Dresden 2006 – wissenschaftliches Programm
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DY: Dynamik und Statistische Physik
DY 22: Quantum Dynamics II
DY 22.1: Vortrag
Dienstag, 28. März 2006, 11:00–11:15, H\"UL 186
"Hermite" states in the quantum interaction of vortices — •Alexey Romanov, Chukbar Konstantin, and Zaburdaev Vasiliy — Russian Research Center ’Kurchatov Institute’, pl. Kurchatova 1, 123182 Moscow
In this paper, we consider transition from classical dynamics of vortices to quantum. Problem of two identical cinematic vortices (each vortex produce 2d velocity field with current function ψ(|r|)) reduces to the Hamiltonian system with Hamilton function H(q,p)= ψ(√q2+p2) (q=x, p=y). We perform transition to quantum vortices dynamic according to the standard rule: q→ q, p→ −iℏ∂/∂ q. Now we start to solve quantum problem with Hamilton operator Ĥ = ψ( √r2 ),r2 = − ℏ 2 ∂ 2 /∂ q2 + q2 . Operator r2 corresponds to quantum oscillator with Hermite eigenfunction, and eigenvalue spectrum (1,3,..,2n+1). So Hamiltonian Ĥ, which describes dynamic of quantum vortex, has Hermite eigenfunctions and eigenvalue spectrum: ψ ( √2n + 1 ). Quantum oscillator has, so called, coherent states, which is stable during quantum evolution. Vortex Hamiltonian doesn’t have such states, because of dispersion of angular frequency. Also we consider transition for system with anisotropic current function ψ = Ax2 − y2 /( x2 + y2 )2 .