References: Shankar, R. (1994), *Principles of Quantum Mechanics*, Plenum Press. Exercise 1.8.12.

We’ll continue our study of the system of two masses coupled by springs. The system is described by the matrix equation of motion:

where

in the basis

In this basis, is the operator whose matrix form is

We found that the solution could be written as

In compact form, we can write this as

where the propagator operator is defined as

From 1, we can operate on both sides of 7 with the operator to get

Since the initial positions are arbitrary and contains no time dependence, the matrix satisfies the differential equation

By direct calculation (I used Maple, but you can do it by hand using the usual rules for matrix multiplication, although it’s quite tedious), we can show that and commute and, since both and are hermitian, they are simultaneously diagonalizable. We already worked out the eigenvectors of :

Since is not degenerate, these must also be the eigenvectors of , so the unitary matrix

can be used to diagonalize according to

This matches the diagonal form for given as equation 1.8.43 in Shankar’s book. The diagonal entries are the eigenvalues of .

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