By Ambegaokar V.
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Extra resources for Quantum Brownian Motion and Its Classical Limit
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 .
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Quantum Brownian Motion and Its Classical Limit by Ambegaokar V.