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:
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
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 .