For each of the following operators, test each function to see if it is an eigenfunction of the operator.

• If it is, what is the eigenvalue?
• If it is not, can you write it as a superposition of functions that are eigenfunctions of that operator?

1. \(\hat{p} = -i\hbar\frac{d}{dx}\)

   • \(\psi_1(x) = Ae^{-ikx}\)

   • \(\psi_2(x) = Ae^{+ikx}\)

   • \(\psi_3(x) = A\sin kx\)

   
   

2. \(\hat{H} = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2}\)

   • \(\psi_1(x) = Ae^{-i\frac{p}{\hbar}x}\)

   • \(\psi_2(x) = Ae^{+i\frac{p}{\hbar}x}\)

   • \(\psi_3(x) = A\sin \frac{px}{\hbar}\)

   
   

3. \(\hat{H} = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2}\)

   • \(\psi_1(x) = A\sin kx\)

   • \(\psi_2(x) = A\cos kx\)

   • \(\psi_3(x) = Ae^{ikx}\)

   
   

4. \(\hat{S_z} \doteq \begin{bmatrix} \frac{\hbar}{2} & 0\\ 0 & -\frac{\hbar}{2}\end{bmatrix}\)

   • \(\left|{\psi_1}\right\rangle \doteq \begin{bmatrix} 1 \\ 0 \end{bmatrix}\)
     

   • \(\left|{\psi_2}\right\rangle \doteq \begin{bmatrix} 0 \\ 1 \end{bmatrix}\)
     

   • \(\left|{\psi_3}\right\rangle \doteq \frac{1}{\sqrt{2}}\begin{bmatrix} 1 \\ 1 \end{bmatrix}\)

Question: Are superpositions of eigenfunctions also eigenfunctions themselves?