. We are talking about classical ideal gas using classical phase space formulation of classical statistical mechanics, but still from a semi-classical point of view we are not ...



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