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Electron in a rotating magnetic field

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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 &#969;, so that at any later time t this field is at an angle &#969;t 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 &#969;.

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

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Solution Preview

The Hamiltonian for an electron in a magnetic field is given by

H = u_b B dot sigma

In this problem we have:

B = B_0 z-hat + B_1 [cos(omega t) x-hat + sin(omega t) z-hat]

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.

If we write the wavefunction as:

|psi(t)> = c_1(t) exp(-E_1 t/hbar) |+> + c_2(t) exp(-E_2 t/hbar) |->

then first order perturbation ...

Solution Summary

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.