1. For each of the following complex numbers \(z\), find \(z^2\), \(\vert z\vert^2\), and rewrite \(z\) in exponential form, i.e. as a magnitude times a complex exponential phase:

   • \(z_1=i\),

   • \(z_2=2+2i\),
   • \(z_3=3-4i\).
2. 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. \[\left|D\right\rangle\doteq \begin{pmatrix} 7e^{i\frac{\pi}{6}}\\ 3e^{i\frac{\pi}{2}}\\ -1\\ \end{pmatrix}\\ \left|E\right\rangle\doteq \begin{pmatrix} i\\ 4\\ \end{pmatrix}\\ \left|F\right\rangle\doteq \begin{pmatrix} 2+2i\\ 3-4i\\ \end{pmatrix} \]

Consider the three quantum states: \[\left\vert \psi_1\right\rangle = \frac{1}{\sqrt{3}}\left\vert +\right\rangle+ i\frac{\sqrt{2}}{\sqrt{3}} \left\vert -\right\rangle\] \[\left\vert \psi_2\right\rangle = \frac{1}{\sqrt{5}}\left\vert +\right\rangle- \frac{2}{\sqrt{5}} \left\vert -\right\rangle\] \[\left\vert \psi_3\right\rangle = \frac{1}{\sqrt{2}}\left\vert +\right\rangle+ i\frac{e^{\frac{i\pi}{4}}}{\sqrt{2}} \left\vert -\right\rangle\] For each of the \(\vert \psi_i\rangle\) above, find the normalized vector \(\vert \phi_i\rangle\) that is orthogonal to it.

The general solution of the homogeneous differential equation

\[\ddot{x}-\dot{x}-6 x=0\]

is

\[x(t)=A\, e^{3t}+ B\, e^{-2t}\]

where \(A\) and \(B\) are arbitrary constants that would be determined by the initial conditions of the problem.

1. Find a particular solution of the inhomogeneous differential equation \(\ddot{x}-\dot{x}-6 x=-25\sin(4 t)\).

2. Find the general solution of \(\ddot{x}-\dot{x}-6 x=-25\sin(4 t)\).

3. Some terms in your general solution have an undetermined coefficients, while some coefficients are fully determined. Explain what is different about these two cases.

4. Find a particular solution of \(\ddot{x}-\dot{x}-6 x=12 e^{-3 t}\)

5. Find the general solution of \(\ddot{x}-\dot{x}-6 x=12 e^{-3 t}-25\sin(4 t)\)

   How is this general solution related to the particular solutions you found in the previous parts of this question?

   Can you add these particular solutions together with arbitrary coefficients to get a new particular solution?

6. Sense-making: Check your answer; Explicitly plug in your final answer in part (e) and check that it satisfies the differential equation.

Consider the arbitrary Pauli matrix \(\sigma_n=\hat n\cdot\vec \sigma\) where \(\hat n\) is the unit vector pointing in an arbitrary direction.

1. Find the eigenvalues and normalized eigenvectors for \(\sigma_n\). The answer is: \[ \begin{pmatrix} \cos\frac{\theta}{2}e^{-i\phi/2}\\{} \sin\frac{\theta}{2}e^{i\phi/2}\\ \end{pmatrix} \begin{pmatrix} -\sin\frac{\theta}{2}e^{-i\phi/2}\\{} \cos\frac{\theta}{2}e^{i\phi/2}\\ \end{pmatrix} \] It is not sufficient to show that this answer is correct by plugging into the eigenvalue equation. Rather, you should do all the steps of finding the eigenvalues and eigenvectors as if you don't know the answer. Hint: \(\sin\theta=\sqrt{1-\cos^2\theta}\).
2. Show that the eigenvectors from part (a) above are orthogonal.
3. Simplify your results from part (a) above by considering the three separate special cases: \(\hat n=\hat\imath\), \(\hat n=\hat\jmath\), \(\hat n=\hat k\). In this way, find the eigenvectors and eigenvalues of \(\sigma_x\), \(\sigma_y\), and \(\sigma_z\).

1. Let \[|\alpha\rangle \doteq \frac{1}{\sqrt{2}} \begin{pmatrix} 1\\ 1 \end{pmatrix} \qquad \rm{and} \qquad |\beta\rangle \doteq \frac{1}{\sqrt{2}} \begin{pmatrix} 1\\ -1 \end{pmatrix}\] Show that \(\left|{\alpha}\right\rangle \) and \(\left|{\beta}\right\rangle \) are orthonormal. (If a pair of vectors is orthonormal, that suggests that they might make a good basis.)
2. Consider the matrix \[C\doteq \begin{pmatrix} 3 & 1 \\ 1 & 3 \end{pmatrix} \] Show that the vectors \(|\alpha\rangle\) and \(|\beta\rangle\) are eigenvectors of C and find the eigenvalues. (Note that showing something is an eigenvector of an operator is far easier than finding the eigenvectors if you don't know them!)
3. A operator is always represented by a diagonal matrix if it is written in terms of the basis of its own eigenvectors. What does this mean? Find the matrix elements for a new matrix \(E\) that corresponds to \(C\) expanded in the basis of its eigenvectors, i.e. calculate \(\langle\alpha|C|\alpha\rangle\), \(\langle\alpha|C|\beta\rangle\), \(\langle\beta|C|\alpha\rangle\) and \(\langle\beta|C|\beta\rangle\) and arrange them into a sensible matrix \(E\). Explain why you arranged the matrix elements in the order that you did.
4. Find the determinants of \(C\) and \(E\). How do these determinants compare to the eigenvalues of these matrices?

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 \(j=1\), \(2\), \(3\).