A question regarding the number of fermions with a certain velocity component

In summary, to prove the number of electrons with a certain velocity component, we can use the Fermi-Dirac Distribution function and the definition of kinetic energy. By integrating out the other velocity components, we can use a 2-D analog of the equation 4 \pi v^2 dv = dv_x dv_y dv_z to obtain the number of particles in the given range.
  • #1
Jopi
14
0

Homework Statement



Prove that the number of electrons whose velocity's x-component is between [x,x+dx] is given by
[tex]
dN = \frac{4\pi V m^2 k_B T}{h^3} ln [exp(\frac{E_F -mv_x^2/2}{k_B T})+1]dv_x
[/tex]

Homework Equations


The Fermi-Dirac Distribution function:
[tex]
\frac{dn}{dE}=\frac{4\pi V \sqrt{2m^3}}{h^3}\frac{\sqrt{E}}{exp \left((E-E_F)/ k_B T\right) +1}
[/tex]

Where E is kinetic energy,
[tex]
E=\frac{1}{2} m v^2.
[/tex]

The Attempt at a Solution


First, I used the definition of kinetic energy above to rewrite the distribution as a function of velocity. I got
[tex]
dn=\frac{V m^{3/2}}{h^3}\frac{1}{exp((\frac{1}{2} m v^2 -E_F)/k_B T) +1} 4 \pi v^2 dv
[/tex]

Now, we know that
[tex]
4 \pi v^2 dv = dv_x dv_y dv_z.
[/tex]

Now I should be able to get the number of particles with vx in the given range by integratin out dvy and dvz. In the assignment it is suggested that I write
[tex]
t^2=v_y^2+v_z^2
[/tex]
and then I can use the formula
[tex]
\int_0^{\infty} (ae^x+1)^{-1}dx=ln(1+\frac{1}{a}).
[/tex]

But I don't know how to do that. What is the differential element dvydvz written with dt?
 
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  • #2
Jopi said:
But I don't know how to do that. What is the differential element dvydvz written with dt?

Think of vy and vz as cartesian coordinates, and t as the radial coordinate in the plane.

You should then be able to derive/remember the 2-D analog of the equation you have already written down:
[tex]
4 \pi v^2 dv = dv_x dv_y dv_z.
[/tex]
 

1. How do you determine the number of fermions with a certain velocity component?

The number of fermions with a certain velocity component can be determined by using the Fermi-Dirac distribution function, which takes into account the energy and temperature of the system. This function can be used to calculate the probability of finding a fermion with a specific energy and velocity, and then the total number of fermions can be calculated by integrating over all possible energies and velocities.

2. What is the significance of studying fermions with a certain velocity component?

Studying fermions with a certain velocity component can provide valuable information about the behavior and properties of a system. This can help in understanding the thermodynamic properties of materials, as well as in the development of new technologies such as quantum computing.

3. Is there a limit to the number of fermions with a certain velocity component?

According to the Pauli exclusion principle, no two fermions can occupy the same quantum state simultaneously. This means that there is a limit to the number of fermions with a certain velocity component, as they will eventually reach a point where all possible quantum states are occupied.

4. How does the number of fermions with a certain velocity component change with temperature?

The number of fermions with a certain velocity component is directly related to the temperature of the system. As the temperature increases, more fermions will have higher energy and velocity components, leading to an increase in the total number of fermions with that specific velocity component. However, at very high temperatures, the distribution of fermions may become more uniform, resulting in a decrease in the number of fermions with a specific velocity component.

5. Can the number of fermions with a certain velocity component be controlled?

In certain systems, such as in a Bose-Einstein condensate, the number of fermions with a certain velocity component can be controlled and manipulated through external factors such as magnetic fields. This allows for precise control and measurement of fermions, which is crucial for various applications in technology and research.

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