Path integrals with arbitrary generators and the eigenfunction problem

Abstract

We generalize the path integral formalism of quantum mechanics to include the use of arbitrary infinitesimal generators, thus providing explicit expressions for solutions of a wide class of differential equations. In particular, we develop a method of calculating the eigenfunctions of a large class of operators.

Time-dependent invariants and Green functions in the probability representation of quantum mechanics

“…We show here a derivation of this result, which is essentially an extension of previous techniques [25][26][27]29,28]. In the mentioned papers the time-dependent invariants of a given system were shown to be in connection with the wave function and with the Green function of the Schrödinger equation.…”

“…In [26,27] it was shown that the Green function G(q, q ′ , t) is a solution of the system I q G(q, q ′ , t) = q′ G(q, q ′ , t) (II.20)…”

“…Also, we give a new definition of coherence, solely in terms of properties of the MDF, which relates the coherent marginal distribution function to invariants of the quantum system. In this regard let us note that the general approach to time-dependent invariants in quantum mechanics and their relation to the wave function were elucidated by Lewis and Riesenfeld in [25], while the connection among integrals of the motion and the quantum propagator was found in [26,27]. In [28] the relation of time-dependent invariants to the Schwinger action principle was established, and in [29] the relation with the Nöther theorem was discussed.…”