Homework for Spins

  1. Have students complete the theory section for the third section of Spins Lab 2.
  2. (SpinOneInterferometerBrief)

  3. (SpinOneInterferometer) This problem is similar to problem 1 above, but with the spin-1 system. This problem is an extension of section 3 from Spins Lab 3.

    Consider a spin 1 interferometer which prepares the state as $\vert 1\rangle$, then sends this state through an $S_x$ apparatus and then an $S_z$ apparatus. Measure the relative probabilities after the final Stern-Gerlach analyzer for the seven possible cases where one beam, a pair of beams, or all three beams from the $S_x$ Stern-Gernach analyzer were used. Compare the simulation results to theory. Make sure that, for the theory section, you explicitly discuss your choice of projection operators.

  4. (LinearOp)

    Show that the matrix $$A\doteq\pmatrix{a_{11} & a_{12}\cr a_{21}& a_{22}}$$ is a linear operator on the space of all vectors, i.e.:

    1. Show that $$A\left(\vert v_1\rangle + \vert v_2\rangle\right) =A\vert v_1\rangle + A\vert v_2\rangle $$

    2. Also show that $$A\left(\lambda\vert v_1\rangle\right) =\lambda A\vert v_1\rangle $$


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