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

Andrzej Korzeniowski ()
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Andrzej Korzeniowski: University of Texas at Arlington

Methodology and Computing in Applied Probability, 1999, vol. 1, issue 3, 277-282

Abstract: 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 $$H = A + V$$ , 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 $$ - V$$ 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 (search for similar items in EconPapers)
Date: 1999
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DOI: 10.1023/A:1010082327048

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