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Confinamiento magnético 185Aharonov-Bohm effect for pedestrian 229and then, eq.21 byˆP 12 Ψ n (r) = (−1) (M+n+1) Ψ n (r), (24)so that if M + n is even/odd |M,n〉 will be triplet/singlet. Then, from eq. 15,we see that if F = 0 the lowest M = 0 will combine with n = 0 yielding a tripletground state. At F = 1/2 it is M = −1 which combines with n = 0 yielding asinglet state, etc. Therefore, we find out again the same picture as in QR exceptfor a shift of half flux unit. The similarities between the 1D QR and AQR remainif we consider H ξ , eq. 22.5. Optic Aharonov-Bohm effect: excitonsAn exciton in a QR is a neutral entity. Then, it should not be sensitive to theapplied magnetic flux. However different masses of electrons and holes yield observableeffects in realistic 3D QR. Namely, dark exciton in some windows of magneticfield [11]. This is the so called optical AB effect [19]. Romer and Raikh [20]employed a short-range e-h attractive potential in a 1D QR and conclude that theAB effects will be present if electron and hole can tunnel in the opposite directionsand meet each other on the opposite side of the ring. However it seems that actualCoulomb terms prevent the ground state oscillations in 1D QR [21–23].The Hamiltonian of an electron and a hole in a 1D QR pierced by a magneticfield reads:Ĥ = − 1 ( ) ∂ 2 ( )1 ∂ 212m ∗ eR 2 + iF −∂φ e 2m ∗ − iF −h R2 ∂φ h 2R|sin φe−φ (25)h2|where m e , m h are the electron/hole effective masses, both considered positive inthis model. If we disregard by the time being the Coulomb attraction, Ψ(φ e ,φ h ) =e i Me φe e i M h φ his the eigenfunction associated to the eigenvalue:λ = E − E g =12m ∗ eR 2(M e + F) 2 1+2m ∗ h R2(M h − F) 2 , (26)where E g is the electron-hole energy gap and M e , M h = 0 ±1±2... The E vs. Fplot shows periodic changes of ground state (M e ,M h ) = (0,0),(−1,1),(−2,2) . ..However, M L = M e +M h is always zero. Then, the selection rule M L = 0 is fulfilledand there are not dark windows for luminescence, i.e., no optic AB effect can beseen. If we take into account that electron and hole have different effective masses,we may think that in a real QR electron and hole will follow different orbits. Avery simple model of a 2D QR where electron and hole follow circular orbits withradii R e ≠ R h pierced by a magnetic field (including the region where the system islocated) has been recently proposed [19]. This allows to have different flux insidethe electron and hole orbits: F e = πReB/Φ 2 0 , F h = πRh 2B/Φ 0. As a result, eq. 26turns into1E = E g +2m ∗ eRe2 (M e + F e ) 2 1+ (M h − F h ) 2 , (27)2m ∗ h R2 h