How Does a Fluctuating Hamiltonian Affect the Expectation Value of Sx?

[Niles]

Homework Statement

Hi

Say I have a Hamiltonian given by H = δS_z acting on my system, where δ is a random variable controlled by some fluctuations in my environment. I have to show that if I start out with <S_x>=½, then the Hamiltonian will reduce <S_x> to

<S_x> = ½<cos(δt)>

where the <> around the cosine means averaged over all values of δ. What I would do is to use

<e^iHtS_x(0)e^-iHt> = <S_x(t)>

but this seems very tedious. Am I on the right path here?


Niles.

 

[Klockan3]

Isn't the easiest way to just get the eigenfunctions of S_z and then project S_x on that?

 

[ideasrule]

where the <> around the cosine means averaged over all values of δ. What I would do is to use

<e^iHtS_x(0)e^-iHt> = <S_x(t)>

but this seems very tedious. Am I on the right path here?

Yes, but it shouldn't be a tedious computation. The expectation value of that time-evolved operator is just equal to x^H*Sx*x, where x is the time-evolved state vector. x is a 2x1 vector (written using the eigenvectors of H as a basis, because that's convenient) and Sx is a 2x2 matrix, so the computation shouldn't be too hard.