group Small Group Activity
30 min.
central forces quantum mechanics eigenstates eigenvalues quantum measurements angular momentum energy hermitian operators probability superposition representations notations degeneracy
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.assignment Homework
Find \(N\).
accessibility_new Kinesthetic
10 min.
Quantum State Vectors Complex Numbers Spin 1/2 Arms Representation
Students, working in pairs, use their left arms to represent each component in a two-state quantum spin 1/2 system. Reinforces the idea that quantum states are complex valued vectors. Students make connections between Dirac, matrix, and Arms representation.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.format_list_numbered Sequence
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_ind Small White Board Question
10 min.
assignment Homework
The general equation for a straight line in polar coordinates is given by: \begin{equation} r(\phi)=\frac{r_0}{\cos(\phi-\delta)} \end{equation} Find the polar equation for the straight lines below. You do NOT need to evaluate any complicated trig or inverse trig functions. You may want to try plotting the general polar equation to figure out the roles of the parameters.
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.
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.
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).
The following are 2 different represtations for the \(\textbf{same}\) state on a quantum ring \begin{equation} \left|{\Phi}\right\rangle = \sqrt\frac{1}{2}\left|{2}\right\rangle -\sqrt\frac{ 1}{4}\left|{0}\right\rangle +i\sqrt\frac{ 1}{4}\left|{-2}\right\rangle \end{equation} \begin{equation} \Phi(\phi) \doteq \sqrt {\frac{1}{8 \pi r_0}} \left( \sqrt{2}e^{i 2 \phi} -1 + ie^{-i 2 \phi} \right) \end{equation}
Write down the matrix representation for the same state.
With all 3 representations, calculate the probability that a measurement of \(L_z\) will yield \(0\hbar\), \(-2\hbar\), \(2\hbar\).
If you measured the \(z\)-component of angular momentum to be \(2\hbar\), what would the state of the particle be immediately after the measurement is made?
What is the probability that a measurement of energy, \(E\), will yield \(0\frac{\hbar^2}{I}\)?,\(2\frac{\hbar^2}{I}\)?,\(4\frac{\hbar^2}{I}\)?
If you measured the energy of the state to be \(2\frac{\hbar^2}{I}\), what would the state of the particle be immediately after the measurement is made?