The following are 2 different representations for the same state on a quantum ring \begin{align} \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{align} \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?
assignment Homework
The following are 3 different representations for the \(\textbf{same}\) state on a quantum ring for \(r_0=1\) \begin{equation} \left|{\Phi_a}\right\rangle = i\sqrt{\frac{ 2}{12}}\left|{3}\right\rangle - \sqrt{\frac{ 1}{12}}\left|{1}\right\rangle +\sqrt{\frac{ 3}{12}}e^{i\frac{\pi}{4}}\left|{0}\right\rangle -i\sqrt{\frac{ 2}{ 12}}\left|{-1}\right\rangle +\sqrt{\frac{ 4}{12}}\left|{-3}\right\rangle \end{equation} \begin{equation} \left| \Phi_b\right\rangle \doteq \left( \begin{matrix} \vdots \\ i\sqrt{\frac{ 2}{12}}\\ 0 \\ -\sqrt{\frac{ 1}{12}} \\ \sqrt{\frac{ 3}{12}}e^{i\frac{\pi}{4}} \\ -i\sqrt{\frac{ 2}{12}}\\ 0 \\ \sqrt{\frac{4}{12} }\\ \vdots \end{matrix}\right) \begin{matrix} \leftarrow m=0 \end{matrix} \end{equation} \begin{equation} \Phi_c(\phi) \doteq \sqrt{\frac{1}{24 \pi}} \left( i\sqrt{2}e^{i 3 \phi} -e^{i\phi} +\sqrt{3}e^{i\frac{\pi}{4}} -i \sqrt{2} e^{-i\phi} + \sqrt{4}e^{-i 3 \phi} \right) \end{equation}
If you measured the energy of the state to be \(\frac{9}{2}\frac{\hbar^2}{I}\), what would the state of the particle be immediately after the measurement is made?
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 \(N\).
assignment Homework
group Small Group Activity
60 min.
assignment Homework
\(z_1=i\),
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.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.