Activities
Students re-represent a state given in Dirac notation in matrix notation
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 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).
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.
Problem
Consider the three quantum states: \[\left\vert \psi_1\right\rangle = \frac{1}{\sqrt{3}}\left\vert +\right\rangle+ i\frac{\sqrt{2}}{\sqrt{3}} \left\vert -\right\rangle\] \[\left\vert \psi_2\right\rangle = \frac{1}{\sqrt{5}}\left\vert +\right\rangle- \frac{2}{\sqrt{5}} \left\vert -\right\rangle\] \[\left\vert \psi_3\right\rangle = \frac{1}{\sqrt{2}}\left\vert +\right\rangle+ i\frac{e^{\frac{i\pi}{4}}}{\sqrt{2}} \left\vert -\right\rangle\]
- For each of the \(\vert \psi_i\rangle\) above, find the normalized vector \(\vert \phi_i\rangle\) that is orthogonal to it.
- Calculate the inner products \(\langle \psi_i\vert \psi_j\rangle\) for \(i\) and \(j=1\), \(2\), \(3\).
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.
Problem
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}
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.
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 \(P_{\ell=0,m=0}/\left(P_{\ell=1,m=-1} + P_{\ell=1,m=0} + P_{\ell=1,m=1}\right)\)At what temperature is the value of this ratio 1?
- 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 \(P_{\ell=0,m=0}/\left(P_{\ell=2,m=-2} + P_{\ell=2,m=-1} + \cdots + P_{\ell=2,m=2}\right)\)
Problem
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).
Compute the entropy.
Work out the heat capacity at constant pressure \(C_p\).
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).
- Compute the internal energy \(U\).
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 \(\varepsilon\).
From the free energy, find expressions for the internal energy \(U\) and entropy \(S\) of the system.
Plot the entropy versus \(T\). Explain its asymptotic behavior as the temperature becomes high.
Plot the \(S(T)\) versus \(U(T)\). Explain the maximum value of the energy \(U\).
Treat the ground state of a quantum particle-in-a-box as a periodic function.
Set up the integrals for the Fourier series for this state.
Which terms will have the largest coefficients? Explain briefly.
Are there any coefficients that you know will be zero? Explain briefly.
Using the technology of your choice or by hand, calculate the four largest coefficients. With screen shots or otherwise, show your work.
- Using the technology of your choice, plot the ground state and your approximation on the same axes.
Students learn how to express Angular Momentum as a vector quantity in polar coordinates, and then in Cylindrical and Spherical Coordinates
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
- Are perpendicular?
- Are parallel?
- Have an angle less than \(\pi/2\) between them?
- Have an angle of more than \(\pi/2\) between them?
Kinesthetic
30 min.
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.
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.
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.
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.
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?
Students use their arms to act out stationary and non-stationary states of a quantum 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.
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.
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.
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.
The instructor and students do a skit where students represent quantum states that are “measured” by the instructor resulting in a state collapse.
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 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.
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.
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).
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 \(\left|{-}\right\rangle _x\):
\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*}
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 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.
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
Student explore the properties of an orthonormal basis using the Cartesian and \(S_z\) bases as examples.
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
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}
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?
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
- 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 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.”
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 \(L_z\).
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).
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 \(\sigma_z\).
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
Consider the three quantum states: \[\left\vert \psi_1\right\rangle = \frac{4}{5}\left\vert +\right\rangle+ i\frac{3}{5} \left\vert -\right\rangle\] \[\left\vert \psi_2\right\rangle = \frac{4}{5}\left\vert +\right\rangle- i\frac{3}{5} \left\vert -\right\rangle\] \[\left\vert \psi_3\right\rangle = -\frac{4}{5}\left\vert +\right\rangle+ i\frac{3}{5} \left\vert -\right\rangle\]
- For each of the \(\left|{\psi_i}\right\rangle \) above, calculate the probabilities of spin component measurements along the \(x\), \(y\), and \(z\)-axes.
- 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.
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.)
- 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.
- Find the Hamiltonian for a 2-D box and show that your eigenstates are indeed eigenstates and find a formula for the possible energies
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. \begin{equation} \psi(x,y)=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\, c_{nm}\; \hbox{eigenfunction}_n(x)\;\hbox{eigenfunction}_m(y) \end{equation} 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*}
- 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.
Students, working in pairs, use their left arms to demonstrate time evolution in spin 1/2 quantum systems.
Problem
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): \begin{equation} n_Q \equiv \left(\frac{MkT}{2\pi\hbar^2}\right)^{\frac32} \end{equation} within a factor of the order of unity.
Problem
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:
- 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.
- 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.
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.
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}
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. \[\left|D\right\rangle\doteq \begin{pmatrix} 7e^{i\frac{\pi}{6}}\\ 3e^{i\frac{\pi}{2}}\\ -1\\ \end{pmatrix}\\ \left|E\right\rangle\doteq \begin{pmatrix} i\\ 4\\ \end{pmatrix}\\ \left|F\right\rangle\doteq \begin{pmatrix} 2+2i\\ 3-4i\\ \end{pmatrix} \]