Activities
Students, working in pairs, use their left arms to demonstrate time evolution in spin 1/2 quantum systems.
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.
The instructor gives a brief lecture about time dependence of energy eigenstates (e.g. McIntyre, 3.1). Notes for the students are attached.
Students answer conceptual questions about time dilation and proper time.
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.
A spin-3/2 particle initially is in the state \(|\psi(0)\rangle = |\frac{1}{2}\rangle\). This particle is placed in an external magnetic field so that the Hamiltonian is proportional to the \(\hat{S}_x\) operator, \(\hat{H} = \alpha \hat{S}_x \doteq \frac{\alpha\hbar}{2}\begin{pmatrix} 0 & \sqrt{3} & 0 & 0\\ \sqrt{3} & 0 & 2 & 0\\ 0 & 2 & 0 & \sqrt{3} \\ 0 & 0 & \sqrt{3} & 0 \end{pmatrix}\)
- Find the energy eigenvalues and energy eigenstates for the system.
- Find \(|\psi(t)\rangle\).
- List the outcomes of all possible measurements of \(S_x\) and find their probabilities. Explicitly identify any probabilities that depend on time.
- List the outcomes of all possible measurements of \(S_z\) and find their probabilities. Explicitly identify any probabilities that depend on time.
Two electrons are placed in a magnetic field in the \(z\)-direction. The initial state of the first electron is \(\frac{1}{\sqrt{2}}\begin{pmatrix} 1\\ i\\ \end{pmatrix}\) and the initial state of the second electron is \(\frac{1}{2}\begin{pmatrix} \sqrt{3}\\ 1\\ \end{pmatrix}\).
- Find the probabilty of measuring each particle to have spin-up in the \(x\)-, \(y\)-, and \(z\)-directions at \(t = 0\).
- Find the probabilty of measuring each particle to have spin-up in the \(x\)-, \(y\)-, and \(z\)-directions at some later time \(t\).
- Calculate the expectation values for \(S_x\), \(S_y\), and \(S_z\) for each particle as functions of time.
- Are there any times when all the probabilities you have calculated are the same as they were at \(t = 0\)?
In this small group activity, students solve for the time dependence of two quantum spin 1/2 particles under the influence of a Hamiltonian. Students determine, given a Hamiltonian, which states are stationary and under what circumstances measurement probabilities do change with time.
Students act out the classic light clock scenario for deriving time dilation.
Students do calculations for time evolution for spin-1.
Students use their arms to act out stationary and non-stationary states of a quantum particle on a ring.
Problem
A particle in an infinite square well potential has an initial state vector \[\left|{\Psi(0)}\right\rangle = A\big(\left|{\phi_1}\right\rangle -\left|{\phi_2}\right\rangle +i\left|{\phi_3}\right\rangle \big)\]
where \(|\phi_n\rangle\) are the energy eigenstates. You have previously found \(\left|{\Psi(t)}\right\rangle \) for this state.
Use a computer to graph the wave function \(\Psi(x,t)\) and probability density \(\rho(x,t)\). Choose a few interesting values of \(t\) to include in your submission.
Use a computer to calculate the probability of measuring the particle to be near the middle of the well (within 1% on either side) as a function of time. Include both your symbolic result and a graph in your submission.
Choose another location in the well, different from the location above. Use a computer to calculate the probability of measuring the particle to be near your chosen location (within 1% on either side) as a function of time. Include both your symbolic result and a graph in your submission.
Are there any locations in the well where the probability is independent of time? Explain how you determined your answer.
- The time dependence for a wave function like this is complicated. Write a lengthy description in words about the major features of this wave function and its probability density, how they change in time, and why they change the way they do. Comment on any interesting features you noticed that you have not already discussed in the questions above and describe any additional things you learned from the process of solving this problem.
Consider a two-state quantum system with a Hamiltonian \begin{equation} \hat{H}\doteq \begin{pmatrix} E_1&0\\ 0&E_2 \end{pmatrix} \end{equation} Another physical observable \(M\) is described by the operator \begin{equation} \hat{M}\doteq \begin{pmatrix} 0&c\\ c&0 \end{pmatrix} \end{equation} where \(c\) is real and positive. Let the initial state of the system be \(\left|{\psi(0)}\right\rangle =\left|{m_1}\right\rangle \), where \(\left|{m_1}\right\rangle \) is the eigenstate corresponding to the larger of the two possible eigenvalues of \(\hat{M}\). What is the expectation value of \(M\) as a function of time? What is the frequency of oscillation of the expectation value of \(M\)?
Mathematica Activity
30 min.
Students see probability density for eigenstates and linear combinations of eigenstates for a particle on a ring. The three visual representations: standard position vs probability density plot, a ring with colormapping, and cylindrical plot with height and colormapping, are also animated to visualize time-evolution.
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.
In an optional extension, students find a series expansion for \(\vec{B}(\vec{r})\) either on the axis or in the plane of the ring, for either small or large values of the relevant geometric variable. Add an extra half hour or more to the time estimate for the optional extension.
Students work in small groups to use Coulomb's Law \[\vec{E}(\vec{r}) =\frac{1}{4\pi\epsilon_0}\int\frac{\rho(\vec{r}^{\,\prime})\left(\vec{r}-\vec{r}^{\,\prime}\right)}{\vert \vec{r}-\vec{r}^{\,\prime}\vert^3} \, d\tau^{\prime}\] to find an integral expression for the electric field, \(\vec{E}(\vec{r})\), everywhere in space, due to a ring of charge.
In an optional extension, students find a series expansion for \(\vec{E}(\vec{r})\) either on the axis or in the plane of the ring, for either small or large values of the relevant geometric variable. Add an extra half hour or more to the time estimate for the optional extension.
This activity starts with a brief lecture introduction to power series and a short derivation of the formula for calculating the power series coefficients.
\[c_n={1\over n!}\, f^{(n)}(z_0)\]
Students use this formula to compute the power series coefficients for a \(\sin\theta\) (around both the origin and (if time allows) \(\frac{\pi}{6}\)). The meaning of these coefficients and the convergence behavior for each approximation is discussed in the whole-class wrap-up and in the follow-up activity: Visualization of Power Series Approximations.
Students work in small groups to use the superposition principle \[V(\vec{r}) =\frac{1}{4\pi\epsilon_0}\int\frac{\rho(\vec{r}^{\,\prime})}{\vert \vec{r}-\vec{r}^{\,\prime}\vert} \, d\tau^{\prime}\] to find an integral expression for the electrostatic potential, \(V(\vec{r})\), everywhere in space, due to a ring of charge.
In an optional extension, students find a series expansion for \(V(\vec{r})\) either on the axis or in the plane of the ring, for either small or large values of the relevant geometric variable. Add an extra half hour or more to the time estimate for the optional extension.
The students will set up a Styrofoam cup with heating element and a thermometer in it. They will measure the temperature as a function of time, and thus the energy transferred from the power supply, from which they compute changes in entropy.
Students solve for the equations of motion of a box sliding down (frictionlessly) a wedge, which itself slides on a horizontal surface, in order to answer the question "how much time does it take for the box to slide a distance \(d\) down the wedge?". This activities highlights finding kinetic energies when the coordinate system is not orthonormal and checking special cases, functional behavior, and dimensions.
Students work in small groups to use the superposition principle \[\vec{A}(\vec{r}) =\frac{\mu_0}{4\pi}\int\frac{\vec{J}(\vec{r}^{\,\prime})}{\vert \vec{r}-\vec{r}^{\,\prime}\vert}\, d\tau^{\prime}\] to find an integral expression for the magnetic vector potential, \(\vec{A}(\vec{r})\), due to a spinning ring of charge.
In an optional extension, students find a series expansion for \(\vec{A}(\vec{r})\) either on the axis or in the plane of the ring, for either small or large values of the relevant geometric variable. Add an extra half hour or more to the time estimate for the optional extension.