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I'm currently going through statistical physics, especially on Fermi energy when I came across a term called "quasi-continuum", what exactly is it?

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"Quasi-something" is "almost something". In particular, quasicontinuum is a discrete set that contains so many entries (typically energy eigenvalues) with so small spacings that it looks like a continuum.

For example, if the energy eigenvalues are $$E_n = n\epsilon$$ where $n$ is integer and $\epsilon$ is $0.0001$ times some natural energy difference, then $E_n$ is almost continuous and the set $\{E_n|n\in Z\}$ is a quasi-continuum. In many physical questions, a quasi-continuum behaves just like a continuum.

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