# Basic principles 2.4
**URL:** https://forum.solidstate.quantumtinkerer.tudelft.nl/t/basic-principles-2-4/3056
**Category:** Physics Q&A
**Created:** [May 20, 2024, 8:25am UTC](https://forum.solidstate.quantumtinkerer.tudelft.nl/t/basic-principles-2-4/3056 "2024-05-20T08:25:59Z")
**Posts on this page:** 6
**Page:** 1
### Author:  [@koent](https://forum.solidstate.quantumtinkerer.tudelft.nl/u/koent)
#### Post date: [May 20, 2024, 8:26am UTC](https://forum.solidstate.quantumtinkerer.tudelft.nl/t/basic-principles-2-4/3056/1 "2024-05-20T08:26:00Z")
It seems that they use \frac{2}{\pi} as the states per unit length, why?
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### Author:  [@t.vandersar](https://forum.solidstate.quantumtinkerer.tudelft.nl/u/t.vandersar)
#### Post date: [May 20, 2024, 8:43am UTC](https://forum.solidstate.quantumtinkerer.tudelft.nl/t/basic-principles-2-4/3056/2 "2024-05-20T08:43:55Z")
We get many questions about solutions, but it is much more useful to try to derive the DOS starting from the question and see if/where you get stuck. If you get stuck you can post your derivation here and I am very happy to discuss it.
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### Author:  [@koent](https://forum.solidstate.quantumtinkerer.tudelft.nl/u/koent)
#### Post date: [May 21, 2024, 8:47am UTC](https://forum.solidstate.quantumtinkerer.tudelft.nl/t/basic-principles-2-4/3056/3 "2024-05-21T08:47:38Z")
I don’t even know where to start. The lecture notes are very clear but it does not seem to be correct. g(E)=\sum \frac{dn}{dk}\frac{dk}{dE}, using k=\frac{1}{a}\sqrt{\frac{E-E\_g}{t\_{cb}}} . We arrive at g(E)=\frac{1}{2a\sqrt{t\_{cb}(E-E\_g)}}\sum \frac{dn}{dk}. I don’t have a formula for n, but in the lecture notes (e.g. tight binding model) it says that in 1D: g(E)=\frac{L}{2\pi}\sum|\frac{dk}{dE}|, which seems to suggest that \frac{dn}{dk}=\frac{L}{2\pi}. But this is incorrect!!
Is there a formula for n(k) that i’m missing?
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### Author:  [@t.vandersar](https://forum.solidstate.quantumtinkerer.tudelft.nl/u/t.vandersar)
#### Post date: [May 21, 2024, 9:09am UTC](https://forum.solidstate.quantumtinkerer.tudelft.nl/t/basic-principles-2-4/3056/4 "2024-05-21T09:09:43Z")
but you are completely right: dn/dk = L/2\pi in 1D, although you still have to multiply by a factor 2 if you want to account for negative k.
To understand how to derive a density of states, I suggest to start by writing the number of states in a certain energy range as a sum over k-values: n\_{states} = \sum\_k = \frac{L}{2\pi}\int\_{-k\_0}^{k\_0} dk = 2\frac{L}{2\pi}\int\_{0}^{k\_0} dk . Then, using the dispersion, you can write this as \int g(E) dE. Note that another factor 2 will enter for the spin degeneracy
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### Author:  [@anton-akhmerov](https://forum.solidstate.quantumtinkerer.tudelft.nl/u/anton-akhmerov)
#### Post date: [March 16, 2025, 12:20pm UTC](https://forum.solidstate.quantumtinkerer.tudelft.nl/t/basic-principles-2-4/3056/5 "2025-03-16T12:20:02Z")
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### Author:  [@anton-akhmerov](https://forum.solidstate.quantumtinkerer.tudelft.nl/u/anton-akhmerov)
#### Post date: [March 16, 2025, 12:24pm UTC](https://forum.solidstate.quantumtinkerer.tudelft.nl/t/basic-principles-2-4/3056/6 "2025-03-16T12:24:25Z")