An electron is in a strong, static, homogenous magnetic field with magnitude B0 in the z direction. At time t=0, the spin of the electron is in the +z direction. At t=0 a weak, homogenous magnetic field with magnitude B1 (where B1 << B0) is turned on. At t=0 this field is pointing in the x direction, but it rotates counterclockwise in the x-z plane with angular frequency ω, so that at any later time t this field is at an angle ωt relative to the x-axis.

Calculate the probability that at a later time tf the electron spin has flipped to the -z direction. Do not assume anything about the particular value of ω.

Please see the attached PDF file for question along with a diagram.

B_1 is much smaller than B_0, so we can treat the terms proportional to B_1 in the Hamiltonian as a perturbation.

The eigenstates of the unperturbed Hamiltonian are the spin up and spin down states in the z-direction, denoted as |+> and |->, respectively. The energy eigenvalues are E_1 = u_b B_0 and E_2 = -u_b B_0, respectively.

We apply time dependent perturbation theory to compute the probability of a spin flip due to a rotating magnetic field. All steps are explained in detail.

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