Unique Hamiltonian Operators via Feynman Path Integrals

Abstract

The old problem of how to represent uniquely a prescribed classical Hamiltonian H as a well-defined quantal operator Ĥ is shown to have a clear answer within Feynman's path-integral scheme (as expanded by Garrod) for quantum mechanics. The computation of Ĥ involves the momentum Fourier transform of a coordinate average of H. A differential equation for a reduced form of the Feynman propagator giving Ĥ from H is found; and the example of polynomial H worked out to give the Born-Jordan ordering rule for Ĥ in this case.

Exact solvability of potentials with spatially dependent effective masses

“…By the way this is even used to define the prescription to evaluate the Feynmann path integral at the mean point [25][26] [27]. The way one chooses the point to evaluate the path integral is shown to be closely related to the problem of ordering ambiguity [28] [29]. In fact the so called Weyl ordering is usually accepted by some textbooks as the correct one [30][31].…”

Observing quantum trajectories: From Mott’s problem to quantum Zeno effect and back

“…We notice that the propagator K(x, x 0 , t, t 0 ) was postulated by Garrod [19], as well as Kerner and Sutcliffe [49], though they failed to prove the estimates (25) and (26). For details we refer to de Gosson's recent works [24,25].…”

“…insertion of this value in (51) yields (49). We are now going to use equation (49) to derive formulas (47) and (48).…”

“…• In Section 2 we review the notion of short-time propagators for the Schrödinger equation using previous works of ours [21,26,28] and of Makri and Miller [52,53]; the main result is that one can replace the Van Vleck propagator -which is accurate to order ∆t 2 -with what we call the Kerner-Sutcliffe propagator [49], which is much easier to use in explicit calculations since it does not require the evaluation of the exact action integral, but only of simply computable ∆t 2 approximation thereof;…”

Unique Translation between Hamiltonian Operators and Functional Integrals

“…This criterion leads to a unique translation into a continuous functional integral. We can also take the perspective from the problem of operator ordering [14][15][16][17]. To think of the easiest example, the different operatorsxp andpx give the same zeroth order contribution in the action.…”