Homework for Spins

  1. (GeneralState) This problem challenges students to make connections between an arbitrary state and the possible resulting measurements that can be made. This problem can be quite tricky at the start for some students because the state given is not normalized. If you feel this is unfair, warn the class ahead of time.

    Use a New Representation: Consider a quantum system with an observable $A$ that has three possible measurement results: $a_1$, $a_2$, and $a_3$.

    1. Write down the three kets $\ket{a_1}$, $\ket{a_2}$, and $\ket{a_3}$, corresponding to these possible results, using matrix notation.

    2. The system is prepared in the state:

      $$\ket{\psi} = 1\ket{a_1}-2\ket{a_2}+5\ket{a_3}$$

      Staying in bra-ket notation, calculate the probabilities of all possible measurement results of the observable $A$. Plot a histogram of the predicted measurement results.

    3. In a different experiment, the system is prepared in the state:

      $$\ket{\psi} = 2\ket{a_1}+3i\ket{a_2}$$

      Write this state in matrix notation and calculate the probabilities of all possible measurement results of the observable $A$. Plot a histogram of the predicted measurement results.

  2. (SpinOneIntro) This problem serves as an introduction to the spin-1 system and shows students how the probabilities between different spin states in the x,y, and z directions typically look.

    Using the Spins simulation, follow the link to switch to the \texttt{Spin-1} case. Set up an experiment for two successive measurements of spin projections.

    1. Measure the probability that a state which starts out with $z$-component of spin equal to $\hbar$ ends up with $z$-component of spin equal to $\hbar$ after the $z$-component of spin is measured. Write your statement in bra-ket language.

    2. Measure the probability that a state which starts out with $z$-component of spin equal to $\hbar$ ends up with $z$-component of spin equal to zero after the $z$-component of spin is measured. Write your statement in bra-ket language. What does this probability tell you about the $z$ basis?

    3. Measure the probability that a state which starts out with $x$-component of spin equal to zero ends up with $z$-component of spin equal to zero after the $z$-component of spin is measured. Write your statement in bra-ket language. What does this probability tell you about the $x$ and $z$ bases?

    4. Use your simulation to find the value of $\vert\braket{1}{-1}_x\vert^2$. State in words what the measured quantity represents. Compare your “measured” value to a theoretical value computed from the Spin Reference Sheet.

  3. (OperatorsSpinThreeHalves)

    If a beam of spin-3/2 particles is input to a Stern-Gerlach analyzer, there are four output beams whose deflections are consistent with magnetic moments arising from spin angular momentum components of $\frac{3}{2}\hbar$, $\frac{1}{2}\hbar$, $-\frac{1}{2}\hbar$, and $-\frac{3}{2}\hbar$. For a spin-3/2 system:

    1. Write down the eigenvalue equations for the $S_z$ operator.

    2. Write down the matrix representation of the $S_z$ eigenstates.

    3. Write down the matrix representation of the $S_z$ operator.

    4. Write down the eigenvalue equations for the $S^2$ operator. (The eigenvalues of the $S^2$ are $\hbar^2s(s+1)$, where $s$ is the spin quantum number, and $S^2$ is proportional to the identify operator.)

    5. Write down the matrix representation of the $S^2$ operator. Check Beasts: Is your operator proportional to the identity operator?


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