Show that the average energy of an electron in a one-dimensional metal is related to the Fermi e

Show that the average energy of an electron in a...

Key Concepts

Fermi Energy

Fermi energy is the energy level at which the highest occupied state exists at absolute zero temperature. It is a critical concept in quantum statistics, defining the distribution of electrons in a metal and serving as an upper limit to the energies of electrons when no thermal excitation is present.

Average Energy

The average energy in a system is the mean value of the energy per particle, obtained by integrating the energy of all occupied states and then dividing by the number of particles. In quantum systems like metals, this concept helps in understanding how the total energy is distributed among electrons and is essential for deriving relations between microscopic and macroscopic properties.

Density of States

The density of states represents the number of quantum states available to electrons per unit energy interval. In one-dimensional systems, the peculiar form of the density of states directly affects the integration process used to determine both total and average energies. This concept is central to connecting the microscopic energy levels with observable properties like the average energy.

One-Dimensional Electron Gas

A one-dimensional electron gas refers to electrons confined to move in a single spatial dimension. This simplification yields distinctive forms for both the density of states and the energy distribution, which in turn result in specific relationships between quantities like the Fermi energy and the average energy. Such systems serve as foundational models in solid-state physics for understanding more complex materials.

Transcript

00:01 Hi there.

00:01 So for this problem, we are told to show that from the fermi distribution that in a metal with the temperature equals to 0 kelvin, the average energy of an electron is equal to 3 divided by 5 times the fermi energy.

00:26 So we know that at the temperature equals to 0 kelvin, the number of particles in terms of the energy can be 1 or 0, 1 when the energy is between 0 but less or equal to the fermi energy and 0 when the energy is greater than the fermi energy.

01:00 Now, hence, the total number of particles is given by the following integral, that is the integral from zero to infinity of the product between the number of particles with the respect energy, well, n, which is the number of moles in this case, so times the number of particles, and this times the differential in energy.

01:31 So in this case, we will have that we are left in this case with the integral from zero to the fermi energy of the number of particles in function of the energy integrated over the energy.

01:50 So we know that in this case, the number of particles is going to be in function of the energy.

02:02 Of the volume and the energy.

02:06 So we can take out everything that is a constant that do not depend on the energy.

02:12 So we will have a times pi times the volume times two times the mass to the three and this elevated to one divided by two and this divided by plants constant to the three.

02:27 The integral from zero to the fermi energy of the square root of the energy integrated over the energy.

02:40 So we know that the integral of the square root of the energy is going to be the energy elevated to 3 divided by 2.

02:48 And a factor that comes from there, which is 2 divided by 3 plus 1, of course.

02:57 3 divided by 2 plus.

02:59 So that's the term that we need to divide this from that integral.

03:05 In this case for n is equal to 16 times pi times the volume times two times the mass to the three and that elevated to one divided by two and this divided by three times plums constant to the three times the fermi energy elevated to three divided by two and we obtained this after just evaluating this integral the result of that integral...

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