Central Forces 2021
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.
What students learn
- Practice time evolution
- To calculate quantum probabilities in Dirac and Wavefunction notation
- To visualize how wave wavefunctions depend on time after time evolution
- To notice that not all probabilities of operators will depend on time
Consider a quantum particle on a ring. At \(t=0\), the particle is in state:
\begin{equation*}
\left|{\Phi(t=0)}\right\rangle =\frac{7i}{10}\left|{-2}\right\rangle -\frac{1}{2}\left|{-1}\right\rangle +\frac{1}{2}\left|{0}\right\rangle -\frac{1}{10}\left|{2}\right\rangle
\end{equation*}
- Find \(\left|{\Phi(t)}\right\rangle \)
- Go to the Ring States GeoGebra applet on the course schedule. Explore changing values of initial coefficents in the applet and see how \(Re(\psi)\), \(Im(\psi)\), and \(|\psi|^2\) change with time.
Then, create the state given above. How do both pieces of the wavefunction and the probability density change with time?
- Calculate the probability that you measure the \(z\)-component of the angular momentum to be \(-2\hbar\) at time \(t\). Is it time dependent?
- Calculate the probability that you measure the energy to be \(\frac{2\hbar^2}{I}\) at time \(t\). Is it time dependent?
- Author Information
- Corinne Manogue, Kerry Browne, Elizabeth Gire, Mary Bridget Kustusch, David McIntyre, Dustin Treece
- Keywords
- central forces quantum mechanics eigenstates eigenvalues angular momentum time dependence hermitian operators probability degeneracy quantum measurements
- Learning Outcomes
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