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Activities

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)

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)

Kinesthetic

10 min.

##### Spin 1/2 with Arms
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.
• Found in: Quantum Fundamentals course(s) Found in: Arms Sequence for Complex Numbers and Quantum States sequence(s)

Kinesthetic

10 min.

##### Using Arms to Represent Overall and Relative Phase in Spin 1/2 Systems
Students, working in pairs, use the Arms representations to represent states of spin 1/2 system. Through a short series of instructor-led prompts, students explore the difference between overall phase (which does NOT distinguish quantum states) and relative phase (which does distinguish quantum states).
• Found in: Quantum Fundamentals course(s) Found in: Arms Sequence for Complex Numbers and Quantum States sequence(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)

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

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.

##### Expectation Values for a Particle on a Ring
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.
• Found in: Central Forces, Theoretical Mechanics course(s) Found in: Quantum Ring Sequence 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)

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)

Kinesthetic

30 min.

##### Time Evolution of a Quantum Particle on a Ring with Arms
Students use their arms to act out stationary and non-stationary states of a quantum particle on a ring.
• Found in: Central Forces course(s) Found in: Arms Sequence for Complex Numbers and Quantum States sequence(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

5 min.

##### Unit Learning Outcomes: Quantum Mechanics on a Ring

In this unit, you will explore the quantum mechanics of a simple system: a particle confined to a one-dimensional ring.

Motivating Questions

• What are the energy eigenstates, i.e. eigenstates of the Hamiltonian?
• What physical properties of the energy eigenstates can be measured?
• What other states are possible and what are their physical properties?
• How do the states change if this system and their physical properties depend on time?

Key Activities/Problems

Unit Learning Outcomes

At the end of this unit, you should be able to:

• Describe the energy eigenstates for the ring system algebraically and graphically.
• List the physical measurables for the system and give expressions for the corresponding operators in bra/ket, matrix, and position representations.
• Give the possible quantum numbers for the quantum ring system and describe any degeneracies.
• For a given state, use the inner product in bra/ket, matrix, and position representations, to find the probability of making any physically relevant measurement, including states with degeneracy.
• Use an expansion in energy eigenstates to find the time dependence of a given state.

Equation Sheet for This Unit

• Found in: Central Forces course(s)

Problem

##### Diagonalization
1. Let $|\alpha\rangle \doteq \frac{1}{\sqrt{2}} \begin{pmatrix} 1\\ 1 \end{pmatrix} \qquad \rm{and} \qquad |\beta\rangle \doteq \frac{1}{\sqrt{2}} \begin{pmatrix} 1\\ -1 \end{pmatrix}$ Show that $\left|{\alpha}\right\rangle$ and $\left|{\beta}\right\rangle$ are orthonormal. (If a pair of vectors is orthonormal, that suggests that they might make a good basis.)
2. Consider the matrix $C\doteq \begin{pmatrix} 3 & 1 \\ 1 & 3 \end{pmatrix}$ Show that the vectors $|\alpha\rangle$ and $|\beta\rangle$ are eigenvectors of C and find the eigenvalues. (Note that showing something is an eigenvector of an operator is far easier than finding the eigenvectors if you don't know them!)
3. A operator is always represented by a diagonal matrix if it is written in terms of the basis of its own eigenvectors. What does this mean? Find the matrix elements for a new matrix $E$ that corresponds to $C$ expanded in the basis of its eigenvectors, i.e. calculate $\langle\alpha|C|\alpha\rangle$, $\langle\alpha|C|\beta\rangle$, $\langle\beta|C|\alpha\rangle$ and $\langle\beta|C|\beta\rangle$ and arrange them into a sensible matrix $E$. Explain why you arranged the matrix elements in the order that you did.
4. Find the determinants of $C$ and $E$. How do these determinants compare to the eigenvalues of these matrices?
• Found in: Quantum Fundamentals course(s)

Problem

5 min.

##### Frequency
Consider a two-state quantum system with a Hamiltonian $$\hat{H}\doteq \begin{pmatrix} E_1&0\\ 0&E_2 \end{pmatrix}$$ Another physical observable $M$ is described by the operator $$\hat{M}\doteq \begin{pmatrix} 0&c\\ c&0 \end{pmatrix}$$ 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 $M$?
• Found in: Quantum Fundamentals course(s)

Problem

##### Orthogonal
Consider the three quantum states: $\left\vert \psi_1\right\rangle = \frac{1}{\sqrt{3}}\left\vert +\right\rangle+ i\frac{\sqrt{2}}{\sqrt{3}} \left\vert -\right\rangle$ $\left\vert \psi_2\right\rangle = \frac{1}{\sqrt{5}}\left\vert +\right\rangle- \frac{2}{\sqrt{5}} \left\vert -\right\rangle$ $\left\vert \psi_3\right\rangle = \frac{1}{\sqrt{2}}\left\vert +\right\rangle+ i\frac{e^{\frac{i\pi}{4}}}{\sqrt{2}} \left\vert -\right\rangle$
1. For each of the $\vert \psi_i\rangle$ above, find the normalized vector $\vert \phi_i\rangle$ that is orthogonal to it.
2. Calculate the inner products $\langle \psi_i\vert \psi_j\rangle$ for $i$ and $j=1$, $2$, $3$.
• Found in: Quantum Fundamentals course(s)

Problem

5 min.

##### Dirac Practice
For this problem, use the vectors $|a\rangle = 4 |1\rangle - 3 |2\rangle$ and $|b\rangle = -i |1\rangle + |2\rangle$.
1. Find $\langle a | b \rangle$ and $\langle b | a \rangle$. Discuss how these two inner products are related to each other.
2. For $\hat{Q}\doteq \begin{pmatrix} 2 & i \\ -i & -2 \end{pmatrix}$, calculate $\langle1|\hat{Q}|2\rangle$, $\langle2|\hat{Q}|1\rangle$, $\langle a|\hat{Q}| b \rangle$ and $\langle b|\hat{Q}|a \rangle$.
3. What kind of mathematical object is $|a\rangle\langle b|$? What is the result if you multiply a ket (for example, $| a\rangle$ or $|1\rangle$) by this expression? What if you multiply this expression by a bra?
• Found in: Quantum Fundamentals course(s)

Small Group Activity

30 min.

##### $|\pm\rangle$ Forms an Orthonormal Basis
Student explore the properties of an orthonormal basis using the Cartesian and $S_z$ bases as examples.
• Found in: Quantum Fundamentals course(s) Found in: Completeness Relations sequence(s)

Problem

##### Eigenvectors of Pauli Matrices
1. Find the eigenvalues and normalized eigenvectors of the Pauli matrices $\sigma_x$, $\sigma_y$, and $\sigma_z$ (see the Spins Reference Sheet posted on the course website).
• Found in: Quantum Fundamentals course(s)

Problem

##### Eigen Spin Challenge
Consider the arbitrary Pauli matrix $\sigma_n=\hat n\cdot\vec \sigma$ where $\hat n$ is the unit vector pointing in an arbitrary direction.
1. Find the eigenvalues and normalized eigenvectors for $\sigma_n$. The answer is: $\begin{pmatrix} \cos\frac{\theta}{2}e^{-i\phi/2}\\{} \sin\frac{\theta}{2}e^{i\phi/2}\\ \end{pmatrix} \begin{pmatrix} -\sin\frac{\theta}{2}e^{-i\phi/2}\\{} \cos\frac{\theta}{2}e^{i\phi/2}\\ \end{pmatrix}$ It is not sufficient to show that this answer is correct by plugging into the eigenvalue equation. Rather, you should do all the steps of finding the eigenvalues and eigenvectors as if you don't know the answer. Hint: $\sin\theta=\sqrt{1-\cos^2\theta}$.
2. Show that the eigenvectors from part (a) above are orthogonal.
3. Simplify your results from part (a) above by considering the three separate special cases: $\hat n=\hat\imath$, $\hat n=\hat\jmath$, $\hat n=\hat k$. In this way, find the eigenvectors and eigenvalues of $\sigma_x$, $\sigma_y$, and $\sigma_z$.
• Found in: Quantum Fundamentals course(s)

Problem

5 min.

##### Spin Matrix
The Pauli spin matrices $\sigma_x$, $\sigma_y$, and $\sigma_z$ are defined by: $\sigma_x= \begin{pmatrix} 0&1\\ 1&0\\ \end{pmatrix} \hspace{2em} \sigma_y= \begin{pmatrix} 0&-i\\ i&0\\ \end{pmatrix} \hspace{2em} \sigma_z= \begin{pmatrix} 1&0\\ 0&-1\\ \end{pmatrix}$ These matrices are related to angular momentum in quantum mechanics.
1. By drawing pictures, convince yourself that the arbitrary unit vector $\hat n$ can be written as: $\hat n=\sin\theta\cos\phi\, \hat x +\sin\theta\sin\phi\,\hat y+\cos\theta\,\hat z$ where $\theta$ and $\phi$ are the parameters used to describe spherical coordinates.
2. Find the entries of the matrix $\hat n\cdot\vec \sigma$ where the “matrix-valued-vector” $\vec \sigma$ is given in terms of the Pauli spin matrices by $\vec\sigma=\sigma_x\, \hat x + \sigma_y\, \hat y+\sigma_z\, \hat z$ and $\hat n$ is given in part (a) above.
• Found in: Quantum Fundamentals course(s)

Problem

5 min.

##### Orthogonal Brief

Consider the quantum state: $\left\vert \psi\right\rangle = \frac{1}{\sqrt{3}}\left\vert +\right\rangle+ i\frac{\sqrt{2}}{\sqrt{3}} \left\vert -\right\rangle$

Find the normalized vector $\vert \phi\rangle$ that is orthogonal to it.

• 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

##### Quantum concentration
Consider one particle confined to a cube of side $L$; the concentration in effect is $n=L^{-3}$. Find the kinetic energy of the particle when in the ground state. There will be a value of the concentration for which this zero-point quantum kinetic energy is equal to the temperature $kT$. (At this concentration the occupancy of the lowest orbital is of the order of unity; the lowest orbital always has a higher occupancy than any other orbital.) Show that the concentration $n_0$ thus defined is equal to the quantum concentration $n_Q$ defined by (63): $$n_Q \equiv \left(\frac{MkT}{2\pi\hbar^2}\right)^{\frac32}$$ within a factor of the order of unity.
• Found in: Thermal and Statistical Physics course(s)

Problem

##### Spin Fermi Estimate
The following two problems ask you to make Fermi estimates. In a good Fermi estimate, you start from basic scientific facts you already know or quantities that you can reasonably estimate based on your life experiences and then reason your way to estimate a quantity that you would not be able guess. You may look up useful conversion factors or constants. Use words, pictures, and equations to explain your reasoning:
1. Imagine that you send a pea-sized bead of silver through a Stern-Gerlach device oriented to measure the z-component of intrinsic spin. Estimate the total z-component of the intrinsic spin of the ball you would measure in the HIGHLY improbable case that every atom is spin up.
2. Protons, neutrons, and electrons are all spin-1/2 particles. Give a (very crude) order of magnitude estimate of the number of these particles in your body.
• Found in: Quantum Fundamentals course(s)

Problem

##### Diagonalization Part II
First complete the problem Diagonalization. In that notation:
1. Find the matrix $S$ whose columns are $|\alpha\rangle$ and $|\beta\rangle$. Show that $S^{\dagger}=S^{-1}$ by calculating $S^{\dagger}$ and multiplying it by $S$. (Does the order of multiplication matter?)
2. Calculate $B=S^{-1} C S$. How is the matrix $E$ related to $B$ and $C$? The transformation that you have just done is an example of a “change of basis”, sometimes called a “similarity transformation.” When the result of a change of basis is a diagonal matrix, the process is called diagonalization.
• Found in: Quantum Fundamentals course(s)

Small White Board Question

30 min.

##### Magnetic Moment & Stern-Gerlach Experiments
Students consider the relation (1) between the angular momentum and magnetic moment for a current loop and (2) the force on a magnetic moment in an inhomogeneous magnetic field. Students make a (classical) prediction of the outcome of a Stern-Gerlach experiment.
• Found in: Quantum Fundamentals course(s)

Lecture

30 min.

##### Determining $|\pm_x\rangle$ and $|\pm_y\rangle$ in the $S_z$ basis

Lecture about finding $\left|{\pm}\right\rangle _x$ and then $\left|{\pm}\right\rangle _y$. There are two conventional choices to make: relative phase for $_x\left\langle {+}\middle|{-}\right\rangle _x$ and $_y\left\langle {+}\middle|{+}\right\rangle _x$.

So far, we've talked about how to calculate measurement probabilities if you know the input and output quantum states using the probability postulate:

$\mathcal{P} = | \left\langle {\psi_{out}}\middle|{\psi_{in}}\right\rangle |^2$

Now we're going to do this process in reverse.

I want to be able to relate the output states of Stern-Gerlach analyzers oriented in different directions to each other (like $\left|{\pm}\right\rangle _x$ and $\left|{\pm}\right\rangle _x$ to $\left|{\pm}\right\rangle$). Since $\left|{\pm}\right\rangle$ forms a basis, I can write any state for a spin-1/2 system as a linear combination of those states, including these special states.

I'll start with $\left|{+}\right\rangle _x$ written in the $S_z$ basis with general coefficients:

$\left|{+}\right\rangle _x = a \left|{+}\right\rangle + be^{i\phi} \left|{-}\right\rangle$

Notice that:

(1) $a$, $b$, and $\phi$ are all real numbers; (2) the relative phase is loaded onto the second coefficient only.

My job is to use measurement probabilities to determine $a$, $b$, and $\phi$.

I'll prepare a state $\left|{+}\right\rangle _x$ and then send it through $x$, $y$, and $z$ analyzers. When I do that, I see the following probabilities:

 Input = $\left|{+}\right\rangle _x$ $S_x$ $S_y$ $S_z$ $P(\hbar/2)$ 1 1/2 1/2 $P(-\hbar/2)$ 0 1/2 1/2

First, looking at the probability for the $S_z$ components:

$\mathcal(S_z = +\hbar/2) = | \left\langle {+}\middle|{+}\right\rangle _x |^2 = 1/2$

Plugging in the $\left|{+}\right\rangle _x$ written in the $S_z$ basis:

$1/2 = \Big| \left\langle {+}\right|\Big( a\left|{+}\right\rangle + be^{i\phi} \left|{-}\right\rangle \Big) \Big|^2$

Distributing the $\left\langle {+}\right|$ through the parentheses and use orthonormality: \begin{align*} 1/2 &= \Big| a\cancelto{1}{\left\langle {+}\middle|{+}\right\rangle } + be^{i\phi} \cancelto{0}{\left\langle {+}\middle|{-}\right\rangle } \Big|^2 \\ &= |a|^2\12pt] \rightarrow a &= \frac{1}{\sqrt{2}} \end{align*} Similarly, looking at $S_z = -\hbar/2$: \begin{align*} \mathcal(S_z = +\hbar/2) &= | \left\langle {-}\middle|{+}\right\rangle _x |^2 = 1/2 \\ 1/2 = \Big| \left\langle {-}\right|\Big( a\left|{+}\right\rangle + be^{i\phi} \left|{-}\right\rangle \Big) \Big|^2\\ 1/2 &= \Big| a\cancelto{0}{\left\langle {-}\middle|{+}\right\rangle } + be^{i\phi} \cancelto{1}{\left\langle {-}\middle|{-}\right\rangle } \Big|^2 \\ &= |be^{i\phi}|^2\\ &= |b|^2 \cancelto{1}{(e^{i\phi})(e^{-i\phi})}\\[12pt] \rightarrow b &= \frac{1}{\sqrt{2}} \end{align*} I can't yet solve for $\phi$ but I can do similar calculations for $\left|{-}\right\rangle _x$:  Input = $\left|{-}\right\rangle _x$ $S_x$ $S_y$ $S_z$ $P(\hbar/2)$ 0 1/2 1/2 $P(-\hbar/2)$ 1 1/2 1/2 \begin{align*} \left|{-}\right\rangle _x &= c \left|{+}\right\rangle + de^{i\gamma} \left|{-}\right\rangle \\ \mathcal(S_z = +\hbar/2) &= | \left\langle {+}\middle|{-}\right\rangle _x |^2 = 1/2\\ \rightarrow c = \frac{1}{\sqrt{2}}\\ \mathcal(S_z = +\hbar/2) &= | \left\langle {-}\middle|{-}\right\rangle _x |^2 = 1/2\\ \rightarrow d = \frac{1}{\sqrt{2}}\\ \end{align*} So now I have: \begin{align*} \left|{+}\right\rangle _x &= \frac{1}{\sqrt{2}} \left|{+}\right\rangle + \frac{1}{\sqrt{2}}e^{i\beta} \left|{-}\right\rangle \\ \left|{-}\right\rangle _x &= \frac{1}{\sqrt{2}} \left|{+}\right\rangle + \frac{1}{\sqrt{2}}e^{i\gamma} \left|{-}\right\rangle \\ \end{align*} I know $\beta \neq \gamma$ because these are not the same state - they are orthogonal to each other: \begin{align*} 0 &= \,_x\left\langle {+}\middle|{-}\right\rangle _x \\ &= \Big(\frac{1}{\sqrt{2}} \left\langle {+}\right| + \frac{1}{\sqrt{2}}e^{i\beta} \left\langle {-}\right| \Big)\Big( \frac{1}{\sqrt{2}} \left|{+}\right\rangle + \frac{1}{\sqrt{2}}e^{i\gamma} \left|{-}\right\rangle \Big)\\ \end{align*} Now FOIL like mad and use orthonormality: \begin{align*} 0 &= \frac{1}{2}\Big(\cancelto{1}{\left\langle {+}\middle|{+}\right\rangle } + e^{i\gamma} \cancelto{0}{\left\langle {+}\middle|{-}\right\rangle } + e^{i\beta} \cancelto{0}{\left\langle {-}\middle|{+}\right\rangle } + e^{i(\gamma - \beta)}\cancelto{1}{\left\langle {-}\middle|{-}\right\rangle } \Big)\\ &= \frac{1}{2}\Big(1 + e^{i(\gamma - \beta} \Big) \\ \rightarrow & \quad e^{i(\gamma-\beta)} = -1 \end{align*} This means that $\gamma-\beta = \pi$. I don't have enough information to solve for $\beta$ and $\gamma$, but there is a one-time conventional choice made that $\beta = 0$ and $\gamma = 1$, so that: \begin{align*} \left|{+}\right\rangle _x &= \frac{1}{\sqrt{2}} \left|{+}\right\rangle + \frac{1}{\sqrt{2}}\cancelto{1}{e^{i0}} \left|{-}\right\rangle \\ \left|{-}\right\rangle _x &= \frac{1}{\sqrt{2}} \left|{+}\right\rangle + \frac{1}{\sqrt{2}}\cancelto{-1}{e^{i\pi}} \left|{-}\right\rangle \\[12pt] \rightarrow \left|{+}\right\rangle _x &= \frac{1}{\sqrt{2}} \left|{+}\right\rangle \color{red}{+} \frac{1}{\sqrt{2}}\left|{-}\right\rangle \\ \left|{-}\right\rangle _x &= \frac{1}{\sqrt{2}} \left|{+}\right\rangle \color{red}{-} \frac{1}{\sqrt{2}}\left|{-}\right\rangle \\[12pt] \end{align*} When $\left|{\pm}\right\rangle _y$ is the input state:  Input = $\left|{+}\right\rangle _y$ $S_x$ $S_y$ $S_z$ $P(\hbar/2)$ 1/2 1 1/2 $P(-\hbar/2)$ 1/2 0 1/2  Input = $\left|{-}\right\rangle _y$ $S_x$ $S_y$ $S_z$ $P(\hbar/2)$ 1/2 0 1/2 $P(-\hbar/2)$ 1/2 1 1/2 The calculations proceed in the same way. The $S_z$ probabilities give me: \begin{align*} \left|{+}\right\rangle _y &= \frac{1}{\sqrt{2}} \left|{+}\right\rangle + \frac{1}{\sqrt{2}}\cancelto{1}{e^{i\alpha}} \left|{-}\right\rangle \\ \left|{-}\right\rangle _y &= \frac{1}{\sqrt{2}} \left|{+}\right\rangle + \frac{1}{\sqrt{2}}\cancelto{-1}{e^{i\theta}} \left|{-}\right\rangle \\ \end{align*} The orthongality between $\left|{+}\right\rangle _y$ and $\left|{-}\right\rangle _y$ mean that $\theta - \alpha = \pi$. But I also know the $S_x$ probabilities and how to write $|ket{\pm}_x$ in the $S_z$ basis. For an input of $\left|{+}\right\rangle _y$: \begin{align*} \mathcal(S_x = +\hbar/2) &= | \,_x\left\langle {+}\middle|{+}\right\rangle _y |^2 = 1/2 \\ 1/2 &= \Big| \Big(\frac{1}{\sqrt{2}} \left\langle {+}\right| + \frac{1}{\sqrt{2}}\left\langle {-}\right|\Big) \Big( \frac{1}{\sqrt{2}}\left|{+}\right\rangle + \frac{1}{\sqrt{2}}e^{i\alpha} \left|{-}\right\rangle \Big) \Big|^2\\ 1/2 &= \Big| \frac{1}{\sqrt{2}} \frac{1}{\sqrt{2}} \cancelto{1}{\left\langle {+}\middle|{+}\right\rangle } + \frac{1}{\sqrt{2}} \frac{1}{\sqrt{2}}e^{i\alpha} \cancelto{1}{\left\langle {-}\middle|{-}\right\rangle } \Big|^2 \\ &= \frac{1}{4}|1+e^{i\alpha}|^2\\ &= \frac{1}{4} \Big( 1+e^{i\alpha}\Big) \Big( 1+e^{-i\alpha}\Big)\\ &= \frac{1}{4} \Big( 2+e^{i\alpha} + e^{-i\alpha}\Big)\\ &= \frac{1}{4} \Big( 2+2\cos\alpha\Big)\\ \frac{1}{2} &= \frac{1}{2} + \frac{1}{2}\cos\alpha \\ 0 &= \cos\alpha\\ \rightarrow \alpha = \pm \frac{\pi}{2} \end{align*} Here, again, I can't solve exactly for alpha (or $\theta$), but the convention is to choose $alpha = \frac{\pi}{2}$ and $\theta = \frac{3\pi}{2}$, making \begin{align*} \left|{+}\right\rangle _y &= \frac{1}{\sqrt{2}} \left|{+}\right\rangle + \frac{1}{\sqrt{2}}\cancelto{i}{e^{i\pi/2}} \left|{-}\right\rangle \\ \left|{-}\right\rangle _y &= \frac{1}{\sqrt{2}} \left|{+}\right\rangle + \frac{1}{\sqrt{2}}\cancelto{-i}{e^{i3\pi/2}} \left|{-}\right\rangle \\ \rightarrow \left|{+}\right\rangle _y &= \frac{1}{\sqrt{2}} \left|{+}\right\rangle \color{red}{+} \frac{\color{red}{i}}{\sqrt{2}} \left|{-}\right\rangle \\ \left|{-}\right\rangle _y &= \frac{1}{\sqrt{2}} \left|{+}\right\rangle \color{red}{-} \frac{\color{red}{i}}{\sqrt{2}} \left|{-}\right\rangle \\ \end{align*} If I use these two convenctions for the relative phases, then I can write down $\left|{\pm}\right\rangle _n$ in an arbitrary direction described by the spherical coordinates $\theta$ and $\phi$ as: Discuss the generalize eigenstates: \begin{align*}\ \left|{+}\right\rangle _n &= \cos \frac{\theta}{2} \left|{+}\right\rangle + \sin \frac{\theta}{2} e^{i\phi} \left|{-}\right\rangle \\ \left|{-}\right\rangle _n &= \sin \frac{\theta}{2} \left|{+}\right\rangle - \cos \frac{\theta}{2} e^{i\phi} \left|{-}\right\rangle \end{align*} And how the $\left|{\pm}\right\rangle _x$ and $\left|{\pm}\right\rangle _y$ are consistent. • Found in: Quantum Fundamentals course(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) Kinesthetic 30 min. ##### Inner Product of Spin-1/2 System with Arms Students use their arms to act out two spin-1/2 quantum states and their inner product. • Found in: Arms Sequence for Complex Numbers and Quantum States sequence(s) Small Group Activity 120 min. ##### Finding Eigenvectors and Eigenvalues Eigenvalues and Eigenvectors Each group will be assigned one of the following matrices. \[ A_1\doteq \begin{pmatrix} 0&-1\\ 1&0\\ \end{pmatrix} \hspace{2em} A_2\doteq \begin{pmatrix} 0&1\\ 1&0\\ \end{pmatrix} \hspace{2em} A_3\doteq \begin{pmatrix} -1&0\\ 0&-1\\ \end{pmatrix}
$A_4\doteq \begin{pmatrix} a&0\\ 0&d\\ \end{pmatrix} \hspace{2em} A_5\doteq \begin{pmatrix} 3&-i\\ i&3\\ \end{pmatrix} \hspace{2em} A_6\doteq \begin{pmatrix} 0&0\\ 0&1\\ \end{pmatrix} \hspace{2em} A_7\doteq \begin{pmatrix} 1&2\\ 1&2\\ \end{pmatrix}$
$A_8\doteq \begin{pmatrix} -1&0&0\\ 0&-1&0\\ 0&0&-1\\ \end{pmatrix} \hspace{2em} A_9\doteq \begin{pmatrix} -1&0&0\\ 0&-1&0\\ 0&0&1\\ \end{pmatrix}$
$S_x\doteq \frac{\hbar}{2}\begin{pmatrix} 0&1\\ 1&0\\ \end{pmatrix} \hspace{2em} S_y\doteq \frac{\hbar}{2}\begin{pmatrix} 0&-i\\ i&0\\ \end{pmatrix} \hspace{2em} S_z\doteq \frac{\hbar}{2}\begin{pmatrix} 1&0\\ 0&-1\\ \end{pmatrix}$

1. Find the eigenvalues.
2. Find the (unnormalized) eigenvectors.
3. Describe what this transformation does.

If you finish early, try another matrix with a different structure, i.e. real vs. complex entries, diagonal vs. non-diagonal, $2\times 2$ vs. $3\times 3$, with vs. without explicit dimensions.

## Instructor's Guide

### Main Ideas

This is a small group activity for groups of 3-4. The students will be given one of 10 matrices. The students are then instructed to find the eigenvectors and eigenvalues for this matrix and record their calculations on their medium-sized whiteboards. In the class discussion that follows students report their finding and compare and contrast the properties of the eigenvalues and eigenvectors they find. Two topics that should specifically discussed are the case of repeated eigenvalues (degeneracy) and complex eigenvectors, e.g., in the case of some pure rotations, special properties of the eigenvectors and eigenvalues of hermitian matrices, common eigenvectors of commuting operators.

### Introduction

Give a mini-lecture on how to calculate eigenvalues and eigenvectors. It is often easiest to do this with an example. We like to use the matrix $A_7\doteq\begin{pmatrix}1&2\cr 9&4\cr\end{pmatrix}$ from the https://paradigms.oregonstate.edu/activities/2179 https://paradigms.oregonstate.edu/activities/2179 Finding Eigenvectors and Eigenvalues since the students have already seen this matrix and know what it's eigenvectors are. Then every group is given a handout, assigned a matrix, and then asked to: - Find the eigenvalues - Find the (unnormalized) eigenvectors - Normalize the eigenvectors - Describe what this transformation does

### Student Conversations

• Typically, students can find the eigenvalues without too much problem. Eigenvectors are a different story. To find the eigenvectors, they will have two equations with two unknowns. They expect to be able to find a unique solution. But, since any scalar multiple of an eigenvector is also an eigenvector, their two equations will be redundant. Typically, they must choose any convenient value for one of the components (e.g. $x=1$) and solve for the other one. Later, they can use this scale freedom to normalize their vector.
• The examples in this activity were chosen to include many of the special cases that can trip students up. A common example is when the two equations for the eigenvector amount to something like $x=x$ and $y=-y$. For the first equation, they may need help to realize that $x=$ “anything” is the solution. And for the second equation, sadly, many students need to be helped to the realization that the only solution is $y=0$.

### Wrap-up

The majority of the this activity is in the wrap-up conversation.

The [[whitepapers:narratives:eigenvectorslong|Eigenvalues and Eigenvectors Narrative]] provides a detailed narrative interpretation of this activity, focusing on the wrap-up conversation.

• Complex eigenvectors: connect to discussion of rotations in the Linear Transformations activity where there did not appear to be any vectors that stayed the same.
• Degeneracy: Define degeneracy as the case when there are repeated eigenvalues. Make sure the students see that, in the case of degeneracy, an entire subspace of vectors are all eigenvectors.
• Diagonal Matrices: Discuss that diagonal matrices are trivial. Their eigenvalues are just their diagonal elements and their eigenvectors are just the standard basis.
• Matrices with dimensions: Students should see from these examples that when you multiply a transformation by a real scalar, its eigenvalues are multiplied by that scalar and its eigenvectors are unchanges. If the scalar has dimensions (e.g. $\hbar/2$), then the eigenvalues have the same dimensions.

• Found in: Quantum Fundamentals course(s) Found in: Matrices & Operators sequence(s)

Computational Activity

120 min.

##### Sinusoidal basis set
Students compute inner products to expand a wave function in a sinusoidal basis set. This activity introduces the inner product for wave functions, and the idea of approximating a wave function using a finite set of basis functions.
• Found in: Computational Physics Lab II course(s) Found in: Computational wave function inner products sequence(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)

Computational Activity

120 min.

##### Kinetic energy
Students implement a finite-difference approximation for the kinetic energy operator as a matrix, and then use numpy to solve for eigenvalues and eigenstates, which they visualize.
• Found in: Computational Physics Lab II course(s) Found in: Computational wave function inner products sequence(s)

Computational Activity

120 min.

##### Position operator
Students find matrix elements of the position operator $\hat x$ in a sinusoidal basis. This allows them to express this operator as a matrix, which they can then numerically diagonalize and visualize the eigenfunctions.
• Found in: Computational Physics Lab II course(s) Found in: Computational wave function inner products sequence(s)

Small Group Activity

10 min.

##### Matrix Representation of Angular Momentum
This activity allows students to puzzle through indexing, the from of operators in quantum mechanics, and working with the new quantum numbers on the sphere in an applied context.
• Found in: Central Forces course(s)

Problem

##### Quantum harmonic oscillator
1. Find the entropy of a set of $N$ oscillators of frequency $\omega$ as a function of the total quantum number $n$. Use the multiplicity function: $$g(N,n) = \frac{(N+n-1)!}{n!(N-1)!}$$ and assume that $N\gg 1$. This means you can make the Sitrling approximation that $\log N! \approx N\log N - N$. It also means that $N-1 \approx N$.

2. Let $U$ denote the total energy $n\hbar\omega$ of the oscillators. Express the entropy as $S(U,N)$. Show that the total energy at temperature $T$ is $$U = \frac{N\hbar\omega}{e^{\frac{\hbar\omega}{kT}}-1}$$ This is the Planck result found the hard way. We will get to the easy way soon, and you will never again need to work with a multiplicity function like this.

• Found in: Thermal and Statistical Physics 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)

Small Group Activity

10 min.

##### Using Tinker Toys to Represent Spin 1/2 Quantum Systems
Students use Tinker Toys to represent each component in a two-state quantum spin system in all three standard bases ($x$, $y$, and $z$). Through a short series of instructor-led prompts, students explore the difference between overall phase (which does NOT change the state of the system) and relative phase (which does change the state of the system). This activity is optional in the Arms Sequence Arms Sequence for Complex Numbers and Quantum States.
• Found in: Arms Sequence for Complex Numbers and Quantum States sequence(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)

Problem

##### Energy fluctuations
Consider a system of fixed volume in thermal contact with a resevoir. Show that the mean square fluctuations in the energy of the system is $$\left<\left(\varepsilon-\langle\varepsilon\rangle\right)^2\right> = k_BT^2\left(\frac{\partial U}{\partial T}\right)_{V}$$ Here $U$ is the conventional symbol for $\langle\varepsilon\rangle$. Hint: Use the partition function $Z$ to relate $\left(\frac{\partial U}{\partial T}\right)_V$ to the mean square fluctuation. Also, multiply out the term $(\cdots)^2$.
• Found in: Thermal and Statistical Physics course(s)

Problem

5 min.

##### Free energy of a two state system
1. Find an expression for the free energy as a function of $T$ of a system with two states, one at energy 0 and one at energy $\varepsilon$.

2. From the free energy, find expressions for the internal energy $U$ and entropy $S$ of the system.

3. Plot the entropy versus $T$. Explain its asymptotic behavior as the temperature becomes high.

4. Plot the $S(T)$ versus $U(T)$. Explain the maximum value of the energy $U$.

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

Lecture

120 min.

##### Ideal Gas
These notes from week 6 of https://paradigms.oregonstate.edu/courses/ph441 cover the ideal gas from a grand canonical standpoint starting with the solutions to a particle in a three-dimensional box. They include a number of small group activities.
• Found in: Thermal and Statistical Physics course(s)

Problem

##### Fluctuations in a Fermi gas
(K&K 7.11) Show for a single orbital of a fermion system that \begin{align} \left<(\Delta N)^2\right> = \left<N\right>(1+\left<N\right>) \end{align} if $\left<N\right>$ is the average number of fermions in that orbital. Notice that the fluctuation vanishes for orbitals with energies far enough from the chemical potential $\mu$ so that $\left<N\right>=1$ or $\left<N\right>=0$.
• Found in: Thermal and Statistical Physics course(s)

Lecture

30 min.

##### Review of Thermal Physics
These are notes, essentially the equation sheet, from the final review session for https://paradigms.oregonstate.edu/courses/ph441.
• Found in: Thermal and Statistical Physics course(s)

Lecture

120 min.

##### Fermi and Bose gases
These lecture notes from week 7 of https://paradigms.oregonstate.edu/courses/ph441 apply the grand canonical ensemble to fermion and bosons ideal gasses. They include a few small group activities.
• Found in: Thermal and Statistical Physics course(s)

Lecture

120 min.

##### Chemical potential and Gibbs distribution
These notes from the fifth week of https://paradigms.oregonstate.edu/courses/ph441 cover the grand canonical ensemble. They include several small group activities.
• Found in: Thermal and Statistical Physics course(s)

Lecture

120 min.

##### Thermal radiation and Planck distribution
These notes from the fourth week of https://paradigms.oregonstate.edu/courses/ph441 cover blackbody radiation and the Planck distribution. They include a number of small group activities.
• Found in: Thermal and Statistical Physics course(s)

Small Group Activity

120 min.

##### Box Sliding Down Frictionless Wedge
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.
• Found in: Theoretical Mechanics 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)