Functional Derivative Approach

Abstract

Let us now leave the path integral formalism temporarily and reformulate operatorial quantum mechanics in a way which will make it easy later on to establish the formal connection between operator and path integral formalism. Our objective is to introduce the generating functional into quantum mechanics. Naturally we want to generate transition amplitudes. The problem confronting us is how to transcribe operator quantum mechanics as expressed in Heisenberg’s equation of motion into a theory formulated solely in terms of c-numbers. This can be achieved either by Schwinger’s action principle or with the aid of a generation functional defined as follows:

$$\displaystyle{ \langle q_{2},t_{2}\mid q_{1},t_{1}\rangle ^{Q,P} = \langle q_{ 2},t_{2}\mid T\big(\text{e}^{ \frac{\text{i}}{ \hslash } \int _{t_{1}}^{t_{2}}dt(Q(t)p(t)+P(t)q(t)) }\big)\mid q_{1},t_{1}\rangle \;. }$$

(18.1)

Here Q(t) and P(t) stand for arbitrary c-number functions (“sources”) and T denotes the time-ordering operation with respect to the Heisenberg operators q(t) and p(t). When acting on a string of operators with different time arguments it orders the operators in such a way that the time arguments increase from the right to the left.