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.
1. << Time Evolution Refresher (Mini-Lecture) | Quantum Ring Sequence | Visualization of Quantum Probabilities for a Particle Confined to a Ring >>
In this activity, your group will carry out calculation on the following quantum state on a ring: \begin{equation*} \left|{\Phi}\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?
Probability v. Probability Density: Students struggle with the two different ways of finding probability: for discrete and continuous measurements. Most recognize that they need to do an integral for a continuous quantity, but are not sure when to square (before integration or after).
\(\left|\int \phi_n^*(x)\Psi(x,t) dx\right|^2\) vs. \(\int\left|\Psi(t) \right|^2 dx\)
In particular, many students will forget to do the squaring for the calculation on the left because \(\int \phi_n^*(x) \Psi(x,t) dx\) looks a lot like \(\int \Psi^*(x,t) \Psi(x,t) dx\).
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 another activity and/or homework from the Quantum Ring Sequence that allows students to practice this more general method.
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Students work in small groups to use the Biot-Savart law \[\vec{B}(\vec{r}) =\frac{\mu_0}{4\pi}\int\frac{\vec{J}(\vec{r}^{\,\prime})\times \left(\vec{r}-\vec{r}^{\,\prime}\right)}{\vert \vec{r}-\vec{r}^{\,\prime}\vert^3} \, d\tau^{\prime}\] to find an integral expression for the magnetic field, \(\vec{B}(\vec{r})\), due to a spinning ring of charge.
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