Homework for Spins

  1. (ExpectationValue)

    A beam of spin-1 particles is prepared in the state $$\ket{\psi}=\frac{2}{\sqrt{29}}\ket{1}+i\frac{3}{\sqrt{29}}\ket{0}-\frac{4}{\sqrt{29}}\ket{-1}.$$

    1. What are the possible results of a measurement of the spin component $S_z$, and with what probabilities would they occur?

    2. What are the possible results of a measurement of the spin component $S_x$, and with what probabilities would they occur?

    3. What are the possible results of a measurement of the spin component $S_x$, and with what probabilities would they occur?

  2. (Magnet) This is a more advanced problem that focuses on the change of probabilities after time evolutions and changes of state due to a magnetic field.

    Consider a spin-1/2 particle with a magnetic moment. At time $t=0$, the state of the particle is $\ket{\psi(t=0)}=\ket{+}$.

    1. If the observable $S_x$ is measured at time $t=0$, what are the possible results and the probabilities of those results?

    2. Instead of performing the above measurement, the system is allowed to evolve in a uniform magnetic field $\vec{B}=B_0\, \hat y$. Calculate the state of the system after a time $t$ using the $S_z$ basis.

    3. At time $t$, the observable $S_x$ is measured. What is the probability that a value $\hbar$/2 will be found?

    4. Using Mathematica (or some other computer software), plot the probability of measuring $S_x = {\hbar \over 2}$ as a function of time.

      Example of Mathematica code is:

      \includegraphics[scale=0.5]{\TOP Figures/magnet.png}

      Your plot should have a title and the axes should be labels and have units. Be sure to include the code you used to create your graph.

      Describe the meaning of your graph in words. How does the probability function behave over time?

  3. (Frequency) This problem is initially tricky because the state given is not in the basis of the Hamiltonian, requiring a projection onto the Hamiltonian basis. Also, the problem gives little information and requires students to keep everything in general. Oftentimes students will get stuck looking for too much information, when all that's important is finding the relative phase in the expectation value for A.

    Consider a two-state quantum system with a Hamiltonian \begin{equation} \hat{H}\doteq\pmatrix{E_1&0\cr 0&E_2} \end{equation} Another physical observable $A$ is described by the operator \begin{equation} \hat{M}\doteq\pmatrix{0&c\cr c&0} \end{equation} where $c$ is real and positive. Let the initial state of the system be $\ket{\psi(0)}=\ket{m_1}$, where $\ket{m_1}$ is the eigenstate corresponding to the larger of the two possible eigenvalues of $\hat{M}$. What is the frequency of oscillation of the expectation value of $M$? Compare this frequency to the Bohr frequency.

  4. (Expectation) This is a well-rounded problem that has students perform a little of everything. Students will have to find the eigenvalues and eigenvectors of the hamiltonian, time evolve, calculate probabilities, and find an expectation value.

    A quantum mechanical system starts out in the state: \begin{eqnarray} \ket{\psi(0)}=C\left(3\ket{a_1}+4\ket{a_2}\right) \end{eqnarray} where $\ket{a_i}$ are the normalized eigenstates of the operator $\hat{A}$ corresponding to the eigenvalues $a_i$. In this $\ket{a_i}$ basis, the Hamiltonian of this system is represented by the matrix: \begin{equation} \hat{H}\doteq E_0\, \pmatrix{2&1\cr 1&2} \end{equation}

    1. What is the state of the system at some later time $t$?

    2. If you measure the energy of this system, what values are possible, and what are the probabilities of measuring those values?

  5. (Uncertainty)

    Consider the state $\vert -1\rangle_y$ in a spin 1 system.


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