Quantum Mechanics II

Asymptotic formula for the quantum entropy of position in energy eigenstates

“…coincides with that obtained by means of the theory of strong asymptotics of Freud orthogonal polynomials, which include Hermite polynomials as a particular case [ 14,15], and also agrees with the exact numerical values of SQ for n » I [ 12,15] . An independent derivation of Eq.…”

“…( 4) has been checked for three simple systems, namely the particle in a box (where the asymptotic formula becomes an exact identity), the linear potential, and the harmonic oscillator. In particular, the asymptotic formula for the harmonic oscillator ( 34) coincides with that obtained by means of the theory of strong asymptotics of orthogonal polynomials [ 14,15]. This result strongly suggests the mathematical equivalence of these two seemingly very different approaches, although the problem of finding a rigorous proof of this equivalence remains open.…”

“…on the whole real axis [ 14,15], but no general result of this kind has been found up to now for arbitrary quantum systems. In this Letter we show that, in the simplest case of a one-dimensional system whose Hamiltonian is of the form p2 H= -…”

“…whose leading term is the WKB approximate value given by ( 14) [ 4]. The resulting numerical values of the classical and quantum entropies, as well as their difference, are displayed for several values of n in Table 1.…”

Curvature of the total electron density at critical coupling: attractive impurity in an electron gas

“…Nevertheless, to prove this cancellation for the general case of an arbitrary finite-range central potential λV(r), we need an auxiliary lemma to express the bound state contribution in terms of the Jost functions. We know [21,23] …”