# Average energy of the electrons at T = 0

• I
• Paolis
In summary: E_F ##.In summary, the quantum mechanical free electron model predicts that the average energy in the 3D case is E=3EF/5. However, in the 1D model, the average energy at T=0 is 0, which is different from the 3D case. This is due to the different ways of calculating the average energy in each dimension. In quantum mechanics, absolute energies are arbitrary and do not play a role. The global energy scale is also arbitrary.
Paolis
According to the quantum mechanical free electron model the average energy is E=3EF/5 for the 3D case. Nevertheless I saw in a specialised physics book that for the 1D model the average energy at T=0 is 0 and wanted to know if it is the same for the 3D case.

You know that the answer is not 0 for the 3D case.

It does not matter, as absolute energies do not play a role in quantum mechanics. The global energy scale is arbitrary anyway.

Paolis,
Let me first derive the formula for the 3 D case. For a free electron model, electron energy is given as ## E = \frac {\hbar^2 k^2}{2m} ##. We can invert the formula and write ## k = \sqrt {\frac {2mE}{\hbar^2}}## or ## k \sim \sqrt E##.
In 3D, the constant energy surface is a surface of a sphere. The number of electrons of energy not grater than at the surface is the reciprocal space volume of a sphere, that is ##N \sim \frac 4 3 \pi k^3 \sim E^ {\frac 3 2}##
that gives the density of states as ##g(E) = \frac {dN}{dE} \sim E^{\frac 1 2} ##

Now, the average kinetic energy is $$\left< E \right> = \frac {\int^{E_F}_0 E\cdot g(E) \, dE} {\int_0^{E_F} g(E)\, dE} = \frac {\int^{E_F}_0 E\cdot E^{\frac 1 2} \, dE} {\int_0^{E_F} E^{\frac 1 2}\, dE} = \frac 3 5 \cdot E_F$$

That's how you got the formula for the average kinetic energy in the 3-D case.
In one dimension, the number of states from zero to ##k_F## is proportional to ##k_F##, that is proportional to ##\sqrt{(E)}##. Differentiating wrt to E, we get the density of states as ## g(E) \sim E^{\frac 1 2} ##
The average kinetic energy can be calculated as before but using a 1-D density of states function, that is
$$\left< E \right> = \frac {\int^{E_F}_0 E\cdot g(E) \, dE} {\int_0^{E_F} g(E)\, dE} = \frac {\int^{E_F}_0 E\cdot E^{-\frac 1 2} \, dE} {\int_0^{E_F} E^{-\frac 1 2}\, dE} = \frac 1 3\cdot E_F$$

So, the average kinetic energy in 1-D measure relative to the bottom of the band is ## \frac 1 3 E_F ##

houlahound

## What is the average energy of electrons at T = 0?

The average energy of electrons at T = 0, also known as the Fermi energy, is the highest energy state occupied by electrons at absolute zero temperature. It is a fundamental property of a material's electronic structure and is typically measured in electron volts (eV).

## How is the average energy of electrons at T = 0 determined?

The average energy of electrons at T = 0 can be determined through various methods, such as theoretical calculations, experimental measurements, or through the use of sophisticated instruments like a scanning tunneling microscope. It is also influenced by factors such as the material's band structure and Fermi level.

## What is the significance of the average energy of electrons at T = 0?

The average energy of electrons at T = 0 is a crucial parameter in understanding the electronic properties of a material. It affects the material's electrical conductivity, thermal conductivity, and other important physical properties. It is also used in various applications, such as in the design of electronic devices and in the development of new materials with specific properties.

## How does the average energy of electrons at T = 0 vary among different materials?

The average energy of electrons at T = 0 can vary significantly among different materials due to their unique electronic structures and compositions. For example, metals tend to have higher Fermi energies compared to insulators, and the Fermi energy can also vary among different types of metals based on their band structures.

## Can the average energy of electrons at T = 0 change under different conditions?

Yes, the average energy of electrons at T = 0 can change under different conditions, such as changes in temperature, pressure, or the material's composition. These changes can also affect the material's electronic properties and can be studied to gain a deeper understanding of the material's behavior and potential applications.

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