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THEORETICAL PHYSICS
SEMINAR
Monday, November 12, 2007
12:00 pm, CP
179
Dr. ALEXANDER TURBINER
CAN SEMICLASSICAL APPROXIMATION
BE
MODIFIED TO STUDY THE GROUND STATE?
ABSTRACT
Semiclassical
(WKB) approximation is one of main methods to study the Schroedinger
equation. Usually, their applicability is limited to highly-excited
states. It is proposed a simple approach based on a combination of WKB
approximation at large distances with perturbation theory at small
distances. It allows to construct uniform approximation of the ground
state eigenfunction for the anharmonic oscillator (AHO) V= m^2 x^2+
x^4 with single well (m^2 > 0) and for the double-well potential
(m^2 < 0). It is shown that if this approximation is treated as
unperturbed problem it leads to an extremely fast convergent
perturbation theory. A possible connection to recent remarkable results
by Eremenko-Gabrielov-Shapiro about complex zeroes of AHO
eigenfunctions is mentioned.
A generalization to different one-dimensional and multidimensional AHO
as well as to the problem of hydrogen in a magnetic field is discussed.
Everyone is welcome