Linear Oscillator with Time-Dependent Frequency

Abstract

Here is another important example of a path integral calculation, namely the time-dependent oscillator whose Lagrangian is given by

$$\displaystyle{ L = \frac{m} {2} \dot{x}^{2} -\frac{m} {2} W(t)x^{2}\;. }$$

(21.1)

Since L is quadratic, we again expand around a classical solution so that later on we will be dealing again with the calculation of the following path integral:

$$\displaystyle{ \int _{x(t_{i})\,=\,0}^{x(t_{f})\,=\,0}[dx(t)]\text{exp}\left \{ \frac{\text{i}} {\hslash }\,\frac{m} {2} \int _{t_{i}}^{t_{f} }dt\left [\left (\frac{dx} {dt} \right )^{\!2} - W(t)x^{2}\right ]\right \}\;. }$$

(21.2)

Using $x(t_{i}) = 0 = x(t_{f}),$ we can integrate by parts and obtain

$$\displaystyle{ S[x(t)] = -\frac{m} {2} \int _{t_{i}}^{t_{f} }dt\left [x(t)\frac{d^{\,2}x} {dt^{2}} + W(t)x^{2}\right ]\;; }$$

(21.3)

i.e.,

$$\displaystyle{ \int _{x(t_{i})\,=\,0}^{x(t_{f})\,=\,0}[dx(t)]\text{exp}\left \{-\frac{\text{i}} {\hslash }\,\frac{m} {2} \int _{t_{i}}^{t_{f} }dt\,x(t)\left [ \frac{d^{2}} {dt^{2}} + W(t)\right ]x(t)\right \}\;. }$$

(21.4)

Here we are dealing with a generalized Gaussian integral. In order to calculate it, we should diagonalize the Hermitean operator,

$$\displaystyle{ \frac{d^{\,2}} {dt^{2}} + W(t)\;. }$$

(21.5)

But at first we shall proceed somewhat differently. Using an appropriate transformation of variables, one can transform the action into that of a free particle.