Homework problems can be found in the back of the first chapter in the textbook: qmch1.pdf

Any problem at the end of chapter 1 is suitable for this section. See especially:

1. (Orthogonal: 1.1)This problem requires that students understand what it means for two 2-dimensional vectors to be orthogonal to each other. It also requires them to know how to normalize complex vectors.

   Consider the three quantum states:

   $$\left\vert \psi_1\right\rangle = \frac{1}{\sqrt{3}}\left\vert +\right\rangle + i\frac{\sqrt{2}}{\sqrt{3}} \left\vert -\right\rangle$$ $$\left\vert \psi_2\right\rangle = \frac{1}{\sqrt{5}}\left\vert +\right\rangle - \frac{2}{\sqrt{5}} \left\vert -\right\rangle$$ $$\left\vert \psi_3\right\rangle = \frac{1}{\sqrt{2}}\left\vert +\right\rangle + i\frac{e^{\frac{i\pi}{4}}}{\sqrt{2}} \left\vert -\right\rangle$$

   1. For each of the $\vert \psi_i\rangle$ above, find the normalized vector $\vert \phi_i\rangle$ that is orthogonal to it.

   2. Calculate the inner products $\langle \psi_i\vert \psi_j\rangle$ for $i$ and $j=1$, $2$, $3$.

2. (Histogram: 1.5)This is the generic problem about quantum states: given a state, find the possible results of the measurement and the probabilities.

   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. 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. (Phase: 1.10)In this problem, students explore the physical meaning (or lack thereof!) of overall and relative phases in quantum states.

   Consider the three quantum states:

   $$\left\vert \psi_1\right\rangle = \frac{4}{5}\left\vert +\right\rangle + i\frac{3}{5} \left\vert -\right\rangle$$ $$\left\vert \psi_2\right\rangle = \frac{4}{5}\left\vert +\right\rangle - i\frac{3}{5} \left\vert -\right\rangle$$ $$\left\vert \psi_3\right\rangle = -\frac{4}{5}\left\vert +\right\rangle + i\frac{3}{5} \left\vert -\right\rangle$$

   1. For each of the $\ket{\psi_i}$ above, calculate the probabilities of spin component measurements along the $x$, $y$, and $z$-axes.

   2. Look For a Pattern (and Generalize): Use your results from $(a)$ to comment on the importance of the overall phase and of the relative phases of the quantum state vector.