Activities
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.
Students use their arms to act out stationary and non-stationary states of a quantum particle on a ring.
In this small group activity, students multiply a general 3x3 matrix with standard basis row/column vectors to pick out individual matrix elements. Students generate the expressions for the matrix elements in bra/ket notation.
This activity reinforces the strategies students have been practicing on each system by letting them create their own matrix operators and columns on the hydrogen atom and do some calculations with them.
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.
For this problem, use the vectors \(|a\rangle = 4 |1\rangle - 3 |2\rangle\) and \(|b\rangle = -i |1\rangle + |2\rangle\).
- Find \(\langle a | b \rangle\) and \(\langle b | a \rangle\). Discuss how these two inner products are related to each other.
- 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 \(\langle b|\hat{Q}|a \rangle\).
- 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?
Students use completeness relations to write a matrix element of a spin component in a different basis.
Students use the completeness relation for the position basis to re-express expressions in bra/ket notation in wavefunction notation.
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).
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.
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.
Problem
Show that if a linear combination of ring energy eigenstates is normalized, then the coefficients must satisfy \begin{equation} \sum_{m=-\infty}^{\infty} \vert c_m\vert^2=1 \end{equation}
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. \[\left|D\right\rangle\doteq \begin{pmatrix} 7e^{i\frac{\pi}{6}}\\ 3e^{i\frac{\pi}{2}}\\ -1\\ \end{pmatrix}\\ \left|E\right\rangle\doteq \begin{pmatrix} i\\ 4\\ \end{pmatrix}\\ \left|F\right\rangle\doteq \begin{pmatrix} 2+2i\\ 3-4i\\ \end{pmatrix} \]
Students, working in pairs, use their left arms to demonstrate time evolution in spin 1/2 quantum systems.
Students use their arms to act out two spin-1/2 quantum states and their inner product.
Students work in small groups to use completeness relations to change the basis of quantum states.
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.
- 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.
- 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.
Problem
- 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.)
- 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!)
- 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.
- Find the determinants of \(C\) and \(E\). How do these determinants compare to the eigenvalues of these matrices?
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.
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.
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.
Students re-represent a state given in Dirac notation in matrix notation
Students calculate probabilities for energy, angular momentum, and position as a function of time for an initial state that is a linear combination of energy/angular momentum eigenstates for a particle confined to a ring written in bra-ket notation. This activity helps students build an understanding of when they can expect a quantity to depend on time and to give them more practice moving between representations.
The instructor gives a brief lecture about time dependence of energy eigenstates (e.g. McIntyre, 3.1). Notes for the students are attached.
This short small group activity introduces students to the Leibniz notation used for partial derivatives in thermodynamics, in which the variables being held constant are given explicitly. Students are guided to associate variables to their proper categories.
Students consider the dimensions of spin-state kets and position-basis kets.
Students practice using inner products to find the components of the cartesian basis vectors in the polar basis and vice versa. Then, students use a completeness relation to change bases or cartesian/polar bases and for different spin bases.
Write out the terms in the following sums that have the lowest energy. Stop when you have at least 9 terms, but only stop at some point that is logical, given the symmetries and degeneracies. Briefly explain why you chose to stop when you did. You may have to guess what system you are working from the form of the sum and which eigenvalue(s) determine the energy. You may assume that low energies correspond to eigenvalues near zero. Clearly state any assumptions that you make. You may use bra/ket notation in your solutions. (If these directions are unclear, check out the solutions below for some examples.)
- \[\sum_{m=-\infty}^{\infty} c_m e^{im\phi}\]
- \[\sum_{m=1}^{\infty}\sum_{n=1}^{\infty} c_{mn} \sin\left(\frac{m\pi x}{L_x}\right)\sin\left(\frac{n\pi y}{L_y}\right)\]
- \[\sum_{n=1}^{\infty}\sum_{\ell=0}^{n-1}\sum_{m=-\ell}^{\ell} c_{n \ell m} \left|{n, \ell, m}\right\rangle \]
Problem
Write (a good guess for) the following series using sigma \(\left(\sum\right)\) notation. (If you only know a few terms of a series, you don't know for sure how the series continues.)
\[1 - 2\,\theta^2 + 4\,\theta^4 - 8\,\theta^6 +\,\dots\]
- \[\frac14 - \frac19 + \frac{1}{16} - \frac{1}{25}+\,\dots\]
Problem
Write out the first four nonzero terms in the series:
\[\sum\limits_{n=0}^\infty \frac{1}{n!}\]
- \[\sum\limits_{n=1}^\infty \frac{(-1)^n}{n!}\]
- \begin{equation} \sum\limits_{n=0}^\infty {(-2)^{n}\,\theta^{2n}} \end{equation}
In this lecture, the instructor guides a discussion about translating between bra-ket notation and wavefunction notation for quantum systems.
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.
Students explore what linear transformation matrices do to vectors. The whole class discussion compares & contrasts several different types of transformations (rotation, flip, projections, “scrinch”, scale) and how the properties of the matrices (the determinant, symmetries, which vectors are unchanged) are related to these transformations.
Problem
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: \begin{equation} g(N,n) = \frac{(N+n-1)!}{n!(N-1)!} \end{equation} 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 \(N-1 \approx N\).
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 \begin{equation} U = \frac{N\hbar\omega}{e^{\frac{\hbar\omega}{kT}}-1} \end{equation} 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.
Problem
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!)
- What are the possible results of a measurement of \(S_x\), with what probabilities?
Repeat part a for measurements of \(S_z\).
- 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?
Set-Up a Sequential Measurement
Add an analyzer to the experiment by:
- Break the links between the analyzer and the counters by clicking on the boxes with up and down arrow labels on the analyzer.
- 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.
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 \(S_z=\frac{-\hbar}{2}\)?
Try all four possible combinations of input/outputs for the second analyzer.
What have you learned from these experiments?
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.
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.
Mathematica Activity
30 min.
Students see probability density for eigenstates and linear combinations of eigenstates for a particle on a ring. The three visual representations: standard position vs probability density plot, a ring with colormapping, and cylindrical plot with height and colormapping, are also animated to visualize time-evolution.
Students, 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.
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.
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 \(\left|{\psi_4}\right\rangle \) )
- 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 \(\left|{-}\right\rangle \).
- 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.
- 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.
- 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?
The instructor and students do a skit where students represent quantum states that are “measured” by the instructor resulting in a state collapse.
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.
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
- Activity: Working with Representations on the Ring
- Problem: Ring Table
- Activity: Visualization of Quantum Probabilities for a Particle Confined to a Ring
- Activity: Time Dependence for a Quantum Particle on a Ring
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
Expectation Values and UncertaintyYou have a system that consists of quantum particles with spin. On this system, you will perform a Stern-Gerlach experiment with an analyzer oriented in the \(z\)-direction.
Consider one of the different initial spin states described below:
A spin 1/2 particle described by:
- \(\left|{+}\right\rangle \)
- \(\frac{i}{2}\left|{+}\right\rangle -\frac{\sqrt{3}}{2}\left|{-}\right\rangle \)
\(\left|{+}\right\rangle _x\)
A spin 1 particle described by:
- \(\left|{0}\right\rangle \)
- \(\left|{-1}\right\rangle _x\)
- \(\frac{2}{3}\left|{1}\right\rangle +\frac{i}{3}\left|{0}\right\rangle -\frac{2}{3}\left|{-1}\right\rangle \)
List the possible values of spin you could measure and determine the probability associated with each value of the z-component of spin.
Plot a histogram of the probabilities.
Find the expectation value of the z-component of spin.
- Find the uncertainty of the z-component of spin.
Introduction
I like to break this activity into two parts:
(1) Calculating expectation values and relating them to the associated distributions of the probabilities of results, and
(2) Calculating the quantum uncertainty of the state and relating the uncertainty to distributions of the probabilities of results.
Therefore, I have my students do the first part of the activity before I introduce quantum uncertainty.
I introduce the activity by reminding students about two ways of calculating the expectation value. Given a quantum state \(\left|{\psi}\right\rangle \), for a measurement of an observable represented by an operator \(\hat{A}\) with eigenstates \(\left|{a_i}\right\rangle \) and eigenvalues \(a_i\): \begin{align*} \langle \hat{A} \rangle &= \sum_{i} a_i\mathcal{P}(a_i) \\ &= \left\langle {\psi}\right|\hat{A}\left|{\psi}\right\rangle \end{align*}
After the students calculate expectation values and we have a whole class discussion about 1 of the examples, then I do a lecture introducing the quantity of quantum uncertainty (relating it to the standard deviation of the distribution of probabilities by spin component value) and deriving the simplified equation:
\[\Delta A = \sqrt{\langle A^2 \rangle - \langle A \rangle ^2}\]
Student Conversations
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 \(0\hbar\).
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.
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 \(\hbar/\sqrt{2}\).
Problem
- 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).
Consider a two-state quantum system with a Hamiltonian \begin{equation} \hat{H}\doteq \begin{pmatrix} E_1&0\\ 0&E_2 \end{pmatrix} \end{equation} Another physical observable \(M\) is described by the operator \begin{equation} \hat{M}\doteq \begin{pmatrix} 0&c\\ c&0 \end{pmatrix} \end{equation} where \(c\) is real and positive. Let the initial state of the system be \(\left|{\psi(0)}\right\rangle =\left|{m_1}\right\rangle \), where \(\left|{m_1}\right\rangle \) is the eigenstate corresponding to the larger of the two possible eigenvalues of \(\hat{M}\). What is the expectation value of \(M\) as a function of time? What is the frequency of oscillation of the expectation value of \(M\)?
Problem
First complete the problem Diagonalization. In that notation:
- 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?)
- 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.
Student explore the properties of an orthonormal basis using the Cartesian and \(S_z\) bases as examples.
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.”
Problem
Consider the arbitrary Pauli matrix \(\sigma_n=\hat n\cdot\vec \sigma\) where \(\hat n\) is the unit vector pointing in an arbitrary direction.
- 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: \(\sin\theta=\sqrt{1-\cos^2\theta}\).
- Show that the eigenvectors from part (a) above are orthogonal.
- 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 \(\sigma_z\).