Statistical Quantum good question

[adelveis]

So here is my question. Have a 2-D electron gas where:

E = P(x)^2/(2m) + P(y)^2/(2m)

Where p^2= p(y)^2 + p(x)^2

1. How many single particle energy states are there with momentum p?

( this may be a really simple question but I need a refresher.)

2. If there are N electrons is the metal, and T=0 , find the fermi energy of the 2-D electron gas

3. Find the ave electron energy of the gas in the 2-D Fermi energy at T=0.

this last one i think I have a better idea since with the 3-D there is a factor of 1/8, the 2-D it would most likely have a factor of 1/4.

Any other help with the first 2 would be awesome!

 

[StatMechGuy]

Just start from the beginning. What's the expression of the average number of particles? From this, what's the expression for the total energy? How does that change from 3D to 2D?

You'll uncover some pretty interesting physics in 2D if you work it out, particularly with the Fermi energy.

 

[denian]

I would be happy to provide a response to your questions.

1. The number of single particle energy states with momentum p can be calculated using the formula:

N(p) = A/(2πħ)^2 * p^2

Where A is the area of the 2-D system. This formula is derived from the fact that in 2-D systems, the density of states is proportional to the square of the momentum.

2. At T=0, the Fermi energy (EF) can be calculated by using the formula:

EF = (ħ^2/2m)(πN/A)^2

Where N is the number of electrons and A is the area of the 2-D system.

3. The average electron energy at EF can be calculated by using the formula:

<E> = (3/5)EF

This formula is derived from the fact that at T=0, the average electron energy is equal to the Fermi energy multiplied by a factor of (3/5) due to the Pauli exclusion principle.

I hope this helps with your understanding of the 2-D electron gas and its properties. If you have any further questions, please don't hesitate to ask. Best of luck with your studies!