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Activities

Problem

##### Measurement Probabilities
A beam of spin-$\frac{1}{2}$ particles is prepared in the initial state $\left\vert \psi\right\rangle = \sqrt{\frac{2}{5}}\; |+\rangle_x - \sqrt{\frac{3}{5}}\; |-\rangle_x$(Note: this state is written in the $S_x$ basis!)
1. What are the possible results of a measurement of $S_x$, with what probabilities?
2. Repeat part a for measurements of $S_z$.

3. Suppose you start with a particle in the state given above, measure $S_x$, and happen to get $+\hbar /2$. You then take that same particle and measure $S_z$. What are the possible results and with what probability would you measure each possible result?
• Found in: Quantum Fundamentals course(s)

Small Group Activity

30 min.

##### Quantum Measurement Play
The instructor and students do a skit where students represent quantum states that are “measured” by the instructor resulting in a state collapse.
• Found in: Quantum Fundamentals course(s)

Problem

##### Phase 2
Consider the three quantum states: $\left\vert \psi_1\right\rangle = \frac{4}{5}\left\vert +\right\rangle+ i\frac{3}{5} \left\vert -\right\rangle$ $\left\vert \psi_2\right\rangle = \frac{4}{5}\left\vert +\right\rangle- i\frac{3}{5} \left\vert -\right\rangle$ $\left\vert \psi_3\right\rangle = -\frac{4}{5}\left\vert +\right\rangle+ i\frac{3}{5} \left\vert -\right\rangle$
1. For each of the $\left|{\psi_i}\right\rangle$ above, calculate the probabilities of spin component measurements along the $x$, $y$, and $z$-axes.
2. Look For a Pattern (and Generalize): Use your results from $(a)$ to comment on the importance of the overall phase and of the relative phases of the quantum state vector.
• Found in: Quantum Fundamentals course(s)

Problem

5 min.

##### Spin One Half Unknowns (Brief)
With the Spins simulation set for a spin 1/2 system, measure the probabilities of all the possible spin components for each of the unknown initial states $\left|{\psi_3}\right\rangle$ and $\left|{\psi_4}\right\rangle$. (Since $\left|{\psi_3}\right\rangle$ has already been covered in class, please only do $\left|{\psi_4}\right\rangle$ )
1. Use your measured probabilities to find each of the unknown states as a linear superposition of the $S_z$-basis states $\left|{+}\right\rangle$ and $\left|{-}\right\rangle$.
2. Articulate a Process: Write a set of general instructions that would allow another student in next year's class to find an unknown state from measured probabilities.
3. Compare Theory with Experiment: Design an experiment that will allow you to test whether your prediction for each of the unknown states is correct. Describe your experiment here, clearly but succinctly, as if you were writing it up for a paper. Do the experiment and discuss your results.
4. Make a Conceptual Connection: In general, can you determine a quantum state with spin-component probability measurements in only two spin-component-directions? Why or why not?
• Found in: Quantum Fundamentals course(s)

Small Group Activity

30 min.

##### Wavefunctions on a Quantum Ring
This activity lets students explore translating a wavefunction that isn't obviously made up of eigenstates at first glance into ket and matrix form. Then students explore wave functions, probabilities in a region, expectation values, and what wavefunctions can tell you about measurements of $L_z$.
• Found in: Central Forces course(s)

Lecture

30 min.

##### Compare & Contrast Kets & Wavefunctions
In this lecture, the instructor guides a discussion about translating between bra-ket notation and wavefunction notation for quantum systems.
• Found in: Completeness Relations sequence(s)

Problem

##### ISW Position Measurement

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.

1. 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.

2. 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.

3. 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.

4. Are there any locations in the well where the probability is independent of time? Explain how you determined your answer.

5. 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.

• Found in: Quantum Fundamentals course(s)

Small Group Activity

30 min.

##### Chain Rule Measurement
This small group activity using surfaces combines practice with the multivariable chain rule while emphasizing numerical representations of derivatives. Students work in small groups to measure partial derivatives in both rectangular and polar coordinates, then verify their results using the chain rule. The whole class wrap-up discussion emphasizes the relationship between a directional derivative in the $r$-direction and derivatives in $x$- and $y$-directions using the chain rule.
• Found in: Vector Calculus I course(s)

Small Group Activity

10 min.

##### Sequential Stern-Gerlach Experiments
1. Set-Up a Sequential Measurement

1. Add an analyzer to the experiment by:

1. Break the links between the analyzer and the counters by clicking on the boxes with up and down arrow labels on the analyzer.
2. Click and drag a new connection from the analyzer to empty space to create a new element. A new analyzer is one of the options.

2. Measure $S_z$ twice in succession.

What is the probability that a particle leaving the first analyzer with $S_z=\frac{+\hbar}{2}$ will be measured by the second analyzer to have $S_z=\frac{-\hbar}{2}$?

3. Try all four possible combinations of input/outputs for the second analyzer.

What have you learned from these experiments?

2. Try All Combinations of Sequential Measurements

In the table, enter the probability of a particle exiting the 2nd analyzer with the spin indicated in row if the particle enters the 2nd analyzer with the spin indicated in each column.

3. You can rotate the Stern-Gerlach analyzers to any direction you want (using spherical coordinates).

Choose an arbitrary direction (not along one of the coordinate axes) for the 1st analyzer and measure the spin along the coordinate directions for the 2nd analyzer.

• Found in: Quantum Fundamentals course(s)

Small Group Activity

30 min.

##### Intro to Stern-Gerlach Experiments 1
Students become acquainted with the Spins Simulations of Stern-Gerlach Experiments and record measurement probabilities of spin components for a spin-1/2 system. Students start developing intuitions for the results of quantum measurements for this system.
• Found in: Quantum Fundamentals course(s)

Lecture

120 min.

##### Gibbs entropy approach
These lecture notes for the first week of https://paradigms.oregonstate.edu/courses/ph441 include a couple of small group activities in which students work with the Gibbs formulation of the entropy.
• Found in: Thermal and Statistical Physics course(s)

Problem

10 min.

##### Coefficients in Wavefunction notation
Students use the completeness relation for the position basis to re-express expressions in bra/ket notation in wavefunction notation.

Problem

5 min.

##### Dimensional Analysis of Kets
Students consider the dimensions of spin-state kets and position-basis kets.
• Found in: Completeness Relations sequence(s)

Problem

##### Gibbs entropy is extensive

Consider two noninteracting systems $A$ and $B$. We can either treat these systems as separate, or as a single combined system $AB$. We can enumerate all states of the combined by enumerating all states of each separate system. The probability of the combined state $(i_A,j_B)$ is given by $P_{ij}^{AB} = P_i^AP_j^B$. In other words, the probabilities combine in the same way as two dice rolls would, or the probabilities of any other uncorrelated events.

1. Show that the entropy of the combined system $S_{AB}$ is the sum of entropies of the two separate systems considered individually, i.e. $S_{AB} = S_A+S_B$. This means that entropy is extensive. Use the Gibbs entropy for this computation. You need make no approximation in solving this problem.
2. Show that if you have $N$ identical non-interacting systems, their total entropy is $NS_1$ where $S_1$ is the entropy of a single system.

##### Note
In real materials, we treat properties as being extensive even when there are interactions in the system. In this case, extensivity is a property of large systems, in which surface effects may be neglected.

• Found in: Thermal and Statistical Physics course(s)

Problem

##### Diatomic hydrogen

At low temperatures, a diatomic molecule can be well described as a rigid rotor. The Hamiltonian of such a system is simply proportional to the square of the angular momentum \begin{align} H &= \frac{1}{2I}L^2 \end{align} and the energy eigenvalues are \begin{align} E_{\ell m} &= \hbar^2 \frac{\ell(\ell+1)}{2I} \end{align}

1. What is the energy of the ground state and the first and second excited states of the $H_2$ molecule? i.e. the lowest three distinct energy eigenvalues.

2. At room temperature, what is the relative probability of finding a hydrogen molecule in the $\ell=0$ state versus finding it in any one of the $\ell=1$ states?
i.e. what is $P_{\ell=0,m=0}/\left(P_{\ell=1,m=-1} + P_{\ell=1,m=0} + P_{\ell=1,m=1}\right)$

3. At what temperature is the value of this ratio 1?

4. At room temperature, what is the probability of finding a hydrogen molecule in any one of the $\ell=2$ states versus that of finding it in the ground state?
i.e. what is $P_{\ell=0,m=0}/\left(P_{\ell=2,m=-2} + P_{\ell=2,m=-1} + \cdots + P_{\ell=2,m=2}\right)$

• Found in: Energy and Entropy course(s)

Small Group Activity

60 min.

##### Going from Spin States to Wavefunctions
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.
• Found in: Quantum Fundamentals course(s) Found in: Completeness Relations, Arms Sequence for Complex Numbers and Quantum States sequence(s)

Small Group Activity

30 min.

##### Working with Representations on the Ring
This activity acts as a reintroduction to doing quantum calculations while also introducing the matrix representation on the ring, allowing students to discover how to index and form a column vector representing the given quantum state. In addition, this activity introduces degenerate measurements on the quantum ring and examines the state after measuring both degenerate and non-degenerate eigenvalues for the state.
• Found in: Central Forces course(s)

Lecture

120 min.

##### Boltzmann probabilities and Helmholtz
These notes, from the third week of https://paradigms.oregonstate.edu/courses/ph441 cover the canonical ensemble and Helmholtz free energy. They include a number of small group activities.
• Found in: Thermal and Statistical Physics course(s)

Computational Activity

120 min.

##### Mean position
Students compute probabilities and averages given a probability density in one dimension. This activity serves as a soft introduction to the particle in a box, introducing all the concepts that are needed.
• Found in: Computational Physics Lab II course(s) Found in: Computational wave function inner products sequence(s)

Small Group Activity

30 min.

##### Superposition States for a Particle on a Ring
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.
• Found in: Quantum Ring Sequence sequence(s)

Small Group Activity

30 min.

##### Time Dependence for a Quantum Particle on a Ring Part 1
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.
• Found in: Central Forces, Theoretical Mechanics course(s) Found in: Quantum Ring Sequence sequence(s)

Mathematica Activity

30 min.

##### Visualization of Quantum Probabilities for a Particle Confined to a Ring
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.
• Found in: Central Forces course(s) Found in: Visualization of Quantum Probabilities, Quantum Ring Sequence sequence(s)

Small Group Activity

30 min.

##### Energy and Angular Momentum for a Quantum Particle on a Ring
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.
• Found in: Theoretical Mechanics course(s) Found in: Quantum Ring Sequence sequence(s)

Mathematica Activity

30 min.

##### Visualizing Combinations of Spherical Harmonics
Students observe three different plots of linear combinations of spherical combinations with probability density represented by color on the sphere, distance from the origin (polar plot), and distance from the surface of the sphere.
• Found in: Central Forces course(s) Found in: Quantum Sphere Sequence, Visualization of Quantum Probabilities, Eigenfunction Sequence sequence(s)

Mathematica Activity

30 min.

##### Visualization of Quantum Probabilities for the Hydrogen Atom
Students use Mathematica to visualize the probability density distribution for the hydrogen atom orbitals with the option to vary the values of $n$, $\ell$, and $m$.
• Found in: Central Forces course(s) Found in: Visualization of Quantum Probabilities sequence(s)

Small White Board Question

5 min.

##### Normalization of the Gaussian for Wavefunctions
Students find a wavefunction that corresponds to a Gaussian probability density.
• Found in: Periodic Systems course(s) Found in: Fourier Transforms and Wave Packets sequence(s)

Small Group Activity

30 min.

##### Time Evolution of a Spin-1/2 System
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.
• Found in: Quantum Fundamentals course(s)