Fluctuation Contribution to Transition Amplitude

The stationary phase method applied to a simple case of an oscillator when

From the conditions 5.147 and 5.148 we consider now the Gaussian path integral (reduced propagator or fluctuation factor) easily evaluated:

This describes an oscillatory motion between the spaces (q_a,t„) and (T and frequency w. This permits us to Fourier series expand y(t) via the harmonic frequency equal a>„ as in the case of free particle seen above:

with, respectively, the conditions of orthonormality and those at ends points of the path:

Letting t_a = 0 for convenience then we select the Fourier coefficients a„ as a discrete set of integration variables:

with each path having its own coefficients a„ and results in a set of coefficients {a,,}.

We evaluate the term proportional to the kinetic energy:

If пф n'y then the integral is equal to zero, while, for n = n' we have

Since,

then

Similarly, the term proportional to the potential energy: or and

This implies a sum over all paths, which is a sum over the coefficients^,,}. So, the expansion ofy(t) in terms of eigenmodes in 5.150 is synonymous to diagonalizing the operator and path integral over ally(f) equal integral over a„. Asy —► a„ then the path integration reduces to an infinite product of finite-dimensional Gaussian integrals that is the path integral which is an infinite number of Fresnel integrals over a„:

Here, / is the Jacobian transformation. Thus, the reduced propagator (fluctuation factor):

To interpret this result in 5.162, we first make sense out of the infinite product that seems divergent

for times equal to —. Besides, the value of the Jacobian J is yet to be properly determined. These difficulty

ties can be resolved by exploiting the transition amplitude of a free particle system corresponding to that of the harmonic oscillator when u> = 0 . This implies, had we computed the transition amplitude for the free particle system via the path integral, we would have obtained the same Jacobian / and, particularly, an infinite product like the one in 5.162, but with w = 0. So, for motion over an integer number of periods then q_b = q„ and the Jacobian / is:

So,

Using the following infinite-product formula [1,14]: then,

This is indeed the transition amplitude fromy(f₀) = 0 loy{t_b) = 0 . Due to time translation invariance the reduced propagator F(T) can only depend on T. For w = 0, then the result for the free particle:

Maslov Correction

We consider an interesting phenomenon [11,12] that is due to a phase-jump of light by-у after passing via a focal point. When a light beam is split into two parts, one part passes via a focal point and the other not. If the two partial waves are recombined, a destructive interference is observed, a phenomenon known as the Maslov correction [11,12] . The expressions 5.162 or 5.166 can only be valid when T is within the range

For larger time moments:

then F(T) acquires an additional factor exp j-/^j [11,12] where the so-called Maslov-Morse index n

counts how many times the trajectory has passed through a focal point where the prefactor diverges. At the focal points the quantum mechanical probability amplitude achieves singularity that in real systems is prevented by anharmonic terms, so that the intensity achieves its maximal value. Spatial accumulation of such points is called a caustic phenomenon. This is often achieved in optics, for example when a water glass is illuminated by sunlight.

So, this can be applied to the transition amplitude for the free and driven oscillator with

The absolute square of the transition amplitude gives the probability per unit length of finding the mass back to where it started at the beginning of an oscillation with period T: