# Solving the one-dimensional Ising chain via mathematical induction: An intuitive approach to the transfer matrix

## TL;DR

This paper introduces an intuitive mathematical induction approach to solve the one-dimensional Ising model, providing a clear alternative to the transfer matrix method and enhancing teaching in statistical mechanics.

## Contribution

It presents a simple, physically transparent formulation using induction that reproduces the transfer matrix results, aiding educational understanding and generalization to other systems.

## Key findings

- Reproduces the partition function of the 1D Ising model
- Offers a computationally friendly alternative to transfer matrix
- Facilitates teaching of statistical mechanics concepts

## Abstract

The aim of this work is to present a formulation to solve the one-dimensional Ising model using the elementary technique of mathematical induction. This formulation is physically clear and leads to the same partition function form as the transfer matrix method, which is a common subject in the introductory courses of statistical mechanics. In this way our formulation is a useful tool to complement the traditional more abstract transfer matrix method. The method can be straightforwardly generalized to other short-range chains, coupled chains and is also computationally friendly. These two approaches provide a more complete understanding of the system, and therefore our work can be of broad interest for undergraduate teaching in statistical mechanics.