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I have a question about the discussion of the free-electron (Fermi) gas in my solid-state physics notes. In the free electron model, you basically have particles in a box, and the state of any particle is described by four quantum numbers, n_{x}, n_{y}, n_{z}, and m_{s}, the spin magnetic quantum number. Furthermore, the wavefunction of a particle is given by:

[tex] \psi(\mathbf{r}) = e^{i\mathbf{k} \cdot \mathbf{r}} [/tex]

wherekis defined as follows:

[tex] k_x = \frac{2 \pi}{l_x}n_x [/tex]

et cetera. I have assume that the box has dimensions

V = l_{x}l_{y}l_{z}

Here is the step I am having trouble understanding:

The number of states associated with an element [itex] d^3k = dk_xdk_ydk_z [/itex] ink-space is then given by

[tex] 2dn_xdn_ydn_z = \left(\frac{V}{8 \pi^3} \right) 2dk_xdk_ydk_z [/tex]

Although this follows if you sort of consider each k component as a function of each corresponding n component, it doesn't make a lot of sense

Questions:

1) [itex] n_x [/itex], [itex] n_y [/itex], and [itex] n_z [/itex] are each [itex] \in \mathbb{Z} [/itex], so why are k and n suddenly being treated as continous variables?

2) Basically, other treatments I have seen dividek-space into aset of blocks or unit cells, each of which is associated with a point (kdiscrete_{x},k_{y},k_{z}). The number of states in each block is then just...2. So what does the statement in bold (the number of states "associated with an element") even?mean

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# Fermi Gas

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