Free and Driven Harmonic Oscillator Transition Amplitude

Following the same procedure as for the free particle, for the functional integral in 5.89, only those paths quite near to the classical path q(t) are important. So, we decompose the path into the classical path q (f) and the fluctuation y(t) as in 5.89, determine the classical action. Hie potential V(q) being a smooth function of the coordinate can be Taylor series expanded about the classical path q (f):

All differentials are evaluated along the classical path q (f), and the prime on U shows differentiation with respect to its coordinate argument. After some transformations we arrive at the following for the Lagrangian:

The contribution of the second term in the action is seen to vanish:

The contribution of the first term to the action is equal to the action of a classical path that has already been evaluated for two cases mentioned above:

Substituting these results into the expression of the transition amplitude 5.76 and considering 5.140, we obtain the following transition amplitude 5.58 with the contribution of the fluctuations determined from the reduced propagator or fluctuation factor:

The relation 5.58 is an approximate formula for the transition amplitude in the stationary phase method. The exponential factor having the classical action is the dominant contribution to the transition amplitude from the classical path. The factor F(T), which is the functional integral, describes the contribution from the quantum corrections of the order o(ft²). Generally, the relation 5.143 in the stationary phase method, imitates the standard quantum mechanics. In the small limit ft -> 0, the “classical” part of the transition amplitude, expj-j-S_C| [0,T]| oscillates rapidly when the classical action

changes, while the prefactor F(T) depends smoothly on ft in this limit. So, in the limit ft -> 0, it is sufficient to evaluate at least the first quantum correction that describes the complete quadratic action in y:

This action provides a Gaussian integral and can be evaluated to the approximation of a normalization constant. The next term in the action should be the nontrivial quantum corrections:

This quantity in F(T) can be evaluated via the perturbation theory. So, the integrand in 5.144 is quadratic in у and, thus, the integral in 5.143 is a Gaussian and may be evaluated conveniently. In this way the stationary phase method evaluates properly the first quantum correction ss ft in the transition amplitude. Finally, dropping orders 5.145 higher than the second in the Taylor expansion of the action of the function F(T) contributing to the fluctuations 5.143, we obtain the Gaussian (or quasi-classical approximation) to the transition amplitude for the quantum oscillator 5.58 where now, the reduced propagator or fluctuation factor becomes