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.
For each of the following operators, test each function to see if it is an eigenfunction of the operator.
\(\psi_1(x) = Ae^{-ikx}\)
\(\psi_2(x) = Ae^{+ikx}\)
\(\psi_3(x) = A\sin kx\)
\(\hat{H} = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2}\)
\(\psi_1(x) = Ae^{-i\frac{p}{\hbar}x}\)
\(\psi_2(x) = Ae^{+i\frac{p}{\hbar}x}\)
\(\psi_3(x) = A\sin \frac{px}{\hbar}\)
\(\hat{H} = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2}\)
\(\psi_1(x) = A\sin kx\)
\(\psi_2(x) = A\cos kx\)
\(\psi_3(x) = Ae^{ikx}\)
\(\left|{\psi_1}\right\rangle \doteq \begin{bmatrix} 1 \\ 0 \end{bmatrix}\)
\(\left|{\psi_2}\right\rangle \doteq \begin{bmatrix} 0 \\ 1 \end{bmatrix}\)
\(\left|{\psi_3}\right\rangle \doteq \frac{1}{\sqrt{2}}\begin{bmatrix} 1 \\ 1 \end{bmatrix}\)
Question: Are superpositions of eigenfunctions also eigenfunctions themselves?
assignment Homework
Any sufficiently smooth spatial wave function inside a 2-D box can be expanded in a double sum of the product wave functions, i.e. \begin{equation} \psi(x,y)=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\, c_{nm}\; \hbox{eigenfunction}_n(x)\;\hbox{eigenfunction}_m(y) \end{equation} Using your expressions from part (a) above, write out all the terms in this sum out to \(n=3\), \(m=3\). Arrange the terms, conventionally, in terms of increasing energy.
You may find it easier to work in bra/ket notation: \begin{align*} \left|{\psi}\right\rangle &=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\, c_{nm}\left|{n}\right\rangle \left|{m}\right\rangle \\ &=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\, c_{nm}\left|{nm}\right\rangle \end{align*}
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assignment Homework
Homogeneous, linear ODEs with constant coefficients were likely covered in your Differential Equations course (MTH 256 or equiv.). If you need a review, please see:
Constant Coefficients, Homogeneous
or your differential equations text.
Answer the following questions for each differential equation below: