User:Mitjp

The homogeneous electron gas is a model in many body physics that aims to describe a physical system of interacting electrons in a background of uniform positive charge density. The theory is a simple example of the use of Hamiltonians in solid state physics. The model is frequently cited as an effort to understand conduction electrons in a metal, particularly when the electron density is sufficiently high to justify approximation schemes for physical quantities such as the ground state energy and dielectric function. Additionally, the electron gas is used in the field of plasma physics as the basis of the so-called One Component Plasma (OCP).

Hamiltonian
By using the methods of so-called 'second quantization' in quantum mechanics and hence formulating the field operators

$$ \Psi = \sum_{j}c_{j}\phi(x) \,\,\,;\,\,\, \Psi^{+} = \sum_{j}c_{j}^{+}\phi^{*}(x)$$

where $$+$$ denotes Hermitian conjugation, the $$c_{j}$$'s are fermionic operators obeying anti-commutation relations and the $$\phi(x)_{j}$$'s form a complete set of orthonormal solutions to the single particle equation $$ H\phi_j(x) = \epsilon_{j}\phi_{j}(x)$$, the Hamiltonian for the electron gas can be written in the form

$$H = \int d^{3}x^{'}\Psi^{+}(\mathbf{x^{'}})\frac{1}{2m}p^{2}\Psi(\mathbf{x^{'}}) + \frac{1}{2}\int d^{3}x^{'}d^{3}y\Psi^{+}(\mathbf{x^{'}})\Psi^{+}(\mathbf{y})V(\mathbf{x^{'}} - \mathbf{y})\Psi(\mathbf{y}\Psi(\mathbf{x^{'}})$$

Making the electron spin explicit in the fermion operators, and selecting plane wave states for the $$\phi(x)_{j}$$'s (wavefunctions for 'free' particles), this can be simplified to

Length scale for the electron gas
It is possible to understand a number of physical properties of the electron gas simply by considering a number of properties of fermion systems. Firstly, define the length parameter $$ r_{s} $$ as the approximate radius of a sphere containing charge $$e$$

$$\frac{4}{3}\pi r_{s}^{3} = \frac{1}{n}$$

Where $$n$$ is the electron number density. $$r_{s}$$ is typically given in atomic units (as a multiple of the bohr radius) and in terms of the fermi wavevector of the system

Ground State Energy
The ground state energy of the Hamiltonian has not been solved exactly. At zero temperature, the ground state energy in the Hartree-Fock approximation is given as

This formula underestimates the ground state energy