- #1
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Hi,
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, nx, ny, nz, and ms, 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]
where k is 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 = lxlylz
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] in k-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 continuous variables?
2) Basically, other treatments I have seen divide k-space into a discrete set of blocks or unit cells, each of which is associated with a point (kx,ky,kz). 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?
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, nx, ny, nz, and ms, 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]
where k is 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 = lxlylz
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] in k-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 continuous variables?
2) Basically, other treatments I have seen divide k-space into a discrete set of blocks or unit cells, each of which is associated with a point (kx,ky,kz). 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?