- #1

patric44

- 296

- 39

- Homework Statement
- i have a question concerning the derivation of the density of states

- Relevant Equations
- g(E) =sqrt(2)/pi^2*(m/hbar^2)^(3/2)*sqrt(E)

hi guys

i have a question about the derivation of the density of states , after solving the Schrodinger equation in the 3d potential box and using the boundary conditions ... etc we came to the conclusion that the quantum state occupy a volume of ##\frac{\pi^{3}}{V_{T}}## in k space

and to count the total number of quantum states its easier to count them in a shell with thickness dk then integrate , so we have :

$$N_{shell} =\frac{\frac{1}{8}4\pi*k^{2}dk}{\frac{\pi^{3}}{V_{T}}}$$

then integrating to get the total number of states give us :

$$N_{T} =\frac{V_{T}}{(2\pi^{2})}\frac{1}{3}k^{3}$$

and translating that expression in terms of the energy

$$N_{T} =\frac{V}{3}(\frac{\sqrt(2)}{\pi^{2}})(\frac{m}{\hbar^{2}})^{3/2}E^{3/2}$$

now dividing by V and E to get the number of quantum states per unit volume and energy give us :

$$g(E) =\frac{1}{3}(\frac{\sqrt(2)}{\pi^{}2})(\frac{m}{\hbar^{2}})^{3/2}E^{1/2}$$

but that expression doesn't look similer to the standerd one in textbooks

$$g(E) =(\frac{\sqrt(2)}{\pi^{}2})(\frac{m}{\hbar^{2}})^{3/2}E^{1/2}$$

what i am doing wrong here ?

i have a question about the derivation of the density of states , after solving the Schrodinger equation in the 3d potential box and using the boundary conditions ... etc we came to the conclusion that the quantum state occupy a volume of ##\frac{\pi^{3}}{V_{T}}## in k space

and to count the total number of quantum states its easier to count them in a shell with thickness dk then integrate , so we have :

$$N_{shell} =\frac{\frac{1}{8}4\pi*k^{2}dk}{\frac{\pi^{3}}{V_{T}}}$$

then integrating to get the total number of states give us :

$$N_{T} =\frac{V_{T}}{(2\pi^{2})}\frac{1}{3}k^{3}$$

and translating that expression in terms of the energy

$$N_{T} =\frac{V}{3}(\frac{\sqrt(2)}{\pi^{2}})(\frac{m}{\hbar^{2}})^{3/2}E^{3/2}$$

now dividing by V and E to get the number of quantum states per unit volume and energy give us :

$$g(E) =\frac{1}{3}(\frac{\sqrt(2)}{\pi^{}2})(\frac{m}{\hbar^{2}})^{3/2}E^{1/2}$$

but that expression doesn't look similer to the standerd one in textbooks

$$g(E) =(\frac{\sqrt(2)}{\pi^{}2})(\frac{m}{\hbar^{2}})^{3/2}E^{1/2}$$

what i am doing wrong here ?