Green's functions in quantum mechanics courses

TL;DR

This paper introduces Green's functions in quantum mechanics education, demonstrating their application to fundamental problems like scattering, potential wells, and density of states, with accessible examples and essential theoretical tools.

Contribution

It presents a pedagogical approach to teaching Green's functions in basic quantum mechanics courses, including practical examples and solution methods.

Findings

Green's functions effectively solve fundamental quantum problems.

Examples include scattering, potential wells, and density of states.

The approach enhances understanding of quantum concepts.

Abstract

Green's functions in Physics have proven to be a valuable tool for understanding fundamental concepts in different branches, such as electrodynamics, solid-state and many -body problems. In quantum mechanics advanced courses, Green's functions usually are explained in the context of the scattering problem by a central force. However, their use for more basic problems is not often implemented. The present work introduces Green's Function in quantum mechanics courses with some examples that can be solved with essential tools. For this, the general aspects of the theory are shown, emphasizing the solution of different fundamental issues of quantum mechanics from this approach. In particular, we introduce the time-independent Green's functions and the Dyson equation to solve problems with an external potential. As examples, we show the scattering by a Dirac delta barrier, where the reflection and transmission coefficients are found. In addition, the infinite square potential well energy levels, and the local density of states, are calculated.