Kinetic Energy of 3D Fermi Gas at Absolute Zero

In summary: I'm glad I could help. In summary, the kinectic energy of a three-dimensional free electrons gas at absolute zero is 3/5 of its total energy.
  • #1
Shawj02
20
0
Show that the kinectic energy of a three-dimensional fermi gas of N free electrons at absolute zero is (Mathematica code used)
u = 3/5 N Subscript[\[Epsilon], F]

Now I know total energy of N particles is this integral

u = \!\(
\*SubsuperscriptBox[\(\[Integral]\), \(0\), \(\[Infinity]\)]\(\
\[Epsilon]\ P[\[Epsilon]] \[DifferentialD]\[Epsilon]\)\)

which is made up of the density of the states and probability of the electron to occupy level with energy \[Epsilon] at temp T.

So P[\[Epsilon]] is this big horrible looking thing. My guess is that there must be a be an easy way to integrate it that comes about from absolute zero tempature because the final answer seems so nice.

Any help, would be nice. thanks!
 
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  • #2


You need to express the Fermi momentum of the system through its number of particles:

[tex]
N = \frac{V}{(2\pi)^{3}} \int_{0}^{k_{F}}{4\pi k^{2} dk}
[/tex]

The Fermi energy [itex]E_{F}[/itex] is defined as the kinetic energy of the particles with Fermi momentum

Once you had done that, the total kinetic energy of the particles is:
[tex]
(E_{\mathrm{kin}})_{\mathrm{tot}} = \frac{V}{(2\pi)^{3}} \, \int_{0}^{k_{F}}{\frac{\hbar^{2} k^{2}}{2 m} \, {4 \pi k^{2} \, dk}
[/tex]

In the final result you need to eliminate [itex]k_{F}[/itex].
 
  • #3


ahhk, yeah that works. you could have explained it more but I got it in the end.
I'll fill in the gaps for anyone else who might look at this in the future.

use Ef=((khbar)^2)/2m to get the it to look right. and the big integral, you need to know that f(E) = 0 for E>Ef and f(E) = 1 for 0<E<Ef

So put the upper limit to Ef (because anything above this range in the integral equals zero) and also anything inside this range "E<Ef" aka 0 to Ef the f(E)=1, so this makes the integral really easy.

Thanks again Dickfore.
 

FAQ: Kinetic Energy of 3D Fermi Gas at Absolute Zero

What is kinetic energy?

Kinetic energy is the energy an object possesses due to its motion. It is dependent on the mass and velocity of the object.

What is a 3D Fermi gas?

A 3D Fermi gas is a system of particles that obey the laws of quantum mechanics and are confined to a three-dimensional space. These particles are fermions, meaning they follow the Pauli exclusion principle and cannot occupy the same quantum state.

What is absolute zero?

Absolute zero is the lowest possible temperature on the Kelvin scale, at which the kinetic energy of particles is at its minimum. It is equivalent to 0 Kelvin or -273.15 degrees Celsius.

How does the kinetic energy of a 3D Fermi gas change at absolute zero?

At absolute zero, the kinetic energy of a 3D Fermi gas is at its minimum, as all particles are in their ground state and have the lowest possible energy. This is known as the Fermi energy or Fermi level.

What is the significance of studying the kinetic energy of a 3D Fermi gas at absolute zero?

Studying the kinetic energy of a 3D Fermi gas at absolute zero allows us to understand the behavior of particles at the lowest possible energy state. This has implications in fields such as condensed matter physics, quantum computing, and superconductivity.

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