Solving the Density of States: Understanding dn/dE

[Addez123]

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.

 

[Mark44]

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?

It's the derivative of n with respect to E. You are given the formula for n as a function of E. This is a fairly simple differentiation problem.

am I suppose to take the derivative of dn first??

No. See above.

If I do im left with
$$\frac{\pi mL^2E}{ \pi^2\hbar^2} \frac{1}{dE}$$

No, that's incorrect.

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.

 

[hutchphd]

Do you understand their "friendly " hint? Do you realize where the 1/8 comes from? You are trying to count the number of degenerateb states as n gets large by taking a continuum approximation.
Youtube formula? How about a book?

 

[Addez123]

It's the derivative of n with respect to E. You are given the formula for n as a function of E. This is a fairly simple differentiation problem.

No. See above.

No, that's incorrect.

I literally can not explain how thankful I am for this response.
I dont know how I couldnt read dn/dE as $\frac{d}{dE}(n)$ but it just never clicked.

One final problem though. Their result has $\sqrt{2}$ in the denominator. I've done it twice but I get it in the numurator, isnt that correct?

 

[Addez123]

Do you understand their "friendly " hint? Do you realize where the 1/8 comes from? You are trying to count the number of degenerateb states as n gets large by taking a continuum approximation.
Youtube formula? How about a book?

The 1/8th is because we calculate all states as if it were in a cartesian coordinate system and since n cant be negative we only cover the first octant.

Its the surface of a sphere in the 1st octant * dn as they explain it.

 

[Mark44]

One final problem though. Their result has $\sqrt{2}$ in the denominator. I've done it twice but I get it in the numurator, isnt that correct?

I get exactly their result. In my work I ended up with 2 in the denominator, and $\sqrt 2$ in the numerator. Simplifying gives $\sqrt 2$ in the denominator.

 

[Addez123]

I get exactly their result. In my work I ended up with 2 in the denominator, and $\sqrt 2$ in the numerator. Simplifying gives $\sqrt 2$ in the denominator.

Ahh yes yes! You get 1/2 from derivating the sqrt(E)!
Now it all makes sense.

Unbelivably grateful, thanks a lot :) :)

 

[Mark44]

You get 1/2 from derivating the sqrt(E)!

Minor nit -- "derivating" is not a word in English, but "differentiating" is.