Homework for Spins

  1. (Euler) This problem should be review for most students, but it's good to be sure that each knows Euler's formula and can recognize cosine and sine in complex polar form.

    Use Euler's formula $e^{i\theta}=\cos\theta+i\sin\theta$ and its complex conjugate to find formulas for $\sin\theta$ and $\cos\theta$. In your physics career, you will often need to read these formula “backwards,” i.e. notice one of these combinations of exponentials in a sea of other symbols and say, “Ah ha! that is $\cos\theta$.” So, pay attention to the result of the homework problem!

  2. (UnknownsOneHalf) This activity is an extension of section 1 from Spins lab 2. This activity will give students practice finding unknown states using only the square of the norm of inner products between two states.

    With the Spins simulation set for a spin 1/2 system, measure the probabilities of all the possible spin components for each of the unknown initial states $\ket{\psi_i}$ $\left( i=1,2,3,4\right)$.

    1. Use your measured probabilities to find each of the unknown states as a linear superposition of the $S_z$-basis states $\ket{+}$ and $\ket{-}$.

    2. Articulate a Process: Write a set of general instructions that would allow another student in next year's class to find an unknown state from measured probabilities.

    3. Compare Theory with Experiment: Design an experiment that will allow you to test whether your prediction for each of the unknown states is correct. Describe your experiment here, clearly but succinctly, as if you were writing it up for a paper. Do the experiment and discuss your results.

  3. (SpinOneUnknown) This problem should only be performed after students have worked with spin-1 systems. This question is also an extension of section 2 of Spins lab 3.

    Using the Spin 1 version of the Spins simulation, find the probabilities for the results of the three spin components for unknowns $\ket{\psi_1}$ and $\ket{\psi_3}$. Use these probabilities to write the unknowns in the $z$-basis.

  4. (ChainedSG)


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