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Journal of Mathematical Sciences

A Probabilistic Approximation of the Evolution Operator exp (t (S∇, ∇)) with a Complex Matrix S

Il'dar Abdullovich Ibragimov, Nataliya V. Smorodina, Михаил Михайлович Фаддеев

Published Jan 9, 2020• 0 citations
Publisher
DOI
10.1007/s10958-020-04652-0

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The coordinate matrix element of the time evolution operator, exp[−iĤt/ℏ], is determined by expanding (its exponent) in a power series in t. Recursion relations are obtained for the expansion coefficients which can be analytically evaluated for any number of degrees of freedom. Numerical application to the tunneling matrix element in a double well potential and to the reactive flux correlation function for a barrier potential show this approach to be a dramatic improvement over the standard short time approximation for the propagator. Its use in a Feynman path integral means that fewer ‘‘time slices’’ in the matrix product exp[(−i/ℏ)ΔtĤ]N, Δt=t/N, will be required. The first few terms in the present expansion constitute a fully quantum version of the short time propagator recently obtained by us using semiclassical methods [Chem. Phys. Lett. 151, 1 (1988)].
In Part II of this note, we argued that if one describes a bound particle in a potential in a statistical manner, like classical statistical mechanics, one is led to a complex, periodic space invariant probability, exp(ipx), for a free particle. We showed that exp(ipx) represents a two vector solution to the vector operated on by a rotation matrix (using cos(px), -sin(px)) set equal to a translation operator 1+delta X d/dx operating on the same vector. In this note, we examine the electromagnetic scenario. The idea of a “wave” describing a photon first emerged with Maxwell and his equations in the 1800s, with interference/diffraction and the speed of light both appearing. It was later found (Einstein and the photoelectric effect) that electromagnetic energy is not spread out all over space like a wave, but is concentrated. Furthermore, if one uses probabilistic ideas to obtain exp(ipx) for a quantum particle, like an electron, which undergoes two slit interference, then why would not two slit interference for photons also be considered in terms of probability? In this note, we examine these ideas.