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Assuming that the operators σi describe the component i of the spin observable for a spin- 1 2 particle, and assuming this particle to be immersed in a time-independent magnetic field B~ such that the Hamiltonian of the system is given by

                    H = − µ σ · B,

with µ > 0 a real constant, show that, in the Heisenberg picture of time evolution, the evolution equation for the operator σi,H(t) is given in 3-vector form compactly by

$\frac{d \sigma_{H}}{dt} = \frac{-2 \mu}{h} B \times \sigma_{H}$

I know I have to have to use

$ \frac{d}{dt} A(t)_{H} = \frac{i}{h}[H , A(t)_{H}] + \frac{\partial}{\partial t}A(t)]_{H} $

So I get $ \frac{d}{dt} \sigma_{H}(t)_{i} = \frac{i}{h} [ H, \sigma_{H}(t)_{i}]$

= $ [H\sigma_{H}(t)_{i} - \sigma_{H}(t)_{i} H ]$

From here,

$ H U^{+} \sigma_{i} U - U^{+} \sigma_{i} U H$

= $ U^{+} [H , \sigma_{i}]U $

Dont know how to proceed from here, any help appreciated!

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    Heisenberg loved the Baby Blue. – 2017-01-14
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    Well you need to insert the definition of H for one thing, also you are missing the partial derivative term – 2017-01-14

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