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Rotations

< Spin precession with the density matrix | Index | Quantum treatment of the rotating frame >

It is useful to find the quantum expression for the operator describing the spin projection along a rotating axis. We can look back at the expression that we used to calculate the spin precession in the density matrix description of the previous section, below eq. (1)

{$ \langle I_x(t) \rangle = Tr(e^{i\omega t I_z}\rho(0) e^{-i\omega t I_z} I_x)$}

which was shown to be equal to a cosine oscillation in time. Since the trace is invariant under cyclic permutations of its operator factors the same result is valid for the quantity {$Tr(\rho(0) e^{-i\omega t I_z} I_x e^{i\omega t I_z})$}, which may be read as the instantaneous value of the new, time dependent operator

{$I_{x'} (t) = e^{-i\omega t I_z} I_x e^{i\omega t I_z}$}

assuming that the state is described by a stationary {$\rho(0)$}, representing a spin along {$ \hat x$}. In order for its expectation value to vary as {$\cos \omega t$} this operator must be the spin projection along an axis, {$ \hat{x'\,}$}, counter-rotating around {$\hat z$} with angular velocity {$\omega$}. Therefore we may write:

{$e^{-i\omega t I_z} I_x e^{i\omega t I_z}=I_x \cos \omega t + I_y \sin \omega t $}

It is evident that this description may provide a quantum analogue of the rotating frame.

< Spin precession with the density matrix | Index | Quantum treatment of the rotating frame >