Bra-Ket Representation of the Inertia Tensor
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Bra-Ket Representation of the Inertia Tensor U-Rae Kim, Dohyun Kim and Jungil Lee∗ KPOPE Collaboration, Department of Physics, Korea University, Seoul 02841, Korea (Received 24 June 2020; accepted 24 July 2020) 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. Keywords: Classical mechanics, Inertia tensor, Bra-ket notation, Diagonalization, Hyperellipsoid DOI: 10.3938/jkps.77.945
I. INTRODUCTION The inertia tensor is one of the essential ingredients in classical mechanics with which one can investigate the rotational properties of rigid-body motion [1]. The symmetric nature of the rank-2 Cartesian tensor guarantees that it is described by three fundamental parameters called the principal moments of inertia Ii , each of which is the moment of inertia along a principal axis. The principal axes are orthogonal so that they construct a natural choice of the Cartesian coordinate system. Additional three parameters of the symmetric tensor can be allocated to the rotation from the natural coordinate system of the principal axes to an arbitrary coordinate system. Thus, the tensor is one of the best tools with which students can train on how to deal with the eigenvalue problem of a Hermitian matrix. The underlying mathematical structure of this problem is closely related to the normal modes. It can further be extended to the Hilbert space in the continuum limit [2]. Hence, we believe that teaching inertia tensors is of great importance in giving students a strong background for their future study of quantum mechanics. Many pedagogical materials involving the general and the inertia tensors are available [3–8], including undergraduate level textbooks [1,2]. Most of them deal with the inertia tensor in explicit matrix representations. In this paper, we employ Dirac’s bra-ket notation [9–11] to define the inertia tensor operator, from which one can project out the corresponding inertia tensor elements by ˆi applying a bra and a ket for the unit basis vectors q ∗ E-mail: [email protected]; Director of the Korea Pragmatist Organization for Physics Education (KPOP E)
pISSN:0374-4884/eISSN:1976-8524
along the principal axes. While the completeness relation reflects the homogeneousness and isotropy of the qi |, the inertia tensor operator Euclidean space 1 = |ˆ qi ˆ acquires the anisotropy of the mass distributio
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