Principles of Quantum Mechanics · Problem 1.8.9

Mathematical Introduction — R. Shankar

Physics — Quantum Mechanics · moment-of-inertia-tensor

Problem

A collection of masses mαm_\alpha, located at rα\mathbf{r}_\alpha and rotating with angular velocity ω\boldsymbol{\omega} around a common axis has an angular momentum l=∑αmα(rα×vα)\mathbf{l} = \sum_\alpha m_\alpha(\mathbf{r}_\alpha \times \mathbf{v}_\alpha) where vα=ω×rα\mathbf{v}_\alpha = \boldsymbol{\omega} \times \mathbf{r}_\alpha is the velocity of mαm_\alpha. By using the identity A×(B×C)=B(A⋅C)−C(A⋅B)\mathbf{A} \times (\mathbf{B} \times \mathbf{C}) = \mathbf{B}(\mathbf{A} \cdot \mathbf{C}) - \mathbf{C}(\mathbf{A} \cdot \mathbf{B}) show that each Cartesian component lil_i of l\mathbf{l} is given by li=∑jMijωjl_i = \sum_j M_{ij} \omega_j where Mij=∑αmα[rα2δij−(rα)i(rα)j]M_{ij} = \sum_\alpha m_\alpha[r_\alpha^2 \delta_{ij} - (\mathbf{r}_\alpha)_i(\mathbf{r}_\alpha)_j] or in Dirac notation ∣l⟩=M∣ω⟩|l\rangle = M|\omega\rangle (1) Will the angular momentum and angular velocity always be parallel? (2) Show that the moment of inertia matrix MijM_{ij} is Hermitian. (3) Argue now that there exist three directions for ω\boldsymbol{\omega} such that l\mathbf{l} and ω\boldsymbol{\omega} will be parallel. How are these directions to be found? (4) Consider the moment of inertia matrix of a sphere. Due to the complete symmetry of the sphere, it is clear that every direction is its eigendirection for rotation. What does this say about the three eigenvalues of the matrix MM?

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