Search Curricular Materials

Expectation Values and Uncertainty

You 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:

1. \(\left|{+}\right\rangle \)
2. \(\frac{i}{2}\left|{+}\right\rangle -\frac{\sqrt{3}}{2}\left|{-}\right\rangle \)

3. \(\left|{+}\right\rangle _x\)

   A spin 1 particle described by:

4. \(\left|{0}\right\rangle \)
5. \(\left|{-1}\right\rangle _x\)
6. \(\frac{2}{3}\left|{1}\right\rangle +\frac{i}{3}\left|{0}\right\rangle -\frac{2}{3}\left|{-1}\right\rangle \)

• 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

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 \(\hbar/\sqrt{2}\).

In this small group activity, students draw components of a vector in Cartesian and polar bases. Students then write the components of the vector in these bases as both dot products with unit vectors and as bra/kets with basis bras.

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.

• How to find area, and volume elements in curvilinear coordinates using geometric methods.

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. Set the first analyzer to measure \(S_z\) and the second analyzer to measure \(S_x\).

      

      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.

   

Use the power series method to solve Legendre's Equation \begin{equation} \frac{d^2 P}{d z^2} -\frac{2z}{1-z^2}\frac{d P}{d z} -\frac{A}{1-z^2} P=0 \label{legendrepolyeqn} \end{equation}

Students compute inner products to expand a wave function in a sinusoidal basis set. This activity introduces the inner product for wave functions, and the idea of approximating a wave function using a finite set of basis functions.

Spherical harmonics are continuous functions on the surface of a sphere.

The \(\ell\) and \(m\) values tell us how the function oscillates across the surface.

Spherical harmonics are complex valued functions.

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 do calculations for time evolution for spin-1.

Gauss's Law: \[ \oint \vec{E}\cdot \hat{n}\, dA = {1\over\epsilon_0}\, Q_{\hbox{enc}} \]

Ampère's Law:

\[ \oint \vec{B}\cdot d\vec{r} = \mu_0 \, I_{\hbox{enc}} \]

Potentials: \begin{eqnarray*} \vec{E}&=&-\vec{\nabla} V\\ \vec{B}&=&\vec{\nabla}\times\vec{A} \end{eqnarray*}

Maxwell's Equations: \begin{eqnarray*} \vec{\nabla}\cdot\vec{E} &=& \frac{\rho}{\epsilon_0}\\ \vec{\nabla}\cdot\vec{B} &=& 0\\ \vec{\nabla}\times\vec{E} &=& 0\\ \vec{\nabla}\times\vec{B} &=& {\mu_0}\, \vec{J} \end{eqnarray*}

Superposition Laws: \begin{eqnarray*} V(\vec{r}) &=& \frac{1}{4\pi\epsilon_0} \int{\rho(\vec{r}')\, d\tau'\over \vert \vec{r}-\vec{r}'\vert}\\ \vec{E}(\vec{r}) &=& \frac{1}{4\pi\epsilon_0} \int{\rho(\vec{r}')(\vec{r}-\vec{r}')\, d\tau'\over \vert \vec{r}-\vec{r}'\vert^3}\\ \vec{A}(\vec{r}) &=& \frac{\mu_0}{4\pi} \int{\vec{J}(\vec{r}')\, d\tau'\over \vert \vec{r}-\vec{r}'\vert}\\ \vec{B}(\vec{r}) &=& \frac{\mu_0}{4\pi} \int{\vec{J}(\vec{r}')\times (\vec{r}-\vec{r}')\, d\tau'\over \vert \vec{r}-\vec{r}'\vert^3}\\ V(B)-V(A)&=&-\int_A^B \vec{E}\cdot d\vec{r} \end{eqnarray*}

Position Vectors \begin{align*} \vec{r} &= x \hat{x} + y\hat{y} + z\hat{z}\\ &= s \hat{s} + z\hat{z}\\ &= r\hat{r} \end{align*}

The distance between two position vectors

1. In cylindrical coordinates: \[\left\vert\vec r -\vec r^{\prime}\right\vert =\sqrt{s^2+s^{\prime\, 2}-2s\, s^{\prime}\cos(\phi- \phi^{\prime}) +(z-z^{\prime})^2}\]
2. In spherical coordinates: \[\left\vert\vec r -\vec r^{\prime}\right\vert =\sqrt{r^2+r^{\prime\, 2}-2r\, r^{\prime}\left[ \sin\theta\sin\theta^{\prime}\cos(\phi-\phi^{\prime}) +\cos\theta\cos\theta^{\prime}\right]}\]

Derivatives in Rectangular Coordinates: \begin{eqnarray*} \vec{\nabla} f &=& \frac{\partial f}{\partial x}\,\hat{x} + \frac{\partial f}{\partial y}\,\hat{y} + \frac{\partial f}{\partial z}\,\hat{z} \\ \vec{\nabla}\cdot\vec{F} &=& \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} \\ \vec{\nabla}\times\vec{F} &=& \left(\frac{\partial F_z}{\partial y}-\frac{\partial F_y}{\partial z}\right)\hat{x} + \left(\frac{\partial F_x}{\partial z} -\frac{\partial F_z}{\partial x}\right)\hat{y} + \left(\frac{\partial F_y}{\partial x} -\frac{\partial F_x}{\partial y}\right)\hat{z} \end{eqnarray*}

Derivatives in Cylindrical Coordinates: \begin{eqnarray*} \vec{\nabla} f &=& \frac{\partial f}{\partial s}\,\hat{s} + \frac{1}{s}\frac{\partial f}{\partial \phi}\,\hat{\phi} + \frac{\partial f}{\partial z}\,\hat{z} \\ \vec{\nabla}\cdot\vec{F} &=& \frac{1}{s}\frac{\partial}{\partial s}\left({s}F_{s}\right) + \frac{1}{s}\frac{\partial F_\phi}{\partial \phi} + \frac{\partial F_z}{\partial z} \\ \vec{\nabla}\times\vec{F} &=& \left( \frac{1}{s}\frac{\partial F_z}{\partial \phi} - \frac{\partial F_\phi}{\partial z} \right) \hat{s} + \left(\frac{\partial F_s}{\partial z}-\frac{\partial F_z}{\partial s}\right) \hat{\phi} + \frac{1}{s} \left( \frac{\partial}{\partial s}\left({s}F_{\phi}\right) - \frac{\partial F_s}{\partial \phi} \right) \hat{z} \end{eqnarray*}

Derivatives in Spherical Coordinates: \begin{eqnarray*} \vec{\nabla} f &=& \frac{\partial f}{\partial r}\,\hat{r} + \frac{1}{r}\frac{\partial f}{\partial \theta}\,\hat{\theta} + \frac{1}{r\sin\theta}\frac{\partial f}{\partial \phi}\,\hat{\phi} \\ \vec{\nabla}\cdot\vec{F} &=& \frac{1}{r^2}\frac{\partial}{\partial r}\left({r^2}F_{r}\right) + \frac{1}{r\sin\theta}\frac{\partial}{\partial \theta}\left({\sin\theta}F_{\theta}\right) + \frac{1}{r\sin\theta}\frac{\partial F_\phi}{\partial \phi} \\ \vec{\nabla}\times\vec{F} &=& \frac{1}{r\sin\theta} \left( \frac{\partial}{\partial \theta} \left({\sin\theta}F_{\phi}\right) - \frac{\partial F_\theta}{\partial \phi} \right) \hat{r} + \frac{1}{r} \left( \frac{1}{\sin\theta} \frac{\partial F_r}{\partial \phi} - \frac{\partial}{\partial r}\left({r}F_{\phi}\right) \right) \hat{\theta} \\ && \quad + \frac{1}{r} \left( \frac{\partial}{\partial r}\left({r}F_{\theta}\right) - \frac{\partial F_r}{\partial \theta} \right) \hat{\phi} \end{eqnarray*}

Lorentz Force Law:

\[\vec{F}=q_{\hbox{test}}\left(\vec{E}+\vec{v}\times\vec{B}\right)\]

Step and Delta Functions: \begin{eqnarray*} \frac{d}{dx} \theta(x-a)&=&\delta(x-a)\\ \int_{-\infty}^{\infty} f(x)\delta(x-a)\, dx&=&f(a) \end{eqnarray*}

Vector Calculus Theorems: \begin{eqnarray*} \oint \vec{F} \cdot d\vec{A} &=& \int \vec{\nabla} \cdot \vec{F} d\tau\\ \oint \vec{F} \cdot d\vec{\ell} &=& \int (\vec{\nabla} \times \vec{F}) \cdot d\vec{A}\\ \end{eqnarray*}

Total Charge and Current: \begin{eqnarray*} Q &=& \int \rho (\vec{r}') d\tau'\\ I &=& \int \vec{J} (\vec{r}') \cdot d\vec{A'}\\ \end{eqnarray*}

In a Stern-Gerlach experiment, the arrival of an atom at a measurement counter is a random process. I would like to use the results of the experiments to answer the question:

What is the probability \(\mathcal{P}\) that an atom will arrive at the top counter?

In the case where all the atoms arrive at the top counter, the probability is 1. However, what if I send 10 atoms through the analyzer and detect 3 atoms in the top counter? How confidently can I conclude that the probability is 0.3? What if I repeat my experiment and send 10 more atoms though the analyzer but detect 4 atoms in the top counter? I probably want to revise my estimate. If I do a bunch of sets of experiments, I will get a distribution of probabilities. Therefore, I'm going to need statistical tools to answer my questions:

1. What is the best estimate of the probability, given the experimental data?
2. How confident am I of that estimate?

To find the best estimate of the probability, I'm going to do a bunch of sets of experiments and take the mean. The mean probability will be my best estimate of the probability.

To determine how confident I am in the estimate, I'm going to consider the shape of the distribution. (For random processes like the Stern-Gerlach experiment - or coin flipping experiments, where there are 2 possible outcomes for each experiment - the underlying distribution is a binomial distribution.) To get a distribution, I can't do just one Stern-Gerlach experiment, or even a one set of Stern-Gerlach experiments - I have to do a bunch of sets of Stern-Gerlach experiments.

Some Definitions

\(\mathcal{P}\) is the “true value” of probability of ending up in the top counter for the physical system (measuring \(S_z = \hbar/2\)). (This probability is the number that I'm trying to experimentally estimate.)

In 1 Stern-Gerlach experiment, as single atom passes through the analyzer and is detected at a counter.

I'm going to do a bunch of experiments and organize them into \(N\) sets. Each individual set \(n\) will include \(M\) particles being sent into an analyzer and counted in a counter.

For example, I can click the "10k" button and send 10,000 particles through the analyzer (i.e., 10,000 experiments). I can record the number of particles in the top counter and then repeat so that I end up with 5 sets of experiments.

\(M\) = the number of Stern-Gerlach experiments in each set. This is the number of particles I send through the analyzer in 1 set. I'll assume that each set has the same number of experiments.

\(x_n\) = the (integer) number of atoms in the top counter after \(M\) Stern-Gerlach experiments

\(\mathcal{P}_n\) is the probability I determine for 1 set of \(M\) Stern-Gerlach experiments.

\(N\) = the number of sets of Stern-Gerlach experiments (note: \(n\) is an index that indicates a single set of experiments)

\(\bar{\mathcal{P}}\) is the mean probability determined from \(N\) sets of \(M\) experiments. This will be my estimation of the true probability.

Best Estimate of the Probability: the Mean

The probability I determine for a set of experiments (like in the table above) is:

\[\mathcal{P}_n = \frac{x_n}{M}\]

The mean of these probabilities is:

\[\bar{\mathcal{P}}= \frac{1}{N}\sum_{n=1}^N \mathcal{P}_n\]

If I want to, I can also write the mean probability in terms of the number of atoms counted: \begin{align*} \bar{\mathcal{P}} &= \frac{1}{N}\sum_{n=1}^N \mathcal{P}_n \\ &= \frac{1}{N}\sum_{n=1}^N \frac{x_n}{M}\\ &= \frac{1}{NM}\sum_{n=1}^N x_n\\ &= \frac{1}{M}\bar{x} \end{align*}

Experimental Uncertainty - the Standard Error

In this section I'm going to argue that the standard error is a sensible thing to report as the experimental uncertainty (and for making statistical inferences). In order to understand the standard error, I'm first going to talk about the variance and the standard deviation.

The Variance

In order to quantify how spread out the distribution is, conceptually I'm tempted to find the average of the difference between each probability \(\mathcal{P}\) and the mean of the distribution. The problem with this approach is that this average should be zero - the average is at the center of all the observations!

\begin{align*} \frac{1}{N}\sum_{n=1}^N (\bar{\mathcal{P}}-\mathcal{P}_n) &= \frac{1}{N}\left(\sum_{n=1}^N \bar{\mathcal{P}}\right)-\left(\frac{1}{N}\sum_{n=1}^N\mathcal{P}_n\right)\\[12pt] &= \frac{1}{N}N\bar{\mathcal{P}}-\bar{\mathcal{P}} \\[12pt] &=\bar{\mathcal{P}}-\bar{\mathcal{P}}\\[6pt] &= 0 \end{align*}

One way to get around this is to square all the differences first. The variance is the squared difference between the probability for one set of SG experiments and the mean probability:

\[var = \frac{1}{N}\sum_{n=1}^{N} (\bar{\mathcal{P}} - \mathcal{P}_n )^2\]

All contributions to the variance are positive, so the variance is greater than zero (though a zero variance is still technically possible if the distribution is one number). The larger the variance, the more spread out the distribution.

The Standard Deviation

The standard deviation is the square root of the variance. \begin{align*} SD &= \sqrt{var}\\ &= \sqrt{\frac{1}{N}\sum_{n=1}^{N} (\bar{\mathcal{P}} - \mathcal{P}_n )^2}\\ &\rightarrow \sqrt{\frac{1}{N-1}\sum_{n=1}^{N} (\bar{\mathcal{P}} - \mathcal{P}_n )^2} \quad \mbox{ for small N} \end{align*}

(For N < 30ish, there are theoretical arguments about how the standard deviation of the sample underestimates the true standard deviation of the system, so the prefactor is front is made a smidge larger.)

• The standard deviation does not decrease with more sets of experiments. The standard deviation does not vary with \(N\). The standard deviation comes from taking an average (you add up N things and then divide by \(N\)). As \(N\) increases, the standard deviation does not change with the number of experiments (it might fluctuate a little because of the random nature of additional experiments, especially if the total number of experiments is small, but if you plot \(SD\) vs. \(N\) (e.g., the number of particles in the top counter), the best fit line should have a near-zero slope). Therefore, the standard deviation is a characteristic of the system.

  

• The standard deviation is a characteristic of the combined physical and measurement system, including information about the distribution of the physical system and sources of random uncertainty during the measurement process.

• Binomial vs “normal” distribution For large numbers of experiments (\(M\)), a binomial distribution is very close to a normal (or Gaussian) distribution. For a normal distribution, 68% of measurements will lie within 1 standard deviation from the mean.

  

  

• The standard deviation is a special kind of average, an rms average. The \(rms\) stands for “root mean square” and describes the order of operations in the calculation (first you square, then you average, then you take a square root). So, the rms average allows me to get a sense of how far away individual probabilities \(\mathcal{P}_n\) are from \(\bar{\mathcal{P}}\) without running into the problem with doing a regular average, as described above.

• Subtle difference between the distributions of number of atoms and probability. The standard deviation does depend on the number of measurements in each set (which conceptually makes sense to me because the standard deviation is a characteristic of the combined physical and measurement system). For binomial distributions, the standard deviation for the distribution of the number of atoms is

  \[SD_{x_n} = \sqrt{M\mathcal{P}(1-\mathcal{P})}\]

  where \(\mathcal{P}\) is the true probability I'm trying to measure. This equation for standard deviation is not general; it is only true for binomial distributions, where each experiment is a coin flip, atom through a Stern-Gerlach analyzer, etc. This equation tells me a system with a characteristic probability \(\mathcal{P}\), the standard deviation will be twice as large if each set includes 100 experiments than if each set includes 25 experiments. (The mean will also be bigger because here I'm counting particles.)

  

  In contrast, the standard deviation of the distribution of the probabilities is different by a factor of \(M\)

  \begin{align*} SD_{\mathcal{P}} &= \sqrt{\frac{1}{N} \sum_{n=1}^N (\bar{\mathcal{P}_n}-\mathcal{P}_n)^2} \\[12pt] &= \sqrt{\frac{1}{N} \sum_{n=1}^N \left(\frac{\bar{x_n}}{M}-\frac{x_n}{M}\right)^2}\\[12pt] &= \sqrt{\frac{1}{M^2N} \sum_{n=1}^N (\bar{\mathcal{x}_n}-\mathcal{x}_n)^2} \\[12pt] &= \frac{1}{M}\sqrt{\frac{1}{N} \sum_{n=1}^N (\bar{x_n}-x_n)^2} \\[12pt] &= \frac{1}{M} s_{x_n} \\[12pt] &= \frac{1}{M}\sqrt{M\mathcal{P}(1-\mathcal{P})}\\[12pt] &= \sqrt{\frac{\mathcal{P}(1-\mathcal{P})}{M}} \end{align*}

  This tells me that the distribution of probabilities will get narrower as the number of experiments in each set gets larger.

  

The Standard Error

The standard error (a.k.a. the standard deviation of the mean) \(\sigma\) is an a measure of how well I know the mean. In this lab, I'm estimating the true value of the probability by doing many (N) sets of Stern-Gerlach experiments and finding the mean of these sets. Now imagine that I repeat this whole process many times (N times) so that I get many means. Each mean is a better estimate of the true value of the probability than any individual probabily I measure, and the distribution of these means is much narrower than the distribution of the probabilities that I determined from each set of Stern-Gerlach experiments. If I compute the standard deviation of the distribution of means, it turns out that:

\[StErr_{\mathcal{P}} = \frac{SD_{\mathcal{P}}}{\sqrt N}\]

(see Taylor, pp. 147-148 for a nice derivation) If I only find one mean (\(\bar{\mathcal{P}}\) from my original \(N\) sets of Stern-Gerlach experiments), I can be confident that there is a 68% chance that my mean lies is within 1 standard error from the true value of the probability.

In the case of Stern-Gerlach experiments (which following a binomial distribution):

\begin{align*} StErr_{\mathcal{P}} &= \frac{\sqrt{\frac{\mathcal{P}(1-\mathcal{P})}{M}}}{\sqrt{N}} \\ &= \sqrt{\frac{\mathcal{P}(1-\mathcal{P})}{MN}} \end{align*}

• The standard error of the probability varies with the total number of experiments. Notice that \(MN\) is the total number of Stern-Gerlach experiments that I run (\(M\) experiments in each set for \(N\) sets). The standard error is inversely proportionally to the square root of the total number of Stern-Gerlach experiments. It doesn't matter how I group them. If I do 10,000 Stern-Gerlach experiments, the standard error is the same as if I do 10 sets of 100 experiments, 20 sets of 50 experiments, or 10,000 sets of 1 experiment. It's hard to tell from looking at the plot alone that the standard error is the same:

  

• The standard error as a measure of uncertainty The standard error tells me about how well my mean probability estimates the true value of the probability. Conceptually, it makes sense that the more experiments I do, the more confidence I should have in my estimate.

Reporting Uncertainty

To answer the question of how confident I am in my estimates, I could choose to report the uncertainty as the standard deviation or the standard error. These two options have different meanings (for this discussion, I'm going to assume that \(M\) is large and we have an approximately normal distribution):

\(\bar{\mathcal{P}} \pm SD_{\mathcal{P}}\)
Meaning: If I do one more set of SG experiments, there is a 68% chance that the probability I measure will fall in this range.

\(\bar{\mathcal{P}} \pm StErr_{\mathcal{P}}\)
Meaning: If I repeat the entire exercise, doing \(N\) sets of \(M\) SG experiments, there is a 68% change that the average probability I determine will fall in this range.

In this case, the standard error of the mean is closer to the thing I mean by my confidence in my estimate.

Comparing Values

If I wanted to compare my estimate of the probability to either (1) someone else's measurement or (2) a theoretically expected answer, both of which I'll call \(\mathcal{P}_{exp}\), I might describe the difference between values in terms of the number of standard errors.

\[ t = \frac{|\bar{\mathcal{P}} - \mathcal{P}_{exp}|}{StErr}\]

A smaller \(t\) corresponds to a higher likelihood that the two values come from the same normal distribution. The boundary between acceptable and unacceptable differences is a matter of opinion, to be decided by the experimenter (and the reader). For normal distributions, many scientists consider differences of:

\(t<2\) to be acceptable (“the discrepancy is insignificant”) and

\(t>2\) to be unacceptable (“the discrepancy between values is significant.”).

Differences that are \(t\approx 2\) (1.9-2.6) are generally considered inconclusive.

For a normal distribution, there is a 95% likelihood that the true values lies with 2 standard errors of mean, meaning \(t<2\).

For Your Information:

Inferential statistical tests can be used to formally compare values, for example:

• One Sample T-Test: A one sample t-test allows us to test whether a sample mean (of a normally distributed variable) significantly differs from a hypothesized value.
• Independent Samples T-Test: An independent samples t-test is used when you want to compare the means of a normally distributed dependent variable for two independent groups.
• Binomial Test: A one sample binomial test can be used to determine whether the proportion of successes on a two-level categorical dependent variable significantly differs from a hypothesized value. (Remember that for large values of \(M\), a binomial distribution approximates a normal distribution, so the first two tests might be applicable.)

For each statistical test, a set of assumptions need to be met in order for the test to give reliable, meaningful results. For example, a one sample t-test assumes that the data are normally distributed.

Students compute both sides of Stokes' theorem.

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.

Systems of Equations: Compare and Contrast

Small Group Directions

1. Solve your assigned system of equations using any algebraic method. Show you work and be ready to explain how you solved it.
2. Also graph the system of equations and show how the solution appears on your graph. You may use graphing technology such as Desmos.

Group Roles

Facilitator: Read the directions out loud and check whether everyone understands each other. “How should we start?” “How do you know?”

Team Captain: Help your team members step up and step back. “How do you know?” “What do you think?”

Resource Manager: Help your group get unstuck. “Is this working?” “What else could we try?” “Should we ask a team question?”

Recorder/Reporter: Be prepared to share out in the whole class discussion. “How should I explain...?”

Problems

1. \[y=-3x\\4x+y=2\]
2. \[y=7x-5\\2x+y=13\]
3. \[x=-5y-4\\x-4y=23\]
4. \[x+y=10\\y=x-4\]
5. \[y=5-x\\4x+2y=10\]
6. \[3x+5y=23\\y=x+3\]
7. \[y=-x-2\\2x+3y=-9\]
8. \[y=2x-3\\-2x+y=1\]
9. \[x=\frac{1}{2}y+\frac{1}{2}\\2x+y=-1\]
10. \[a=2b+4\\b-2a=16\]
11. \[y=3-2x\\4x+2y=6\]
12. \[y=x+1\\x-y=1\]

(Adapted from CPM Core Connections)

Whole Class Directions

1. Each group will share out how you solved your system of equations.
2. Listen to each group and think about similarities and differences.
3. Ask questions about anything you do not understand or you disagree with.
4. You do not need to write anything during the whole class discussion, but you will have an exit ticket to see what you learned from the discussion.

Exit Ticket: Systems of Equations Compare and Contrast

Sheila missed class today. She tried to solve Problem 8 on her own, but she thinks she made a mistake because -3 does not equal 1. \begin{align} &y=2x-3\\ &-2x+y=1 \end{align} \begin{align} &-2x+(2x-3)=1\\ &-2x+2x-3=1\\ &0-3=1\\ &-3=1 \end{align}

Explain to Sheila what happened, using as much detail as possible to help her understand this type of problem.

Introduction

This Compare and Contrast activity is based on the College Preparatory Mathematics (CPM) Core Connections Algebra Parent Guide with Extra Practice, freely available here. CPM is a problem based curriculum with many conceptual problems for students to work on in small groups in class. The parent guide provides examples, exercises, and solutions for students to work alone and/or with parent support if they miss class or need extra practice. As such, the parent guide is one aspect of the CPM curriculum most focused on practice of procedures. The attached problem set is copied exactly from the CPM Parent Guide; the surrounding student instructions were written by Alyssa Sayavedra.

Special Cases of note

Problems 8 and 12 have no solution while Problem 11 has infinite solutions. It is important to include these problems, but be prepared for small groups to get tripped up by them. Many students, when solving equations, expect the “answer” to be a value. They may struggle to interpret an equation that is always or never true.

All other problems have one solution with integer coordinates.

Some problems in this set are easier than others. If any group finishes early, they can be encouraged to complete a second problem. Problem 1 is the most straightforward since y is equal to only one term. The next easiest problems are 2, 3, 4, and 8, because they do not require distribution after substitution.

Problems 1, 2, 3, 4, 5, 8, 9 and 11 can be solved using the Equal Values Method without introducing new fractions. The Equal Values Method is a variant of substitution in which students solve both equations for the same variable, then set the equations equal to each other, resulting in a single equation in one variable. This method is easier for many students because it results in a simpler one variable equation and is less prone to distribution errors. But it is usually not worth introducing fractions into the problem in order to use this method.

Problems 5, 9 and 11 can be simplified by either multiplying or dividing an entire equation by 2. It is unusual for students to think of this strategy at this stage, but it can be a helpful preview of the elimination method. This method also removes the fractions in Problem 9.

Small variations in notation can easily trip students up. Problem 10 uses a and b instead of x and y. Problems 3 and 9 have one equation solved for x instead of y. Problems 4 and 6 have the second equation solved for y instead of the first. Do not be surprised if some students still solve the first equation for y and plug it into the second.

Suggestions for Facilitating Small Group Work

Remind students of class norms for productive and respectful group work. Assign one problem to each group, including at least problems 2, 4, 6, 8, 11 and 12. Walk once or twice around the class within the first five minutes to make sure all small groups understand how to get started and are making progress. Make sure students understand the directions and have started to dig into the mathematics, but avoid giving strategic suggestions at this stage. The purpose of the small group time is for students to wrestle with the tricky bits of one problem. If a group chooses an inefficient strategy or makes an error, monitor their frustration level, but try to allow them to pursue it in some detail before suggesting there may be an easier method. The first 3 questions (from Schoenfeld) assist students with metacognitive monitoring of their own problem solving process. Whenever possible, allow students to check their own work using graphing technology and/or substitution of their answers rather than checking it for them.

Some good questions to ask groups during this time are:

• “What are you doing?”
• “Why are you doing that?”
• “Is it working?”
• “Are you done?”
• “Have you found values for all the unknowns?”
• “How could you check your work?”
• “Can you graph the problem to check your work?”
• “Can you substitute these numbers back in to check your work?”
• “What would you expect to see on the graph?”

Suggestions for Facilitating Whole Class Discussion

Remind students of their norms for active listening during presentations, respect for presenters and treating mistakes as learning opportunities. Ask the reporters from at least 4-6 groups to share out their work (the reporter role should rotate regularly, even every class period). If not all groups will present, give priority to students or groups who present less often but who have done excellent work, to groups that have tried innovative strategies or made important revisions, and to the most important special cases. When sequencing the presentations, start with easier and/or typical examples. Often, it should work well to simply present the examples you choose in numerical order. Close with an exit ticket like “Explain one way you revised your work or thinking today” or “Use Jorge's method to solve this new problem.” You can also create an exit ticket in advance, such as the one attached.

1. A right circular cone has circular base of radius R and height H, both measured in feet.
   1. What is the volume of the cone?
   2. Write down as many different integrals as you can for computing this volume.
   3. Do at least two of these integrals.

For some integrals, you may wish to use the fact that \[\cos(2\alpha)=2\cos^2\!\alpha-1=1-2\sin^2\!\alpha\]

Students set up and compute a scalar surface integral.

This small group activity is designed to help students visual the process of chopping, adding, and multiplying in single integrals. Students work in small groups to determine the volume of a cylinder in as many ways as possible. The whole class wrap-up discussion emphasizes the equivalence of different ways of chopping the cylinder.

A short improvisational role-playing skit based on the Star Trek series in which students explore the definition and notation for position vectors, the importance of choosing an origin, and the geometric nature of the distance formula. \[\vert\vec{r}-\vec{r}^\prime\vert=\sqrt{(x-x^\prime)^2+(y-y^\prime)^2-(z-z^\prime)^2}\]

Students compute surface integrals and explore their interpretation as flux.

1. Consider the rectangle in the first quadrant of the \(xy\)-plane as in the figure with thick black lines.
   • Label the bottom horizontal edge of the rectangle \(y=c\).
   • Label the sides of the rectangle \(\Delta x\) and \(\Delta y\).
   • What is the area of the rectangle?
   • There are also 2 rectangles whose base is the \(x\)-axis, the larger of which contains both the smaller and the original rectangle. Express the area of the original rectangle as the difference between the areas of these 2 rectangles.
2. On the grid below, draw any simple, closed, piecewise smooth curve \(C\), all of whose segments \(C_i\) are parallel either to the \(x\)-axis or to the \(y\)-axis. Your curve should not be a rectangle. Pick an origin and label it, and assume that each square is a unit square.
   
   • Compute the area of the region \(D\) inside \(C\) by counting the number of squares inside \(C\).

   • Evaluate the line integral \(\displaystyle \oint_C y\,\boldsymbol{\hat{x}}\cdot d\boldsymbol{\vec{r}}\) by noticing that along each segment either \(x\) or \(y\) is constant, so that the integral is equal to \(\sum_{C_i} y\,\Delta x\).

     Can you relate this to Problem 1?
     

   • Are your answers to the preceding two calculations the same?
   • Would any of your answers change if you replaced \(y\,\boldsymbol{\hat{x}}\) by \(x\,\boldsymbol{\hat{y}}\) in part (b)?

Main ideas

• Understanding different ways of expressing area using integration.
• Concrete example of Area Corollary to Green's/Stokes' Theorem.

We originally used this activity after covering Green's Theorem; we now skip Green's Theorem and do this activity shortly before Stokes' Theorem.

Prerequisites

• Familiarity with line integrals.
• Green's Theorem is not a prerequisite!

Warmup

• The first problem is a good warmup.

Props

• whiteboards and pens
• a planimeter if available

Wrapup

• Emphasize the magic -- finding area by walking around the boundary!
• Point out that this works for any closed curve, not just the rectangular regions considered here.
• Demonstrate or describe a planimeter, used for instance to measure the area of a region on a map by tracing the boundary.

Details

In the Classroom

• Make sure students use a consistent orientation on their path.
• Make sure students explicitly include all segments of their path, including those which obviously yield zero.
• Students in a given group should all use the same curve.
• Students should be discouraged from drawing a curve whose longest side is along a coordinate axis.
• Students may need to be reminded that \(\oint\) implies the counterclockwise orientation. But it doesn't matter what orientation students use so long as they are consistent!
• A geometric argument that the orientation should be reversed when interchanging \(x\) and \(y\) is to rotate the \(xy\)-plane about the line \(y=x\). (This explains the minus sign in Green's Theorem.)
• Students may not have seen line integrals of this form (see below).
• Students do very well on this lab, particularly after working in groups for several weeks. Resist the urge to intervene.
• Make sure everyone sees the reason \(y\,\boldsymbol{\hat{x}}\cdot d\boldsymbol{\vec{r}}\) is zero on vertical pieces.
• The issue of the negative will come up. Suggest students make a quick sketch of the vector field.
• It is well worthwhile to do an example with a circle together as a class. The line integral should pose no trouble for them and the area of a circle is something they believe.
• Emphasize the connection between the boundary and the interior. This is a concrete display of this relationship.

Subsidiary ideas

• Orientation of closed paths.

• Line integrals of the form \(\int P\,dx+Q\,dy\).

  We do not discuss such integrals in class! Integrals of this form almost always arise in applications as \(\int\boldsymbol{\vec{F}}\cdot d\boldsymbol{\vec{r}}\).

Homework

Determine the area of a triangle or an ellipse by integrating along the boundary.

Essay questions

Describe times in your life when you needed to know area (or imagine such a time). Maybe buying carpet or painting a room. What is the first step in computing area? How does this lab truly differ, if at all?

Enrichment

• Write down Green's Theorem.
• Go to 3 dimensions --- bend the curve out of the plane and stretch the region like a butterfly net or rubber sheet. This is the setting for Stokes' Theorem!

• The gradient is perpendicular to the level curves.
• The gradient is a local quantity, i.e. it only depends on the values of the function at infinitesimally nearby points.
• Although students learn to chant that "the gradient points uphill," the gradient does not point to the top of the hill.
• The gradient path is not the shortest path between two points.

Students work in groups to measure the steepest slope and direction at a given point on a plastic surface and to compare their result with the gradient vector, obtained by measuring its components (the slopes in the coordinate directions).

Students work in groups to measure the steepest slope and direction on a plastic surface, and to compare their result with the gradient vector, obtained by measuring its components (the slopes in the coordinate directions).