Quantum Fundamentals: NoTerm-2023
HW 3 : Due Day 8 W 2/22 Mathbits

1. Histogram S0 4634S A beam of spin-\(\frac{1}{2}\) particles is prepared in the state: \[\left\vert \psi\right\rangle = \frac{2}{\sqrt{13}}\left\vert +\right\rangle+ i\frac{3}{\sqrt{13}} \left\vert -\right\rangle\]
   1. What are the possible results of a measurement of the spin component \(S_z\), and with what probabilities would they occur? Check Beasts: Check that you have the right “beast.”
   2. What are the possible results of a measurement of the spin component \(S_x\), and with what probabilities would they occur? Check Beasts: Check that you have the right “beast.”
   3. Use Another Representation: Plot histograms of the predicted measurement results from parts \((a)\) and \((b)\). Your histogram should be a vertical bar chart with the value of the spin component on the horizontal axis and the probability on the vertical axis.
2. Chained Stern-Gerlach S0 4634S

   Consider the Stern-Gerlach set-up shown, with a thermal oven” source, and some state vectors associated with various analyzer outputs:
   

   1. How many particles were released from the oven? Calculate it exactly but also make a more conceptual/quick way to determine the (approximate, likely) answer. Describe both ways.

   2. Write down Dirac notation expressions for \(|\Psi_a \rangle\), \(|\Psi_b \rangle\), \(|\Psi_c \rangle\) and \(|\Psi_d \rangle\) in the \(S_z\)-basis.

   3. Student A says, “While going through the y-analyzer (Analyzer 2), the y-analyzer changes its incoming particles from being all z+ to an output which is a mixture of half of the particles being in the z+ state and half being in the z- state. That's why the farthest right z-analyzer (call it Analyzer 3) measures half z+ and half z-.”
      

      Student B says, “No that's not right, all the particles entering Analyzer 3 are in the same state: they are in both the z+ and z- at the same time, they are all in a state that is a specific superposition of z+ and z- states, which is \(|\Psi_b\rangle\).”
      

      Student C says, “There's no way to tell the difference between those two cases. So your argument doesn't matter in practice.”
      

      Student D states, “Student B is right (sorry for the formality, B), and A and C are wrong, especially C.”
      

      Describe a simple experiment (or experiments) and use a clear logical argument to show that Student D is correct (including the idea that C is wrong). Draw diagrams to make your case easier to comprehend. (Hint: Figure 1.12 in McIntyre might be helpful.)

3. Finding Orthogonal Vectors (Brief Version) S0 4634S

   Consider the quantum state: \[\left\vert \psi\right\rangle = \frac{1}{\sqrt{3}}\left\vert +\right\rangle+ i\frac{\sqrt{2}}{\sqrt{3}} \left\vert -\right\rangle\]

   Find the normalized vector \(\vert \phi\rangle\) that is orthogonal to it.

4. Phase in Quantum States S0 4634S

   In quantum mechanics, it turns out that the overall phase for a state does not have any physical significance. Therefore, you will need to become quick at rearranging the phase of various states. For each of the vectors listed below, rewrite the vector as an overall complex phase times a new vector whose first component is real and positive. \[\left|D\right\rangle\doteq \begin{pmatrix} 7e^{i\frac{\pi}{6}}\\ 3e^{i\frac{\pi}{2}}\\ -1\\ \end{pmatrix}\\ \left|E\right\rangle\doteq \begin{pmatrix} i\\ 4\\ \end{pmatrix}\\ \left|F\right\rangle\doteq \begin{pmatrix} 2+2i\\ 3-4i\\ \end{pmatrix} \]