Students calculate probabilities for a particle on a ring using three different notations: Dirac bra-ket, matrix, and wave function. After calculating the angular momentum and energy measurement probabilities, students compare their calculation methods for notation.
1. Quantum Ring Sequence | Superposition States for a Particle on a Ring >>
group Small Group Activity
30 min.
assignment Homework
assignment Homework
(2 points each)
You know that the normalized spatial eigenfunctions for a particle in a 1-D box of length \(L\) are \(\sqrt{\frac{2}{L}}\sin{\frac{n\pi x}{L}}\). If you want the eigenfunctions for a particle in a 2-D box, then you just multiply together the eigenfunctions for a 1-D box in each direction. (This is what the separation of variables procedure tells you to do.)
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*}
assignment Homework
Find the upward pointing flux of the electric field \(\vec E =E_0\, z\, \hat z\) through the part of the surface \(z=-3 s^2 +12\) (cylindrical coordinates) that sits above the \((x, y)\)--plane.
group Small Group Activity
30 min.
central forces quantum mechanics eigenstates eigenvalues hermitian operators quantum measurements degeneracy expectation values time dependence
Students calculate the expectation value of energy and angular momentum as a function of time for an initial state for a particle on a ring. This state is a linear combination of energy/angular momentum eigenstates written in bra-ket notation.group Small Group Activity
30 min.
central forces quantum mechanics eigenstates eigenvalues angular momentum time dependence hermitian operators probability degeneracy quantum measurements
Students calculate probabilities for energy, angular momentum, and position as a function of time for an initial state that is a linear combination of energy/angular momentum eigenstates for a particle confined to a ring written in bra-ket notation. This activity helps students build an understanding of when they can expect a quantity to depend on time and to give them more practice moving between representations.group Small Group Activity
60 min.
Wavefunctions quantum states probability amplitude histograms matrix notation of quantum states Arms representation
Students review using the Arms representation to represent states for discrete quantum systems and connecting the Arms representation to histogram and matrix representation. The student then extend the Arms representation to begin exploring the continuous position basis.assignment Homework
Use integration to find the total mass of the icecream in a packed cone (both the cone and the hemisphere of icecream on top).
assignment Homework
Recall that, if you take an infinite number of terms, the series for \(\sin z\) and the function itself \(f(z)=\sin z\) are equivalent representations of the same thing for all real numbers \(z\), (in fact, for all complex numbers \(z\)). This is not always true. More commonly, a series is only a valid, equivalent representation of a function for some more restricted values of \(z\). The technical name for this idea is convergence--the series only "converges" to the value of the function on some restricted domain, called the “interval” or “region of convergence.”
Find the power series for the function \(f(z)=\frac{1}{1+z^2}\). Then, using the Mathematica worksheet from class (vfpowerapprox.nb) as a model, or some other computer algebra system like Sage or Maple, explore the convergence of this series. Where does your series for this new function converge? Can you tell anything about the region of convergence from the graphs of the various approximations? Print out a plot and write a brief description (a sentence or two) of the region of convergence. You may need to include a lot of terms to see the effect of the region of convergence. Keep adding terms until you see a really strong effect!
Note: As a matter of professional ettiquette (or in some cases, as a legal copyright requirement), if you use or modify a computer program written by someone else, you should always acknowledge that fact briefly in whatever you write up. Say something like: “This calculation was based on a (name of software package) program titled (title) originally written by (author) copyright (copyright date).”
In this activity, your group will carry out calculations on each of the following normalized abstract quantum states on a ring: \begin{equation} \left|{\Phi_a}\right\rangle = \sqrt\frac{ 2}{12}\left|{3}\right\rangle + \sqrt\frac{ 1}{12}\left|{2}\right\rangle +\sqrt\frac{ 3}{12}\left|{0}\right\rangle +\sqrt\frac{ 2}{ 12}\left|{-1}\right\rangle +\sqrt\frac{ 1}{12}\left|{-3}\right\rangle +\sqrt\frac{ 3}{12}\left|{-4}\right\rangle \end{equation} \begin{equation} \left| \Phi_b\right\rangle \doteq \left( \begin{matrix} 0 \\ \sqrt\frac{ 2}{12}\\ \sqrt\frac{ 1}{12} \\ 0 \\ \sqrt\frac{ 3}{12} \\ \sqrt\frac{ 2}{12}\\ 0 \\ \sqrt\frac{1}{12} \\ \sqrt\frac{3}{12} \\ \end{matrix}\right) \end{equation} \begin{equation} \Phi_c(\phi) \doteq \sqrt {\frac{1}{24 \pi r_0}} \left( \sqrt{2}e^{i 3 \phi} +e^{i 2\phi} +\sqrt{3} + \sqrt{2} e^{-i 1 \phi} + e^{-i 3 \phi}+\sqrt{3}e^{-i 4 \phi} \right) \end{equation}
For each question state the postulate of quantum mechanics you use to complete the calculation and show explicitly how you use the postulates to answer the question.
- For each state above, what is the probability that you would measure the \(z\)-component of angular momentum to be \(-4\hbar\)? \(0\hbar\)? \(-2\hbar\)? \(3\hbar\)?
- What other possible values for the \(z\)-component of angular momentum could you have obtained with non-zero probability?
- For each state, what is the probability that you would measure the energy to be \(\displaystyle \frac{16\hbar^2}{2 I}\)? \(0\)? \(\displaystyle\frac{4 \hbar^2}{2 I}\)? \(\displaystyle \frac{9 \hbar^2}{2 I}\)?
- If you measured the energy, what other possible values could you obtain with non-zero probability?
- How are the calculations you made for the different state representations similar and different from each other? Be prepared to compare and contrast the calculations you made for each of the different representations (ket, matrix, eigenfunction).
This activity flows naturally from a lecture in which the eigenstates for energy and angular momentum on a ring are found. Many of the calculations done here are similar to calculations they have done before, but this activity emphasizes the different representations we use for quantum calculations and highlights when each representation is most useful.
Remind the students that that an arbitrary state \(|\Phi\rangle\) can be written in the \(L_z\) eigenbasis as
\[ \eqalign{\left| \Phi\right\rangle &\doteq \pmatrix{\vdots \cr \langle 2|\Phi\rangle \cr \langle 1|\Phi\rangle \cr \langle 0|\Phi\rangle \cr \langle -1|\Phi\rangle \cr \langle -2|\Phi\rangle \cr \vdots} = \pmatrix{\vdots \cr a_{2} \cr a_{1} \cr a_{0} \cr a_{-1} \cr a_{-2} \cr \vdots}} \]
Including this in the introduction to this activity should help students avoid confusion about the ordering of the elements in the column vectors used in this activity.
Students readily grasp the strategy of finding probability amplitudes “by inspection” when they are given an initial state written as a sum of eigenstates. We find that students then find it extremely difficult to find probability amplitudes of wavefunctions that are not written this way (i.e. using an integral to find the expansion coefficients of a function). This activity should be followed up with other activities in the Quantum Ring Sequence and homework that allow students to practice this more general method.
Associated Homework Problem: QM Ring Compare