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Exploring fermion exchange and Coulomb interactions

Following my previous post on single particle fermions operators here in my latest work I address the non-interacting fermion systems under the influence of Coulomb forces.

Introduction

Quantum mechanics often deals with scenarios where particles are not isolated but interact through various forces. The Coulomb interaction, which describes the force between two charged particles, is particularly significant in systems of electrons. In this post, I’ll discuss how these interactions are represented within the framework of operator formalism and what implications arise from them.

Fermion operators in non-interacting systems

Initially, consider a system of multiple non-interacting particles, where the fermionic nature of particles requires a unique approach to describing their states. The wavefunction for such a system, including N particles, is given by:

\hat{\psi}(\mathbf{r}_1, \mathbf{r}_2, \ldots, \mathbf{r}_N) = \frac{1}{\sqrt{N}} \sum_{1, 2, \ldots, N} \phi_1(\mathbf{r}_1) \phi_2(\mathbf{r}_2) \dots \phi_n(\mathbf{r}_N) \mathbf{b}_{1} \mathbf{b}_2 \dots \mathbf{b}_N

This expression illustrates the antisymmetric property required by fermions, encapsulating the Pauli Exclusion Principle fundamentally.

Interaction through Coulomb forces

When extending this to consider two electrons interacting via Coulomb forces, the Hamiltonian becomes:

\mathbf H \left(\mathbf r_1, \mathbf r_2\right) = -\frac{\hbar}{2m}\left(\nabla^2_{\mathbf r_1} + \nabla^2_{\mathbf r_2}\right) + \frac{e^2}{4\pi\varepsilon_0|\mathbf r_1 - \mathbf r_2|}

This Hamiltonian encapsulates both the kinetic energy of the electrons and their electrostatic potential energy due to their charge.

Energy expectation and exchange energy

The core of my analysis focuses on the expectation value of the energy in a state where two particles occupy different quantum states:

\langle E \rangle = \frac{1}{2} \left(H_{jiji} + H_{ijij} + H_{ijji} + H_{jiij} \right)

This leads us to confront the concept of exchange energy, an exclusively quantum mechanical phenomenon with no classical counterpart. Exchange energy arises due to the antisymmetry of the fermionic wavefunction, affecting the total energy calculation significantly.

Conclusion

The formalism developed here provides a robust framework for analyzing interactions in fermionic systems, revealing deeper insights into their quantum behavior. These insights are vital for anyone exploring quantum mechanics at an advanced level.

For more insights into this topic, you can find the details here.