 #1
Addez123
 198
 21
 Homework Statement:

$$E = \frac{(n_x^2 + n_y^2 +n_z^2) \pi^2 \hbar^2}{2mL^2}$$
Find density of state
 Relevant Equations:
 Quantum mechanics
$$n = \sqrt{n_x^2 + n_y^2 +n_z^2}$$
$$E = \frac{n^2 \pi^2 \hbar^2}{2mL^2}$$
$$n = \sqrt{ \frac{2mL^2E}{\pi^2 \hbar^2} }$$
This is all given by the textbook.
It's even as friendly as to say
$$\text{differential number of states in dE} = \frac{1}{8}4 \pi n^2 dn$$
$$D(E) = \frac{ \text{differential number of states in dE} }{dE} = \frac{1}{8}4 \pi n^2 \frac{dn}{dE}$$
Everything written above is what my textbook says when it tries to explain density of state.
Then it says: "Its left to the reader to show that this equation becomes:"
$$D(E) = \frac {m^{3/2}L^3}{\pi^2 \hbar^3 \sqrt{2}} E^{1/2}$$
What is dn/dE?
am I suppose to take the derivative of dn first??
If I do im left with
$$\frac{\pi mL^2E}{ \pi^2\hbar^2} \frac{1}{dE}$$
Now what?
Divide by a derivative?! What does that even mean!?
What am I suppose to do with the ##\frac{1}{dE}## term?
I've been stuck at this point for days now. No single youtube formula can explain the steps because everyone does it differently and involves other constants such as k etc. It's all very confusing.
$$E = \frac{n^2 \pi^2 \hbar^2}{2mL^2}$$
$$n = \sqrt{ \frac{2mL^2E}{\pi^2 \hbar^2} }$$
This is all given by the textbook.
It's even as friendly as to say
$$\text{differential number of states in dE} = \frac{1}{8}4 \pi n^2 dn$$
$$D(E) = \frac{ \text{differential number of states in dE} }{dE} = \frac{1}{8}4 \pi n^2 \frac{dn}{dE}$$
Everything written above is what my textbook says when it tries to explain density of state.
Then it says: "Its left to the reader to show that this equation becomes:"
$$D(E) = \frac {m^{3/2}L^3}{\pi^2 \hbar^3 \sqrt{2}} E^{1/2}$$
What is dn/dE?
am I suppose to take the derivative of dn first??
If I do im left with
$$\frac{\pi mL^2E}{ \pi^2\hbar^2} \frac{1}{dE}$$
Now what?
Divide by a derivative?! What does that even mean!?
What am I suppose to do with the ##\frac{1}{dE}## term?
I've been stuck at this point for days now. No single youtube formula can explain the steps because everyone does it differently and involves other constants such as k etc. It's all very confusing.