To be used in future
 Section 5.4: General Gaussian integrals Up Chapter 5: Path Integrals Section 5.6: Time-dependent oscillator 

5.5 Harmonic oscillator

Now let us calculate the path integral for the harmonic oscillator. We use the trapezium rule for the integral of the potential energy, so that we get half the weight at the two end points, We rewrite the this as We now isolate the part only containing , We now need to diagonalise . That may feel like a difficult job; fortunately, we know quite a lot about this matrix, since this structure appears quite regularly in mathematical physics problems. Let us first show that we can simplify the problem further without damage: The matrix can be written as where is the matrix with ’s on the diagonal, and ’s above and below it. The matrix has the same eigenvectors as , since the identity matrix commutes with it, but the eigenvalues of are shifted by . This is useful since the eigenvalues and eigenvectors of can be shown to be
This can be checked easily from
The eigenvalues of are thus . We use this to change the integration variables in two steps: first we diagonalise (now use matrix notation, ) Next we shift integration variables to and thus We can now perform the integration over the variables , and find We shall now show that the exponent is the “classical action”, and evaluate the pre-factor that contains all the quantum information.

5.5.1 The determinant

Let us write and thus The determinant of is thus This is a known identity, and can also be found in standard formula tables.
The proof proceeds as follows. We realise that are the th roots of unity. We find that (here you can think of as ) and thus where we made use of the fact that , , , as well as the definition of the roots of unity.

5.5.2 The classical action

We still need to calculate the three sums in the exponential; there are actually only two different ones: the sums multiplying and are equal—as you can see from the symmetry of the original problem. Let’s deal with these sums first; once again we get “standard sums”:
and (using )
This still requires a simple derivation; I can find these in table books (or similar relations), but I really need a simple proof, as the one given above! Any suggestions appreciated...
The exponential part can be expressed in terms of the classical action, This is result is exact for any , but in the limit we find that, with and that This is the classical action for a harmonic oscillator, i.e., the action for a path starting at and ending at , following the classical equations of motion.

5.5.3 The complete continuum limit

We still need to simplify the determinant in the limit . We find that
Combine The same expression can also be obtained by a Fourier expansion, see the example sheet for a discussion. Also, in the limit it goes to the result for the free particle, Eq. () [Pleae check!]
 Section 5.4: General Gaussian integrals Up Chapter 5: Path Integrals Section 5.6: Time-dependent oscillator