Textbook absolutely fails at teaching density of state, leaves out vital steps

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.

Mentor
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.

Last edited:
Homework Helper
2022 Award
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.

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?

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.
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.

hutchphd
Mentor
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.