Bra-Ket Representation of the Inertia Tensor

TL;DR

This paper introduces a bra-ket notation-based method to define and compute the inertia tensor, simplifying calculations and providing an intuitive approach for both classical and quantum mechanics applications.

Contribution

It develops a bra-ket operator framework for the inertia tensor, enabling basis-independent and geometrically straightforward diagonalization.

Findings

Simplifies inertia tensor computation using bra-ket notation.

Provides a basis-independent method for principal axes determination.

Demonstrates application to an N-dimensional ellipsoid.

Abstract

We employ Dirac's bra-ket notation to define the inertia tensor operator that is independent of the choice of bases or coordinate system. The principal axes and the corresponding principal values for the elliptic plate are determined only based on the geometry. By making use of a general symmetric tensor operator, we develop a method of diagonalization that is convenient and intuitive in determining the eigenvector. We demonstrate that the bra-ket approach greatly simplifies the computation of the inertia tensor with an example of an $N$-dimensional ellipsoid. The exploitation of the bra-ket notation to compute the inertia tensor in classical mechanics should provide undergraduate students with a strong background necessary to deal with abstract quantum mechanical problems.