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Activities

Small Group Activity

60 min.

Multiple Representations of a Quantum State
Students re-represent a state given in Dirac notation in matrix notation
• Found in: Quantum Fundamentals course(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.

Curvilinear Basis Vectors
Students use their arms to depict (sequentially) the different cylindrical and spherical basis vectors at the location of their shoulder (seen in relation to a specified origin of coordinates: either a set of axes hung from the ceiling of the room or perhaps a piece of furniture or a particular corner of the room).
• Found in: Static Fields, Central Forces, AIMS Maxwell, Surfaces/Bridge Workshop, Problem-Solving course(s) Found in: Geometry of Vector Fields Sequence, Curvilinear Coordinate Sequence sequence(s)

Small White Board Question

5 min.

Representations of Vectors
Students each recall a representation of vectors that they have seen before and record it on an individual whiteboard. The instructor uses these responses to generate a whole class discussion that compares and contrasts the features of the representations. If appropriate for the class, the instructor introduces bra/ket notation as a new, but valuable representation.
• Found in: AIMS Maxwell, Static Fields, Surfaces/Bridge Workshop, Problem-Solving 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.

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

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)

Problem

Using Gibbs Free Energy

You are given the following Gibbs free energy: \begin{equation*} G=-k T N \ln \left(\frac{a T^{5 / 2}}{p}\right) \end{equation*} where $a$ is a constant (whose dimensions make the argument of the logarithm dimensionless).

1. Compute the entropy.

2. Work out the heat capacity at constant pressure $C_p$.

3. Find the connection among $V$, $p$, $N$, and $T$, which is called the equation of state (Hint: find the volume as a partial derivative of the Gibbs free energy).

4. Compute the internal energy $U$.

• Found in: Energy and Entropy 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)

Problem

5 min.

Fourier Series for the Ground State of a Particle-in-a-Box.
Treat the ground state of a quantum particle-in-a-box as a periodic function.
1. Set up the integrals for the Fourier series for this state.

2. Which terms will have the largest coefficients? Explain briefly.

3. Are there any coefficients that you know will be zero? Explain briefly.

4. Using the technology of your choice or by hand, calculate the four largest coefficients. With screen shots or otherwise, show your work.

5. Using the technology of your choice, plot the ground state and your approximation on the same axes.
• Found in: Oscillations and Waves, None course(s)

Small Group Activity

10 min.

Angular Momentum in Polar Coordinates
Students learn how to express Angular Momentum as a vector quantity in polar coordinates, and then in Cylindrical and Spherical Coordinates
• Found in: Central Forces course(s)

Problem

5 min.

Vectors

Let \begin{align} \boldsymbol{\vec a} &= \boldsymbol{\hat x}-3\boldsymbol{\hat y}-\boldsymbol{\hat z}\\ \boldsymbol{\vec b} &= \boldsymbol{\hat x}+\boldsymbol{\hat y}+2\boldsymbol{\hat z}\\ {\boldsymbol{\vec c}} &= -2\boldsymbol{\hat x}-\boldsymbol{\hat y}+\boldsymbol{\hat z}\\ \boldsymbol{\vec d} &= -\boldsymbol{\hat x}-\boldsymbol{\hat y}+\boldsymbol{\hat z} \end{align}

Use the dot product to determine which pairs (if any) of these vectors

1. Are perpendicular?
2. Are parallel?
3. Have an angle less than $\pi/2$ between them?
4. Have an angle of more than $\pi/2$ between them?

• Found in: AIMS Maxwell, Static Fields, Problem-Solving course(s)

Kinesthetic

30 min.

Using Arms to Visualize Transformations of Complex Two-Component Vectors (MathBits)
Students, working in pairs, represent two component complex vectors with their left arms. Through a short series of instructor led prompts, students move their left arms to show how various linear transformations affect each complex component.
• Found in: Quantum Fundamentals course(s) Found in: Arms Sequence for Complex Numbers and Quantum States 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)

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.

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)

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)

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.

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)

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)

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

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

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.

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

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.

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)

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)

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)

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:

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

Student Conversations

1. One could have each group report out, or the instructor could discuss a few key examples.

For expectation value, I like to talk about Case 2: $\frac{i}{2}\left|{+}\right\rangle -\frac{\sqrt{3}}{2}\left|{-}\right\rangle$, where the probabilities of the two outcomes are not equal to show how the weighting plays out. Also, the expectation value is not a possible measurement value, and I like to talk about that. “Expectation” value is a misleading name for this quantity - it characterizes the distribution and is not necessarily a result of an individual measurement.

I also like to discuss an example like Case 5: $\left|{1}\right\rangle _x$ where the distribution is symmetric around $0\hbar$.

2. I think it's important to encourage students to calculate expectation values both ways (with probabilities and as a bracket with matrix notation) while the teaching team is available to help them.

3. For quantum uncertainty, I like to talk about an example like Case 3: $\left|{+}\right\rangle _x$ where all the individual measurements are the same ”distance” away from the expectation value as a sensemaking exercise to connect to a conceptual interpretation of physics.

I also like to discuss an example like Case 5: $\left|{-1}\right\rangle _x$, where the fact that we're taking an rms average is apparent: half the measurements are $\hbar$ away from the expectation value and the other half are $0\hbar$ away, but the uncertainty is $\hbar/\sqrt{2}$.

• Found in: Quantum Fundamentals course(s)

Problem

Normalization of Quantum States
Show that if a linear combination of ring energy eigenstates is normalized, then the coefficients must satisfy $$\sum_{m=-\infty}^{\infty} \vert c_m\vert^2=1$$
• Found in: Central Forces 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)

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

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)

Small Group Activity

30 min.

Heat capacity of N2
Students sketch the temperature-dependent heat capacity of molecular nitrogen. They apply the equipartition theorem and compute the temperatures at which degrees of freedom “freeze out.”
• Found in: Contemporary Challenges 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)

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

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.

Quantum Particle in a 2-D Box
You know that the normalized spatial eigenfunctions for a particle in a 1-D box of length $L$ are $\sqrt{\frac{2}{L}}\sin{\frac{n\pi x}{L}}$. If you want the eigenfunctions for a particle in a 2-D box, then you just multiply together the eigenfunctions for a 1-D box in each direction. (This is what the separation of variables procedure tells you to do.)
1. Find the normalized eigenfunctions for a particle in a 2-D box with sides of length $L$ in the $x$-direction and length $W$ in the $y$-direction.
2. Find the Hamiltonian for a 2-D box and show that your eigenstates are indeed eigenstates and find a formula for the possible energies
3. Any sufficiently smooth spatial wave function inside a 2-D box can be expanded in a double sum of the product wave functions, i.e. $$\psi(x,y)=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\, c_{nm}\; \hbox{eigenfunction}_n(x)\;\hbox{eigenfunction}_m(y)$$ Using your expressions from part (a) above, write out all the terms in this sum out to $n=3$, $m=3$. Arrange the terms, conventionally, in terms of increasing energy.

You may find it easier to work in bra/ket notation: \begin{align*} \left|{\psi}\right\rangle &=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\, c_{nm}\left|{n}\right\rangle \left|{m}\right\rangle \\ &=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\, c_{nm}\left|{nm}\right\rangle \end{align*}

4. Find a formula for the $c_{nm}$s in part (c). Find the formula first in bra ket notation and then rewrite it in wave function notation.
• Found in: Central Forces, None course(s)

Kinesthetic

10 min.

Using Arms to Represent Time Dependence in Spin 1/2 Systems
Students, working in pairs, use their left arms to demonstrate time evolution in spin 1/2 quantum systems.
• Found in: Quantum Fundamentals course(s) Found in: Arms Sequence for Complex Numbers and Quantum States sequence(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)

Lecture

5 min.

1-D Particle-in-a-box

Hamiltonian: \begin{align} \hat{H} &\doteq -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} \end{align} Eigenstates: \begin{align} \left|{n}\right\rangle &\doteq\sqrt{\frac{2}{L}}\, \sin\frac{n\pi x}{L}\\ n&=\left\{1, 2, 3, \dots\right\} \end{align}

Eigenvalue Equations: \begin{align} \hat{H}\left|{n}\right\rangle &=\frac{\pi^2\hbar^2}{2\mu L^2}\, n^2 \left|{n}\right\rangle \\ \end{align}

Particle-on-a-Ring

Hamiltonian: \begin{align} \hat{H} &\doteq -\frac{\hbar^2}{2I}\frac{\partial^2}{\partial \phi^2} \end{align} Eigenstates: \begin{align} \left|{m}\right\rangle &\doteq\frac{1}{\sqrt{2\pi r_0}}\, e^{im\phi}\\ m&=\left\{\dots 2, 1, 0, -1, -2, \dots\right\} \end{align} Eigenvalue Equations: \begin{align} \hat{H}\left|{m}\right\rangle &=\frac{\hbar^2}{2I}\, m^2 \left|{m}\right\rangle \\ \hat{L}^2\left|{m}\right\rangle &=\hbar^2\, m^2 \left|{m}\right\rangle \\ \hat{L}_z\left|{m}\right\rangle &=\hbar\, m \left|{m}\right\rangle \end{align}

2-D Particle-in-a-Box

Hamiltonian: \begin{align} \hat{H} &\doteq -\frac{\hbar^2}{2m}\Big(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}) \end{align} Eigenstates: \begin{align} \left|{mn}\right\rangle &\doteq\sqrt{\frac{2}{L_x}}\sqrt{\frac{2}{L_y}}\, \sin\frac{m\pi x}{L_x}\sin\frac{n\pi y}{L_y}\\ m&=\left\{1, 2, 3, \dots\right\}\\ n&=\left\{1, 2, 3, \dots\right\} \end{align} Eigenvalue Equations: \begin{align} \hat{H}\left|{mn}\right\rangle &=\frac{\pi^2\hbar^2}{2\mu}\, \left(\frac{m^2}{L_x^2}+\frac{n^2}{L_y^2}\right) \left|{mn}\right\rangle \\ \end{align}

Particle-on-a-Sphere

Hamiltonian: \begin{align} \hat{H} &\doteq -\frac{\hbar^2}{2I} \Big[\frac{1}{\sin\theta}\frac{\partial}{\partial \theta} \Big(\sin\theta \frac{\partial}{\partial \theta} \big) + \frac{1}{\sin^2\theta}\frac{\partial^2}{\partial \phi^2} \Big] \end{align} Eigenstates: \begin{align} \left|{\ell m}\right\rangle &\doteq Y_{\ell}^m(\theta, \phi)\\ &=(-1)^{\frac{m+|m|}{2}}\sqrt{\frac{2\ell+1}{4\pi}\frac{(\ell-m)!}{(\ell+m)!}} \,P_{\ell}^m(\cos\theta)\, e^{im\phi}\\ \ell&=\left\{0, 1, 2, \dots\right\}\\ m&=\left\{\ell, \dots , 0, \dots,-\ell\right\} \end{align} Eigenvalue Equations: \begin{align} \hat{H}\left|{\ell m}\right\rangle &=\frac{\hbar^2}{2I}\, \ell(\ell+1) \left|{\ell m}\right\rangle \\ \hat{L}^2\left|{\ell m}\right\rangle &=\hbar^2\, \ell(\ell+1) \left|{\ell m}\right\rangle \\ \hat{L}_z\left|{\ell m}\right\rangle &=\hbar\, m \left|{\ell m}\right\rangle \end{align}

Hydrogen Atom

Hamiltonian: \begin{align} \hat{H} &\doteq -\frac{\hbar^2}{2\mu} \nabla^2 - \frac{ke^2}{r} \\ &\doteq -\frac{\hbar^2}{2\mu r^2} \Big[\frac{\partial}{\partial r} \Big( r^2 \frac{\partial}{\partial r} \Big) + \frac{1}{\sin\theta}\frac{\partial}{\partial \theta} \Big(\sin\theta \frac{\partial}{\partial \theta} \big) + \frac{1}{\sin^2\theta}\frac{\partial^2}{\partial \phi^2} \Big] - \frac{ke^2}{r} \end{align} Eigenstates: \begin{align} \left|{n\ell m}\right\rangle &\doteq R_{n\ell}(r)\, Y_{\ell}^m(\theta, \phi)\\ &=-\sqrt{\left(\frac{2Z}{na_0}\right)^3 \frac{(n-\ell-1)!}{2n[(n+\ell)!]^3}} \left(\frac{2\rho}{n}\right)^{\ell}\, e^{-\frac{\rho}{n}}\, L_{n+\ell}^{2\ell+1}{\scriptstyle{\left(\frac{2\rho}{n}\right)}} (-1)^{\frac{m+|m|}{2}} \sqrt{\frac{2\ell+1}{4\pi}\frac{(\ell-m)!}{(\ell+m)!}} \,P_{\ell}^m(\cos\theta)\, e^{im\phi}\\ \rho&=\frac{Zr}{a_0}\\ n&=\left\{1, 2, 3,\dots\right\}\\ \ell&=\left\{0, 1, 2, \dots, n-1\right\}\\ m&=\left\{\ell, \dots , 0, \dots,-\ell\right\} \end{align} Eigenvalue Equations: \begin{align} \hat{H}\left|{n\ell m}\right\rangle &=-\frac{1}{2}\left(\frac{Ze^2}{4\pi\epsilon_0}\right)^2 \frac{\mu}{\hbar^2}\,\frac{1}{n^2}\, \left|{n \ell m}\right\rangle \\ &=-13.6 \text{eV}\,\frac{1}{n^2}\, \left|{n \ell m}\right\rangle \\ \hat{L}^2\left|{n \ell m}\right\rangle &=\hbar^2\, \ell(\ell+1) \left|{n \ell m}\right\rangle \\ \hat{L}_z\left|{n \ell m}\right\rangle &=\hbar\, m \left|{n \ell m}\right\rangle \end{align}

• Found in: Central Forces course(s)

Problem

5 min.

Phase in Quantum States

In quantum mechanics, it turns out that the overall phase for a state does not have any physical significance. Therefore, you will need to become quick at rearranging the phase of various states. For each of the vectors listed below, rewrite the vector as an overall complex phase times a new vector whose first component is real and positive. $\left|D\right\rangle\doteq \begin{pmatrix} 7e^{i\frac{\pi}{6}}\\ 3e^{i\frac{\pi}{2}}\\ -1\\ \end{pmatrix}\\ \left|E\right\rangle\doteq \begin{pmatrix} i\\ 4\\ \end{pmatrix}\\ \left|F\right\rangle\doteq \begin{pmatrix} 2+2i\\ 3-4i\\ \end{pmatrix}$

• Found in: Quantum Fundamentals course(s)