Density of States - Statistical Physics - University Physics

Link to my Patreon page: patreon.com/PazzyBoardmanPhysicsTutorials
The density of states is a concept that's very weird, and in all honesty after learning it many times in my degree I still don't think I have a perfect understanding. In this video I try and explain it as best I can, and derive what the DOS is for a three dimensional system.
In this video we draw inspiration from the one dimension quantum well (or "Particle in a box"). What was found was once the particle was confined, it could only take discrete energy states. The other observation was that there were no degenerate energy states for this system (that is to say, one or more states with the same energy), however in most systems there are many degenerate energy levels.
The density of states quantifies this degeneracy in terms of a smooth analytical function, ρ(E). This function determines what energies each particle in the system has, and how states are filled by particles. In this system we assume the system is isotropic and hence perform an integral in spherical polar coordinates in k-space.
A good exercise to practice is deriving what the DOS is for different system dimensionalities, along with different dispersion relation (i.e photons in 2D, confined systems in 3D, etc). For isotropic systems, instead of a sphere in 3D it will be an integral over a circle in 2D and a line in 1D due to the different symmetries in different dimensions.
If anyone has any questions, let me know in the comments and I'll try to answer as best and as quickly as I can!

[@dbf72829]

One day this is gonna be the most watched video on yt on statistical mechanics

[@Jake-fk5zw]

Thank you so much for taking your time when covering this.

[@honorcolling5556]

Saving me yet again, this is such a good video thankyou for putting so much time into it

[@spacedee6968]

Just wow, thanks for making this!

[@blackhole1222]

vey useful, thanks!

[@xgozulx]

this helped a lot, I also didn't really get the density of states calculation, yours is much clearer, in class what we did i think is maybe more general, but is such a mess

[@pazzy768]

Thanks! Yea you can swap things out for different dimensionalities and different dispersion relations!

[@dbf72829]

@@pazzy768 Mr pazzy what book do you follow for statistical mechanics?

[@melanogaster4586]

tnq so much

[@CanErtugay]

Thanks for the really clear explanations, but you wrote k=2 pi n/L, I think it must be pi n/L ?

[@TryhardGenos]

hi, thanks for this amazing video. U really make me love physics way more than my uni xd. Could u please explain why for example electrons we have to multiply by 2 for the dos? I understand that they have 2 different spins so they have 2 states for 3 same quantum numbers.Dont we take those 2 different states into account while we measure the states when we count them with Σ..... and later with integral so we can find that way the dos?

[@yagmurerhan8132]

hey thanks a lot for these videos! shouldn't the quantization condition be k*L= pi*L instead of 2pi?

[@MathEnth-rz3tt]

22:00 isnt k=npi/L?

[@Rafid_Ahmed101]

same confusion.... T_T

[@CanErtugay]

please let me ask a question, The reason that we take into account only 1/8 of sphere is all the components of k equal npi/L, and n=1,2,3... ???

[@pazzy768]

So we take into account all states, including ones which have n = -1,-2,-3 as they correspond to electrons moving in the opposite direction! These are also valid solutions to the standing wave equation!

[@dbf72829]

Please upload more videos on statistical mechanics

[@matthewt7883]

Stat mech video en route ;)

[@dbf72829]

I'm waiting Mr praxxy We want stat mech We want stat mech

[@matthewt7883]

We want stat mech we want stat mech

[@dbf72829]

I'm waiting

[@dbf72829]

Waiting on your new video's on stat mech

[@matthewt7883]

A few more messages and I reckon we will get one

[@dbf72829]

​@@matthewt7883haha mate we got ig

[@paladin1147]

Ama be real, your derivation doesn't always work for different conditions, or I am really stupid. I really like your derivation, it is super clean but it doesn't always work.

[@pazzy768]

What conditions are you referring to exactly?

[@paladin1147]

@@pazzy768 Like the prefactor doesn't come out right in 1D usually, it is missing 2 or 4 due to counting left and right states, sometimes in 2D

[@paladin1147]

@pazzy768 Ayyy, I love your videos, tho I did pretty well on the test and was just sad that the way you derive it doesn't work sometimes, or I am doing something wrong