In the paper, "Density-functional theory for time-dependent systems" Physical Review Letters 52 (12): 997 the authors mentioned that the action
$$ A= \int_{t_0}^{t_1} \mathrm dt \langle \Phi(t) | i \hbar\;\partial / \partial t – \hat{H}(t) | \Phi(t) \rangle \tag{1} $$
provides the solution of time-dependent Schrödinger equation at its stationary point. Wikipedia called (1) as the Dirac action without further reference.
If I do a variation, indeed the stationary point of action (1) gives
$$ i \hbar\;\partial / \partial t | \Phi(t) \rangle = \hat{H}(t) | \Phi(t) \rangle $$
However, from path-integral point of view, the least action principle is only a limiting case when $\hbar \rightarrow 0$. In general, there is no least action principle in quantum mechanics.
My question is, how to reconcile these two aspects? What does vary of action (1) mean?
Best Answer
There definitely is a least action principle in quantum mechanics, indeed, the path-integral method is based on it. Feynman's doctoral thesis is titled:" the least action principle in quantum mechanics". Please see, e.g., http://cds.cern.ch/record/101498/files/?ln=en