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Now what can we do with it? We put a spin state (A) into this environment and find out what happens to it. The way to find out is to calculate the commutator, which is the matrix:

Because matrix multiplication is often different depending on the order of the matrices in the product, the commutator is a measure of whether the two matrices commute. If they commute, the order of multiplication does not matter and the commutator is zero. This means that the spin state A is a stationary state: it is happy in the environment described by the Hamiltonian and it does not change. If the commutator is not zero, then the spin state A is not stationary and will oscillate between state A and a new state B described by the commutator:

where i is the imaginary number, ^/—l. The oscillation is described by our familiar sine and cosine terms: A ^ A cos + B sin, where the argument of the cos and sin functions is the quantity in front of the Hamiltonian multiplied by t, the time variable—in this case, a (t) = Acos(yB1r) + Bsin(yB1r )

Let's find out what the spin state B is if we start with A=I? + Ib, the equilibrium state:

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