In some books the degeneracy of one Landau level in a two-dimensional gas of free electrons is calculated in the following way:
Note: The electron spin is not considered.
Number of states of a free electron gas
First we consider a two-dimensional free electron gas without any magnetic field. With periodic boundary conditions we get a quantization of $k_x$ and $k_y$: $$k_i = 2\pi n_i /L_i.$$ Thus one state has the area $$ \frac{4\pi^2}{L_x L_y}$$ in the $k$-space. The number of levels $\tilde{N}$ depending on the maximum wave number $k$ is then: $$ \tilde{N}(k) = \frac{\pi k^2}{\frac{4\pi^2}{L_x L_y}} = \frac{k^2}{4\pi}L_x L_y.$$ The number per area $N$ is $$N(k)=\frac{k^2}{4\pi}.$$
Degeneracy of Landau Level
This calculation is based on the argument, that the number of states is conserved, when the magnetic field is turned on. Thus the states between two Landau rings have to condensate on them.
According to this the degeneracy $D$ is: $$ D = N(k^2_{n+1})-N(k^2_n)$$
According to the formulas for Landau quantization the following holds: $$ k_n^2 = \frac{2mE_n}{\hbar^2} = \frac{2m\omega_c}{\hbar} (n + \frac{1}{2}).$$
Inserting this in the previous formula we get: $$ D = \frac{m\omega_c}{h} [(n+1 + \frac{1}{2})-(n + \frac{1}{2})] = \frac{m\omega_c}{h}=\frac{eB}{h}.$$ We used the definition of the cyclotron frequency in the last step.
My question: What is the degeneracy of the first Landau level with $n=0$? According to this calculation it is the half of the degeneracy of the other levels ( $\frac{eB}{2h}$), isn't it?
Answer to Kyle Arean-Raines question: Here is the way, I calculate the degeneracy of the first level: The first level has $n=0$, thus $D=N(k_0^2)-N(0)$ holds. The result is $D=\frac{1}{2}\frac{2m\omega_c}{\hbar}\cdot \frac{1}{4\pi}=\frac{1}{2} \frac{m \omega_c}{h}=\frac{1}{2} \frac{eB}{h}$. Am I wrong?
Answer to Kyle Arean-Raines second question: I interpret the formula for the degeneracy as the formula for the degeneracy of the level $n+1$: $$ D_{n+1} = N(k^2_{n+1})-N(k^2_n)$$ In my opinion $N(0)$ makes sense, it is the number of states in a free electron gas with no magnetic field with a maximum wave number of $0$, thus $N(0)=0$ holds. I used this fact in my previous answer. I assume, that the levels of the gas without a magnetic field, which are inside a Landau ring (and outside of the ring with a lower $n$) condensate on this ring. Is this correct?
Relevance of this question: The reason, why I am asking this question, is the following: This calculation makes it plausible, that the degeneracy of the first level is just the half of the degeneracy of the other ones. But for calculating the Hall resistance in the case of the quantum hall effect a constant degeneracy is used. So there are three possible ways out:
- These Hall resistance calculations are not true for the first level.
- My calculation is wrong, at least for $D_0$.
- The assumption of conversavtion of states is wrong.
