Students are asked to:
- Test to see if one of the given functions is an eigenfunction of the given operator
- See if they can write the functions that are found not to be eigenfunctions as a linear combination of eigenfunctions.
Operators & FunctionsFor 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?
- \(\hat{p} = -i\hbar\frac{d}{dx}\)
\(\psi_1(x) = Ae^{-ikx}\)
\(\psi_2(x) = Ae^{+ikx}\)
\(\psi_3(x) = A\sin kx\)
\(\hat{H} = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2}\)
\(\psi_1(x) = Ae^{-i\frac{p}{\hbar}x}\)
\(\psi_2(x) = Ae^{+i\frac{p}{\hbar}x}\)
\(\psi_3(x) = A\sin \frac{px}{\hbar}\)
\(\hat{H} = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2}\)
\(\psi_1(x) = A\sin kx\)
\(\psi_2(x) = A\cos kx\)
\(\psi_3(x) = Ae^{ikx}\)
- \(\hat{S_z} \doteq \begin{bmatrix} \frac{\hbar}{2} & 0\\ 0 & -\frac{\hbar}{2}\end{bmatrix}\)
\(\left|{\psi_1}\right\rangle \doteq \begin{bmatrix} 1 \\ 0 \end{bmatrix}\)
\(\left|{\psi_2}\right\rangle \doteq \begin{bmatrix} 0 \\ 1 \end{bmatrix}\)
\(\left|{\psi_3}\right\rangle \doteq \frac{1}{\sqrt{2}}\begin{bmatrix} 1 \\ 1 \end{bmatrix}\)
Question: Are superpositions of eigenfunctions also eigenfunctions themselves?