The goal of the problem is to solve these second-order differential
equations to obtain the functions *x*_{1}(*t*) and *x*_{2}(*t*) describing the
motion of the two masses at any given time. Since they are second-order
differential equations, we need two initial conditions for each variable,
i.e.,
,
and
.

Our two differential equations are clearly *coupled*, since
depends not only on *x*_{1}, but also on *x*_{2} (and likewise for
). This makes the equations difficult to solve! The solution was
to write the differential equations in matrix form, and then diagonalize
the matrix to obtain the eigenvectors and eigenvalues.

In matrix form, we have

where . Since this 2x2 matrix is real and symmetric, it must also be Hermitian, so we know that it has real eigenvalues, and that the eigenvectors will be linearly independent and can be made to form an orthonormal basis.

Equation 3 is a particular form of the more
general equation (in Dirac notation)

where we have picked a basis set which we will call , where

(5) |

represents a unit displacement for coordinate

(6) |

represents a unit displacement for coordinate

(7) |

(as in eq. 3), which can always be decomposed into our basis as

= | (8) |

or

(9) |

Hence, eq. 3 can be considered a representation of the more general eq. 4 in the basis.

If we assume
the initial velocities are zero, then we should be able to predict
*x*_{1}(*t*) and *x*_{2}(*t*) directly from *x*_{1}(0) and *x*_{2}(0). Thus, we
seek a solution of the form

where

Again, the strategy is to diagonalize
.
The point of diagonalizing
is that, as you can see from
eq. 3, the coupling between *x*_{1} and *x*_{2} *goes away* if
becomes a diagonal matrix. You can easily
verify that the eigenvectors and their corresponding eigenvalues,
which we will label with Roman numerals I and II, are

(11) | |||

(12) |

This new basis, the

so that now depends only on , and depends only on . The equations are uncoupled! Note that we are now expanding the solution in the basis, so the components in this basis are now and instead of

(14) |

Of course it is possible to switch between the
basis and the
basis.
If we define our basis set transformation matrix as that obtained by
making each column one of the eigenvectors of
,
we obtain

(15) |

which is a unitary matrix (it has to be since is Hermitian). Vectors in the two basis sets are related by

(16) |

In this case,

The matrix equation 13 is of course equivalent to the
two simple equations

= | (17) | ||

= | (18) |

and you can see that valid solutions (assuming that the initial velocities are zero) are

where we have defined

= | (21) | ||

= | (22) |

So, the basis is very special, since any motion of the system can be decomposed into two

So, are we done? If we are content to work everything in the
basis, yes. However, our
original goal was to find the propagator
**G**(*t*) (from
eq. 10) in the original
basis. But notice that we already have
**G**(*t*) in the
basis! We can simply
rewrite equations 19 and 20 in matrix
form as

= | (23) |

So, the propagator in the basis is just

To obtain

(25) |

noting that this is the reverse transform from that needed to bring from the original to the eigenvector basis (so that

Finally, let us step back to the more general Dirac notation to point
out that the general form of the solution is

(26) |

and actual calculation just requires choosing a particular basis set and figuring out the components of and and the matrix elements of operator in that basis. Another representation of operator is clearly

as you can check by evaluating the matrix elements in the basis to get eq. 24. Thus

= | |||

= | (28) |