Quantum Fundamentals: NoTerm-2022
HW 6 : Due Day 15 F 2/25

1. Completeness Relation Change of Basis S0 4369S

   1. Given the polar basis kets written as a superposition of Cartesian kets \begin{eqnarray*} \left|{\hat{s}}\right\rangle &=& \cos\phi \left|{\hat{x}}\right\rangle + \sin\phi \left|{\hat{y}}\right\rangle \\ \left|{\hat{\phi}}\right\rangle &=& -\sin\phi \left|{\hat{x}}\right\rangle + \cos\phi \left|{\hat{y}}\right\rangle \end{eqnarray*}

      Find the following quantities: \[\left\langle {\hat{x}}\middle|{\hat{s}}\right\rangle ,\quad \left\langle {\hat{y}}\middle|{{\hat{s}}}\right\rangle ,\quad \left\langle {\hat{x}}\middle|{\hat{\phi}}\right\rangle ,\quad \left\langle {\hat{y}}\middle|{\hat{\phi}}\right\rangle \]

   2. Given a vector written in the polar basis \[\left|{\vec{v}}\right\rangle = a\left|{\hat{s}}\right\rangle + b\left|{\hat{\phi}}\right\rangle \] where \(a\) and \(b\) are known. Find coefficients \(c\) and \(d\) such that \[\left|{\vec{v}}\right\rangle = c\left|{\hat{x}}\right\rangle + d\left|{\hat{y}}\right\rangle \] Do this by using the completeness relation: \[\left|{\hat{x}}\right\rangle \left\langle {\hat{x}}\right| + \left|{\hat{y}}\right\rangle \left\langle {\hat{y}}\right| = 1\]
   3. Using a completeness relation, change the basis of the spin-1/2 state \[\left|{\Psi}\right\rangle = g\left|{+}\right\rangle + h\left|{-}\right\rangle \] into the \(S_y\) basis. In otherwords, find \(j\) and \(k\) such that \[\left|{\Psi}\right\rangle = j\left|{+}\right\rangle _y + k\left|{-}\right\rangle _y\]
2. Spin One Intro S0 4369S

   The OSP Spins Laboratory simulation can also be used to explore spin-1 systems. The components of spin for these systems can be measured to be:

   \(\hbar\) (corresponding to the “+” port)

   \(0\hbar\) (corresponding to the “0” port)

   \(-\hbar\) (corresponding to the “-” port)

   

   To switch the simulation to a spin-1 system, find the hyperlink about halfway down the page that says “Click here to switch”.

   1. Draw and label a diagram of an experimental setup that would allow you to prepare a set of spin-1 particles to be in the \(|1\rangle_x\) state and than measure the \(z\) component of spin for these particles.

   2. Using the simulation, prepare a set of particles to be in the \(|1\rangle_x\) state and measure the \(x\), \(y\), and \(z\) components of spin of these particles. Draw probability histograms of the results for each spin-component-direction \(S_x\), \(S_y\), and \(S_z\).

3. General State S0 4369S

   Use a New Representation: Consider a quantum system with an observable \(A\) that has three possible measurement results: \(a_1\), \(a_2\), and \(a_3\). States \(\left|{a_1}\right\rangle \), \(\left|{a_2}\right\rangle \), and \(\left|{a_3}\right\rangle \) are eigenstates of the operator \(\hat{A}\) corresponding to these possible measurement results.

   1. Using matrix notation, express the states \(\left|{a_1}\right\rangle \), \(\left|{a_2}\right\rangle \), and \(\left|{a_3}\right\rangle \) in the basis formed by these three eigenstates themselves.

   2. The system is prepared in the state:

      \[\left|{\psi_b}\right\rangle = N\left(1\left|{a_1}\right\rangle -2\left|{a_2}\right\rangle +5\left|{a_3}\right\rangle \right)\]

      1. Staying in bra-ket notation, find the normalization constant.
      2. Calculate the probabilities of all possible measurement results of the observable \(A\). Check “beasts.”
      3. Use a New Representation: Plot a histogram of the predicted measurement results.

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

      \[\left|{\psi_c}\right\rangle = N\left(2\left|{a_1}\right\rangle +3i\left|{a_2}\right\rangle \right)\]

      1. Write this state in matrix notation and find the normalization constant.
      2. Calculate the probabilities of all possible measurement results of the observable \(A\). Check “beasts”.
      3. Use a New Representation: Plot a histogram of the predicted measurement results.

4. Spin One Interferometer Brief S0 4369S

   Consider a spin 1 interferometer which prepares the state as \(| 1\rangle\), then sends this state through an \(S_x\) apparatus and then an \(S_z\) apparatus. For the four possible cases where a pair of beams or all three beams from the \(S_x\) Stern-Gernach analyzer are used, calculate the probabilities that a particle entering the last Stern-Gerlach device will be measured to have each possible value of \(S_z\). Compare your theoretical calculations to results of the simulation. Make sure that you explicitly discuss your choice of projection operators.

   Note: You do not need to do the first case, as we have done it in class.