The time-dependent variational principle (TDVP)

Abstract

In its usual derivation, the time-dependent variational principle is formulated as stemming from an action functional given by t 2 S = I dt L'(9,~) (2.1) t 1 where the Lagrangian is taken as L' (¢,~) = The Dirac bracket in the above formula is meant to imply integration over all the degrees of freedom of the system (usually position, spin and isospin coordinates). This principle is appropriate when the wave function I¢(t)> is also required to be normalized at all times. In that case, iS/~t corresponds to an hermitian operator and therefore, the Lagrangian L' and the action S are real. It is easy to show that arbitrary independent variations of S with respect to 19> and <91 yield the time-dependent SchrSdinger equation (and its complex conjugate) which can be interpreted as *) functional Hamiltonian equations for the fields It(t)> and <9(t)l. For the applications that we have in mind , it will be eonvenien to lift the restriction on the normalization of the wave function. In that case, the Lagrangian L' becomes complex. Although this is not an essential complication, because the imaginary part occurs through a total time derivative that has no effect on the variational principle, it is more convenient to have a real Lagrangian because otherwise the time-dependent phase and normalization become inextricably mixed in with the wave function. In order to achieve equations of motion that are independent of the phase and normalization of the wave function, we define the new real Lagrangian i <915>~ ~ (2.2) L(9,~) = 2 <919> " <919> ~) Compare FR 34 p.253