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Ideal gas in potential step

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There was a mistake in the problem:

Z_{g} = Sum over N of Z(N) Exp(beta mu N)

and NOT:

Z_{g} = Sum over N of Z(N) Exp(- mu N)
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usepackage{amsmath,amssymb}
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newcommand{rhaak}[1]{!left [#1right]}
newcommand{lhaak}[1]{left | #1right |}
newcommand{ahaak}[1]{!left{#1right}}
newcommand{gem}[1]{leftlangle #1rightrangle}
newcommand{gemc}[2]{leftlangleleftlangleleft. #1right | #2
rightranglerightrangle}
newcommand{geml}[1]{leftlangle #1right.}
newcommand{gemr}[1]{left. #1rightrangle}
newcommand{haakl}[1]{left(#1right.}
newcommand{haakr}[1]{left.#1right)}
newcommand{rhaakl}[1]{left[#1right.}
newcommand{rhaakr}[1]{left.#1right]}
newcommand{lhaakl}[1]{left |#1right.}
newcommand{lhaakr}[1]{left.#1right |}
newcommand{ket}[1]{lhaakl{gemr{#1}}}
newcommand{bra}[1]{lhaakr{geml{#1}}}
newcommand{brak}[2]{gem{#1lhaakl{#2}}}
newcommand{braket}[3]{gem{#1lhaak{#2}#3}}
newcommand{floor}[1]{leftlfloor #1rightrfloor}
newcommand{half}{frac{1}{2}}
newcommand{kwart}{frac{1}{4}}
renewcommand{imath}{text{i}}

begin{document}
title{Partition function}
date{}
author{}
maketitle
I'll denote ...

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