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Path-Integral Calculations for the First Excited State of Hydrogen Atom

Authors:	Korzeniowski Andrzej

Institution:	(1) Department of Mathematics, University of Texas at Arlington, USA

Abstract:	Consider a path-integral $E_x e^{\int_0^t {V(X)(s))ds} } \varphi (X(t))$ which is the solution to a diffusion version of the generalized Schro¨dinger's equation $\frac{{\partial u}}{{\partial t}} = Hu,u(0,x) = \varphi (x)$ . Here , where A is an infinitesimal generator of a strongly continuous Markov Semigroup corresponding to the diffusion process $\{ X(s),0 \leqslant s \leqslant t,X(0) = x\}$ . For $A = \frac{1}{2}\Delta$ and V replaced by one obtains $\overline H = - H = - \frac{1}{2}\Delta + V$ , which represents a quantum mechanical Hamiltonian corresponding to a particle of mass 1 (in atomic units) subject to interaction with potential V. This paper is concerned with computer calculations of the second eigenvalue of $- \frac{1}{2}\Delta - \frac{1}{{\sqrt {x^2 + y^2 + z^2 } }}$ by generating a large number of trajectories of an ergodic diffusion process.

Keywords:	Brownian motion ergodic diffusion Schro¨ dinger's equation second eigenvalue
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