# 3D Electron Gas Derivation

• phrygian
In summary, the 3 dimensional electron gas in a solid state book is derived using the Schrodinger equation where the free-electron mass is represented by m. The solutions of the equation are labeled by the wavevector k and are described by plane waves, phi(r)=1/(2pi)^3 Exp(ik.r). However, since Exp(ikx) is not normalizable from -infinity to infinity, the normalization constant cannot be determined. A possible solution is to use identity 303 on the Wikipedia page on Fourier transforms, which would give a normalization constant of 1/(2pi)^(3/2). This is based on the contributions of each x, y, and z wave, which is each represented by 1

## Homework Statement

In a solid state book I am reading the 3 dimensional electron gas is derived. It says, "An unconfined electron in free space is described by the Schrodinger equation where m is the free-electron mass.

The solutions of the equation, phi(r)=1/(2pi)^3 Exp(ik.r) are plane waves labelled by the wavevector k=(kx,ky,kz)."

## The Attempt at a Solution

I know that Exp(ikx) is not normalizble from -infinity to infinity, so how can you determine the normalization constant?

Thanks for the help

But wouldn't that give 1/(2pi)^(3/2) since the x, y, and z waves each contribute 1/(2pi)^(1/2)?

## What is a 3D electron gas?

A 3D electron gas refers to a collection of electrons that are confined to three dimensions, such as in a solid or a semiconductor material. These electrons are free to move within the material and are not bound to any particular atom.

## Why is the derivation of 3D electron gas important?

The derivation of 3D electron gas is important because it helps us understand the behavior of electrons in a three-dimensional system, which is crucial for understanding the properties and applications of materials such as semiconductors and metals.

## What is the formula for 3D electron gas?

The formula for 3D electron gas is given by the Fermi-Dirac distribution function, which describes the probability of finding an electron at a given energy level in a 3D system. The formula is given by f(E) = 1 / (1 + e^((E-E_F)/kT)), where E is the energy level, E_F is the Fermi energy level, k is the Boltzmann constant, and T is the temperature.

## How is the 3D electron gas derived?

The 3D electron gas is derived using quantum mechanics and statistical mechanics principles. This involves solving the Schrödinger equation for a particle in a 3D box and applying the Fermi-Dirac distribution function to determine the probability of finding an electron at a given energy level.

## What are some applications of 3D electron gas?

The understanding of 3D electron gas is crucial for many applications in technology, such as in the development of transistors and other electronic devices. It is also important in the study of materials for energy storage, optoelectronics, and quantum computing.