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Quantum Primer
Published in Thomas M. Nordlund, Peter M. Hoffmann, Quantitative Understanding of Biosystems, 2019
Thomas M. Nordlund, Peter M. Hoffmann
The expression of the uncertainty principle in terms of momentum uncertainty Δp and position uncertainty Δx says that if we confine a particle to a small region Δx, the matter wave will have a large range of momentum components: Δp is large. If the confinement of the particle is somehow released, an initially more confined particle will expand more rapidly because of its larger range of momentum components.
Vector vortex solitons in two-component Bose–Einstein condensates with modulated nonlinearities and a harmonic trap
Published in Journal of Modern Optics, 2018
Si-Liu Xu, Ze-Qiang Wang, Jun-Rong He, Li Xue, Milivoj R. Belić
From this solution, one observes the essential difference of the two-component soliton solutions, as compared to the single-component solitons: It is contained in the square-root term that describes the dependence on the component-specific Gij coefficients in the stationary NLSEs for the components. This difference makes the existence of pairs of solitons possible, and provides for novel matter wave solutions in BECs. Now, we demonstrate that Equation (6) can support any number of exact vector pairs with the localized nonlinearities. Obviously, 0 ≤ R ≤ ∞ as r varies in the whole domain. When considering the localization of gij, ρ must behave as r−n with as r → 0, and diverge as r → ∞. To obtain localized solutions of Equation (2), we consider the boundary condition ψ(r → 0) = ψ(r → ∞) = 0 for the localized solution, so that κ should satisfy , where is the complete elliptic integral of the first kind, and n = 2, 4, 6,… is an even integer denoting the radial quantum number. Thus, as long as one chooses the appropriate values of parameters m, n, q, a1, a2 and η, one obtains the exact localized nonlinear vector solutions of the coupled 2D GPEs (2).