Advanced Quantum Mechanics II PHYS 40202

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To be used in future

Prev Section 5.4: General Gaussian integrals Up Chapter 5: Path Integrals Section 5.6: Time-dependent oscillator Next

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

)

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!]

Prev Section 5.4: General Gaussian integrals Up Chapter 5: Path Integrals Section 5.6: Time-dependent oscillator Next