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*}
keyboard Computational Activity
120 min.
finite difference hamiltonian quantum mechanics particle in a box eigenfunctions
Students implement a finite-difference approximation for the kinetic energy operator as a matrix, and then usenumpy
to solve for eigenvalues and eigenstates, which they visualize.
keyboard Computational Activity
120 min.
quantum mechanics operator matrix element particle in a box eigenfunction
Students find matrix elements of the position operator \(\hat x\) in a sinusoidal basis. This allows them to express this operator as a matrix, which they can then numerically diagonalize and visualize the eigenfunctions.computer Mathematica Activity
30 min.
central forces quantum mechanics angular momentum probability density eigenstates time evolution superposition mathematica
Students see probability density for eigenstates and linear combinations of eigenstates for a particle on a ring. The three visual representations: standard position vs probability density plot, a ring with colormapping, and cylindrical plot with height and colormapping, are also animated to visualize time-evolution.group Small Group Activity
30 min.
central forces quantum mechanics eigenstates eigenvalues quantum measurements angular momentum hermitian operators probability superposition
Students calculate probabilities for a particle on a ring whose wavefunction is not easily separated into eigenstates by inspection. To find the energy, angular momentum, and position probabilities, students perform integrations with the wavefunction or decompose the wavefunction into a superposition of eigenfunctions.assignment Homework
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: