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RN , t): ⎛ ⎞ N i ˜ 1 , r2 , . . , rN , t) Ψ(r1 , r2 , . . , rN , t) = exp ⎝ θ(rj , t)⎠ Ψ(r (46) j=1 where rj denotes the position of particle j in the laboratory frame and rj its position in the moving frame, with constant velocity v0 given by: v0 = E∧B . B2 (47) Therefore, we have the relation: rj = rj − v0 t. It is possible to choose the phase ˜ satisfies the time-dependent Schr¨odinger equation θ(r, t) in such a way that Ψ ˜ deduced from H by removing the associated to the simplified Hamiltonian H potential term −e j V (rj ).

The Landau level index n plays the same role as the representation index j in the rotation group, and it can be recovered as the eigenvalue of a Casimir operator: C = 2b+ b/q + L. The states within each Landau level can be labeled by their angular momentum m ≤ n. When two particles of positive charge q1 and q2 are restricted to their respec+ tive lowest Landau level, we can form the operators b+ = b+ 1 + b2 , b = b1 + b2 and the total angular momentum L = L1 + L2 . These operators obey the commutation relations of the algebra (83) with the charge q = q1 + q2 .

M. Hong, Phys. Rev. Lett. 62, 2724 (1989). U. D. Stone, Phys. Rev. Lett. 63, 414 (1989). J. Beenakker, and H. van Houten, Phys. Rev. Lett. 63, 1857 (1989). B. I. I. Glazman, Phys. Rev. B 46, 4026 (1992). [26] E. Ahlswede, P. Weitz, J. Weis, K. von Klitzing, and K. Eberl, Physica B 298, 562 (2001). -G. Wen, Int. Journ. Mod. Phys. B 6, 1711 (1992). V. J. Goldman, e-preprint, cond-mat/0110193. [29] M. B¨ uttiker, P. V. Sukhorukov, e-preprint, condmat/0404123. M. Haldane, Chap. [1]. I. K. Jain in ref.

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Quantum Brownian Motion and Its Classical Limit by Ambegaokar V.


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