- #1

kuchenvater

- 9

- 0

## Homework Statement

I'm trying to solve problem 1.8.9 -part 3 of "The Principles of Quantum Mechanics" by R Shankar. Here's the problem:

Given the values of M

_{ij}(see next section for equations) which is Hermitian, argue that there exist three directions such that the angular velocity and angular momentum will be parallel.

## Homework Equations

Given

M

_{ij}=∑

_{a}=m

_{a}[ r

^{2}

_{a}δ

_{ij}- (r

_{a})

_{i}(r

_{a})

_{j}]

here i denotes the i

^{th}Cartesian component.

|l>=M|ω>

## The Attempt at a Solution

It's known that M is Hermitian. l and ω will be parallel if ω is an eigenvector of l. So I have to prove that the number of eigenvalues of M is 3. Somehow I can't figure out a way to do that. It's obvious that all eigenvalues will be real and the number of eigenvalues is equal to or less than the rank of a matrix. I was thinking I could consider only the diagonal elements of a diagonalized version of M or just the trace of M(which yields the sum of all eigenvalues) by setting i=j and then I'd have i as the index of sigma. If I can count the number of values of i, I'd have the number of eigenvalues. And since M is Hermitian and has real eigenvalues and this describes a rotating system of mass(i.e. it's moment of inertia), it must be defined in the real plane and that has 3 axes-x,y and z, one for each value of i. So i varies across 3 values, hence the diagonalized matrix has 3 diagonal elements and hence 3 eigenvalues. The question also mentions the mass's cartesian component, so that automatically means x, y, z.

Is that correct reasoning?