Hello,
Is it possible to clarify why it is necessary to use \( g(E) = g(k) \frac{dk}{dE} \)?
Because if we express the number of quantum states below a given energy Ek:
Is it possible to clarify why it is necessary to use \( g(E) = g(k) \frac{dk}{dE} \)?
Because if we express the number of quantum states below a given energy Ek:
\( \Phi (E_{k}) \frac{2\pi}{L} = k \times 2 \)
We multiply by 2 if we take into account the Pauli principle.
\( \Phi (E_{k}) =\frac{L}{2\pi} k \times 2 \)
Then,
Because of the boundary condition, we know that we can express k as:
\( k = \sqrt{\frac{2mE}{\hbar^{2}}} \)
So,
\( \Phi (E_{k}) =\frac{L}{2\pi} \sqrt{\frac{2mE}{\hbar^{2}}} \times 2 \)
Then to have the DOS we do:
\( g(E) = \frac{d\Phi (E_{k})}{dE} = \frac{L}{2\pi} \sqrt{\frac{2m}{\hbar^{2}}} \frac{d(\sqrt{E})}{dE} \times 2 \)
\( g(E) = \frac{L}{2\pi} \sqrt{\frac{2m}{\hbar^{2}}} \frac{1}{2\sqrt{E}} \times 2 \)
Which is exactly the same results as the one in the correction.
So, what is the part of the of my reasoning that failed and needs to take into account the fact that g(E) = g(k) dk/dE?
Thank you in advance for your reply.