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    Time evolution of spin

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    See attached file for equations and answer the following:

    (a) Calculate the matrix representing in the { | + >, | - > } basis, the operator H, the Hamiltonian of the system.
    (b) Calculate the eigenvalues and the eigenvalues of H
    (c) The system at the time t = 0 is in the state | _ >. What values can be found if the energy is measured, and with what probabilities?
    (d) Calculate the state vector at time t. At this instant, S, is measured; what is the mean value of the reuslts that can be obtained? Give a geometrical interpretation.

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    Solution Preview

    The magnetic field is:
    The spin operator is a vector sum as well:
    The magnetic moment of the particle is:
    Where is the mass of the electron, e is the charge and c is the speed of light. In other words we can write the magnetic moment as
    Where is a constant.
    The energy (Hamiltonian) of the interaction between the field and the spin is
    The matrix representation of the spin operators in the basis of are:
    Thus the matrix representation of the Hamiltonian is:

    The eigenvalues of the matrix are:

    Thus, the eigenvalues (energy levels) of the Hamiltonian are
    Where the ...

    Solution Summary

    This step-by-step solution demonstrates how to use vector mechanics to calculate the time evolution and expectation value of the spin.