Activities
Students work in small groups to use the superposition principle \[\vec{A}(\vec{r}) =\frac{\mu_0}{4\pi}\int\frac{\vec{J}(\vec{r}^{\,\prime})}{\vert \vec{r}-\vec{r}^{\,\prime}\vert}\, d\tau^{\prime}\] to find an integral expression for the magnetic vector potential, \(\vec{A}(\vec{r})\), due to a spinning ring of charge.
In an optional extension, students find a series expansion for \(\vec{A}(\vec{r})\) either on the axis or in the plane of the ring, for either small or large values of the relevant geometric variable. Add an extra half hour or more to the time estimate for the optional extension.
Systems of Equations: Compare and ContrastSmall Group Directions
- Solve your assigned system of equations using any algebraic method. Show you work and be ready to explain how you solved it.
- 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
(Adapted from CPM Core Connections)
- \[y=-3x\\4x+y=2\]
- \[y=7x-5\\2x+y=13\]
- \[x=-5y-4\\x-4y=23\]
- \[x+y=10\\y=x-4\]
- \[y=5-x\\4x+2y=10\]
- \[3x+5y=23\\y=x+3\]
- \[y=-x-2\\2x+3y=-9\]
- \[y=2x-3\\-2x+y=1\]
- \[x=\frac{1}{2}y+\frac{1}{2}\\2x+y=-1\]
- \[a=2b+4\\b-2a=16\]
- \[y=3-2x\\4x+2y=6\]
- \[y=x+1\\x-y=1\]
Whole Class Directions
- Each group will share out how you solved your system of equations.
- Listen to each group and think about similarities and differences.
- Ask questions about anything you do not understand or you disagree with.
- 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.
In this lecture, the instructor guides a discussion about translating between bra-ket notation and wavefunction notation for quantum systems.
For \(\ell=1\), the operators that measure the three components of angular momentum in matrix notation are given by: \begin{align} L_x&=\frac{\hbar}{\sqrt{2}}\left( \begin{matrix} 0&1&0\\ 1&0&1\\ 0&1&0 \end{matrix} \right)\\ L_y&=\frac{\hbar}{\sqrt{2}}\left( \begin{matrix} 0&-i&0\\ i&0&-i\\ 0&i&0 \end{matrix} \right)\\ L_z&=\;\;\;\hbar\left( \begin{matrix} 1&0&0\\ 0&0&0\\ 0&0&-1 \end{matrix} \right) \end{align}
Show that:
- Find the commutator of \(L_x\) and \(L_y\).
- Find the matrix representation of \(L^2=L_x^2+L_y^2+L_z^2\).
- Find the matrix representations of the raising and lowering operators \(L_{\pm}=L_x\pm iL_y\). (Notice that \(L_{\pm}\) are NOT Hermitian and therefore cannot represent observables. They are used as a tool to build one quantum state from another.)
- Show that \([L_z, L_{\pm}]=\lambda L_{\pm}\). Find \(\lambda\). Interpret this expression as an eigenvalue equation. What is the operator?
- Let \(L_{+}\) act on the following three states given in matrix representation. \begin{equation} \left|{1,1}\right\rangle =\left( \begin{matrix} 1\\0\\0 \end{matrix} \right)\qquad \left|{1,0}\right\rangle =\left( \begin{matrix} 0\\1\\0 \end{matrix} \right)\qquad \left|{1,-1}\right\rangle =\left( \begin{matrix} 0\\0\\1 \end{matrix} \right) \end{equation} Why is \(L_{+}\) called a “raising operator”?
Instructor's Guide
Introduction
This activity is meant to lay the foundation of what raising and lowering oporators are and how they can be used. This material will become very important for students' study of symmetry matrices in PH427 and the Quantum Harmonic Oscillator in the Quantum Capstone.Student Conversations
At this stage, students will not have seen commutators or done much matrix multiplication in a while, so students may progress lower here than you'd expect. It will be important for the teaching team to be on the look out for groups that are confused at the beginning since some will forget that a commutator can have the form \([A,B]=AB-BA\), which is necessary to progress.
Making sure the teaching team has a good handle on the results of each calculation so they can help trouble shoot errors made during matrix multiplication which are hard to catch in the act and usually can most easilty be inferred from an erronous result (which the students themselves won't usually recognize).
Wrap-up
It is a good idea to reinforce the patterns seen in orbital angular momentum to their experiences with spin angular momentum, such as that cross product-like relationship between commutators of cartesian directed angular momenta. Then it becomes easy to contrast those patterns with that of the raising and lower operators and emphasize that these are not observables which correspond to measures of angular momentum but a different object entirely.
While their importance should be emphasized for study of periodic systems and the quantum harmonic oscilator, it should also be mentioned these operators will not be a major focus of this course or our study of the Hydrogen atom as we head into the home stretch of the course. This content is largely a very important detour.
Problem
In economics, the term utility is roughly related to overall happiness. Many things affect your happiness, including the amount of money you have and the amount of coffee you drink. We cannot directly measure your happiness, but we can measure how much money you are willing to give up in order to obtain coffee or bagels. If we assume you choose wisely, we can thus determine that your happiness increases when you decrease your amount of money by that amount in exchange for increasing your coffee consumption. Thus money is a (poor) measure of happiness or utility.
Money is also a nice quantity because it is conserved---just like energy! You may gain or lose money, but you always do so by a transaction. (There are some exceptions to the conservation of money, but they involve either the Fed, counterfeiters, or destruction of cash money, and we will ignore those issues.)
In this problem, we will assume that you have bought all the coffee and bagels you want (and no more), so that your happiness has been maximized. Thus you are in equilibrium with the coffee shop. We will assume further that you remain in equilibrium with the coffee shop at all times, and that you can sell coffee and bagels back to the coffee shop at cost.^{*}
Thus your savings \(S\) can be considered to be a function of your bagels \(B\) and coffee \(C\). In this problem we will also discuss the prices \(P_B\) and \(P_C\), which you may not assume are independent of \(B\) and \(C\). It may help to imagine that you could possibly buy out the local supply of coffee, and have to import it at higher costs.
The prices of bagels and coffee \(P_B\) and \(P_C\) have derivative relationships between your savings and the quantity of coffee and bagels that you have. What are the units of these prices? What is the mathematical definition of \(P_C\) and \(P_B\)?
Write down the total differential of your savings, in terms of \(B\), \(C\), \(P_B\) and \(P_C\).
- Solve for the total differential of your net worth. Your net worth \(W\) is the sum of your total savings plus the value of the coffee and bagels that you own. From the total differential, relate your amount of coffee and bagels to partial derivatives of your net worth.
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 construct two different rectangular coordinate systems and corresponding vector bases, then compare computations done with each.
Students use known algebraic expressions for vector line elements \(d\boldsymbol{\vec{r}}\) to determine all simple vector area \(d\boldsymbol{\vec{A}}\) and volume elements \(d\tau\) in cylindrical and spherical coordinates.
This activity is identical to Scalar Surface and Volume Elements except uses a vector approach to find directed surface and volume elements.
Students use known algebraic expressions for length elements \(d\ell\) to determine all simple scalar area \(dA\) and volume elements \(d\tau\) in cylindrical and spherical coordinates.
This activity is identical to Vector Surface and Volume Elements except uses a scalar approach to find surface, and volume elements.
There are two versions of this activity:
As a whole class activity, the instructor cuts a pumpkin in order to produce a small volume element \(d\tau\), interspersing their work with a sequence of small whiteboard questions. This version of the activity is described here.
As a small group activity, students are given pineapple rounds and pumpkin wedges to explore area and volume elements in cylindrical and spherical coordinate systems. In this version of the activity, the fruit is distributed to the students with appropriate children's pumpkin cutting equipment, as part of activities Vector Differential--Curvilinear, Scalar Surface and Volume Elements, or Vector Surface and Volume Elements.
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).
Use geometry to find formulas for velocity and acceleration in polar coordinates.
This very quick lecture reviews the content taught in https://paradigms.oregonstate.edu/courses/ph423, and is the first content in https://paradigms.oregonstate.edu/courses/ph441.
Students write python programs to compute the potential due to a square of surface charge, and then to visualize the result. This activity can be used to introduce students to the process of integrating numerically.
These lecture notes for the second week of https://paradigms.oregonstate.edu/courses/ph441 involve relating entropy and temperature in the microcanonical ensemble, using a paramagnet as an example. These notes include a few small group activities.
These notes, from the third week of https://paradigms.oregonstate.edu/courses/ph441 cover the canonical ensemble and Helmholtz free energy. They include a number of small group activities.
These notes from the fourth week of https://paradigms.oregonstate.edu/courses/ph441 cover blackbody radiation and the Planck distribution. They include a number of small group activities.
These lecture notes from week 7 of https://paradigms.oregonstate.edu/courses/ph441 apply the grand canonical ensemble to fermion and bosons ideal gasses. They include a few small group activities.
Eigenvalues and EigenvectorsEach group will be assigned one of the following matrices.
\[ A_1\doteq \begin{pmatrix} 0&-1\\ 1&0\\ \end{pmatrix} \hspace{2em} A_2\doteq \begin{pmatrix} 0&1\\ 1&0\\ \end{pmatrix} \hspace{2em} A_3\doteq \begin{pmatrix} -1&0\\ 0&-1\\ \end{pmatrix} \]
\[ A_4\doteq \begin{pmatrix} a&0\\ 0&d\\ \end{pmatrix} \hspace{2em} A_5\doteq \begin{pmatrix} 3&-i\\ i&3\\ \end{pmatrix} \hspace{2em} A_6\doteq \begin{pmatrix} 0&0\\ 0&1\\ \end{pmatrix} \hspace{2em} A_7\doteq \begin{pmatrix} 1&2\\ 1&2\\ \end{pmatrix} \]
\[ A_8\doteq \begin{pmatrix} -1&0&0\\ 0&-1&0\\ 0&0&-1\\ \end{pmatrix} \hspace{2em} A_9\doteq \begin{pmatrix} -1&0&0\\ 0&-1&0\\ 0&0&1\\ \end{pmatrix} \]
\[ S_x\doteq \frac{\hbar}{2}\begin{pmatrix} 0&1\\ 1&0\\ \end{pmatrix} \hspace{2em} S_y\doteq \frac{\hbar}{2}\begin{pmatrix} 0&-i\\ i&0\\ \end{pmatrix} \hspace{2em} S_z\doteq \frac{\hbar}{2}\begin{pmatrix} 1&0\\ 0&-1\\ \end{pmatrix} \]For your matrix:
- Find the eigenvalues.
- Find the (unnormalized) eigenvectors.
- Describe what this transformation does.
- Normalize your eigenstates.
If you finish early, try another matrix with a different structure, i.e. real vs. complex entries, diagonal vs. non-diagonal, \(2\times 2\) vs. \(3\times 3\), with vs. without explicit dimensions.
Instructor's Guide
Main Ideas
This is a small group activity for groups of 3-4. The students will be given one of 10 matrices. The students are then instructed to find the eigenvectors and eigenvalues for this matrix and record their calculations on their medium-sized whiteboards. In the class discussion that follows students report their finding and compare and contrast the properties of the eigenvalues and eigenvectors they find. Two topics that should specifically discussed are the case of repeated eigenvalues (degeneracy) and complex eigenvectors, e.g., in the case of some pure rotations, special properties of the eigenvectors and eigenvalues of hermitian matrices, common eigenvectors of commuting operators.
Students' Task
Introduction
Give a mini-lecture on how to calculate eigenvalues and eigenvectors. It is often easiest to do this with an example. We like to use the matrix \[A_7\doteq\begin{pmatrix}1&2\cr 9&4\cr\end{pmatrix}\] from the https://paradigms.oregonstate.edu/activities/2179 https://paradigms.oregonstate.edu/activities/2179 Finding Eigenvectors and Eigenvalues since the students have already seen this matrix and know what it's eigenvectors are. Then every group is given a handout, assigned a matrix, and then asked to: - Find the eigenvalues - Find the (unnormalized) eigenvectors - Normalize the eigenvectors - Describe what this transformation doesStudent Conversations
- Typically, students can find the eigenvalues without too much problem. Eigenvectors are a different story. To find the eigenvectors, they will have two equations with two unknowns. They expect to be able to find a unique solution. But, since any scalar multiple of an eigenvector is also an eigenvector, their two equations will be redundant. Typically, they must choose any convenient value for one of the components (e.g. \(x=1\)) and solve for the other one. Later, they can use this scale freedom to normalize their vector.
- The examples in this activity were chosen to include many of the special cases that can trip students up. A common example is when the two equations for the eigenvector amount to something like \(x=x\) and \(y=-y\). For the first equation, they may need help to realize that \(x=\) “anything” is the solution. And for the second equation, sadly, many students need to be helped to the realization that the only solution is \(y=0\).
Wrap-up
The majority of the this activity is in the wrap-up conversation.
The [[whitepapers:narratives:eigenvectorslong|Eigenvalues and Eigenvectors Narrative]] provides a detailed narrative interpretation of this activity, focusing on the wrap-up conversation.
- Complex eigenvectors: connect to discussion of rotations in the Linear Transformations activity where there did not appear to be any vectors that stayed the same.
- Degeneracy: Define degeneracy as the case when there are repeated eigenvalues. Make sure the students see that, in the case of degeneracy, an entire subspace of vectors are all eigenvectors.
- Diagonal Matrices: Discuss that diagonal matrices are trivial. Their eigenvalues are just their diagonal elements and their eigenvectors are just the standard basis.
- Matrices with dimensions: Students should see from these examples that when you multiply a transformation by a real scalar, its eigenvalues are multiplied by that scalar and its eigenvectors are unchanges. If the scalar has dimensions (e.g. \(\hbar/2\)), then the eigenvalues have the same dimensions.
Commutation Relations for Spin OperatorsA commutator of two observables is defined as:
\[[\hat{A},\hat{B}] = \hat{A}\hat{B} - \hat{B}\hat{A}\]
Determine the results of the following commutators:
- \([\hat{S}_x,\hat{S}_y]\)
- \([\hat{S}_y,\hat{S}_z]\)
- \([\hat{S}_z,\hat{S}_x]\)
- \([\hat{S}_y,\hat{S}_x]\)
- \([\hat{S}_z,\hat{S}_y]\)
- \([\hat{S}_x,\hat{S}_z]\)
Remember that the matrix representation of the spin operators written in the \(S_z\) basis is: \begin{align*} \hat{S}_x \doteq \frac{\hbar}{2}\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \quad \hat{S}_y \doteq \frac{\hbar}{2}\begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix} \quad \hat{S}_z \doteq \frac{\hbar}{2}\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \end{align*}
Activity: Introduction
Divide students into groups to work out whether the spin operators commute.Activity: Wrap-up
Groups should find that none of the quantum operators commute and therefore do not share the same basis for their respective eigenvectors. Because of this, it provides mathematical evidence for many properties that have so far been only observed. Since none of them commute, none of them have the same basis, nor can the spin operators be measured simultaneously.
In this introduction to heat capacity, students determine a derivative that indicates how much the internal energy changes as the temperature changes when volume is held constant.
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.
You have a system that consists of two identical (fair) six-sided dice. Imagine that you will perform an experiment where you roll the pair of dice together and record the observable: the norm of the difference between the values displayed by the two dice.
What are the possible results of the observable for each roll?
What is the theoretical probability of measuring each of those results? Assume the results are fair.
Plot a probability histogram. Use your histogram to make a guess about where the average value is and the standard deviation.
Use your theoretical probabilities to determine a theoretical average value of the observable (the expectation value)? Indicate the expectation value on your histogram.
Use your theoretical probabilities to determine the standard deviation (the uncertainty) of the distribution of possible results. Indicate the uncertainty on your histogram.
Challenge: Use
- Dirac bra-ket notation
- matrices
to represent:
the possible states of the dice after a measurement is made;
the state of the dice when you're shaking them up in your hand;
an operator that represents the norm of the difference of the dice.
Choose a vector field \(\boldsymbol{\vec{F}}\) from the first column below. Choose a small loop \(C\) (that is, a simple, closed, positively-oriented curve) which does not go around the origin.
- Is \(\oint\boldsymbol{\vec{F}}\cdot d\boldsymbol{\hat{r}}\) positive, negative, or zero?
- Will a paddlewheel spin if placed inside your loop, and, if so, which way?
Do you think \(\nabla\times\boldsymbol{\vec{F}}\) is zero or nonzero inside your loop?
Explain.
- Compute \(\nabla\times\boldsymbol{\vec{F}}\). Did you guess right? Explain.
- Is \(\oint\boldsymbol{\vec{F}}\cdot\boldsymbol{\hat{n}}\,ds\) positive, negative, or zero? (\(\boldsymbol{\hat{n}}\) is the outward pointing normal vector to \(C\).)
- Is the net flow outwards across your loop positive, negative, or zero?
- Do you think \(\nabla\cdot\boldsymbol{\vec{F}}\) is zero or nonzero inside your loop? Explain.
- Compute \(\nabla\cdot\boldsymbol{\vec{F}}\). Did you guess right? Explain.
- Repeat the above steps for vector fields \(\boldsymbol{\vec{G}}\) and \(\boldsymbol{\vec{H}}\) chosen from the second and third columns.
\(-y\,\boldsymbol{\hat{x}}+x\,\boldsymbol{\hat{y}}\) \((x+y)\,\boldsymbol{\hat{x}}+(y-x)\,\boldsymbol{\hat{y}}\) \(e^{-y^2}\,\boldsymbol{\hat{y}}\) \(x\,\boldsymbol{\hat{x}}+y\,\boldsymbol{\hat{y}}\) \((y-x)\,\boldsymbol{\hat{x}}-(x+y)\,\boldsymbol{\hat{y}}\) \(e^{-x^2}\,\boldsymbol{\hat{y}}\) Main ideas
- Visualization of divergence and curl.
Prerequisites
- Definition of divergence and curl.
- Geometry of divergence and curl, either through a geometric definition or through Stokes' Theorem and the Divergence Theorem.
Warmup
- Students may need to be reminded what circulation is.
- Students may not have seen flux in 2 dimensions.
- Students may only have seen \(\boldsymbol{\hat{n}}\) for surfaces, not curves. Some students will set \(\boldsymbol{\hat{n}}=\boldsymbol{\hat{z}}\)! Emphasize that \(\boldsymbol{\hat{n}}\) is horizontal (and that \(ds\ne\boldsymbol{d\vec{S}}\)).
Props
- whiteboards and pens
- formula sheet for div and curl in spherical and cylindrical coordinates (Each group may need its own copy.)
- divergence and curl transparency
- blank transparencies and pens
Wrapup
- Discuss the effect of choosing loops of different shapes, especially those adapted to the given vector field.
- Talk about the geometry of sinks and sources (for divergence) and paddlewheels (for curl).
Details
In the Classroom
- While students are working on this activity, draw the vector fields on the board to use during the wrapup. Alternatively, bring an overhead transparency showing the vector fields (and blank transparencies for students to write on).
- Students like this lab; it should flow smoothly and quickly.
- Students may need to be reminded what \(\oint\) means, and that the positive orientation in the plane is counterclockwise.
- Yes, two pairs of questions are really the same.
- Make sure the paths do not go around the origin.
- Encourage each group to work on at least two vector fields, which are in different rows and columns. Include one vector field from the third column if time permits.
- Encourage each group to consider, for a single vector field, moving their loop to another location. This is especially effective (and in fact essential) for the two vector fields in the third column.
- See the discussion of using transparencies for Group Activity The Hill.
- Students may eventually realize that the vector fields in the middle column are linear combinations of the vector fields in the first column, which are in turn “pure curl” and “pure divergence”, respectively.
Subsidiary ideas
- Divergence and curl are not just about the behavior near the origin. Derivatives are about change --- the difference between nearby vectors.
Homework
(MHG refers to McCallum, Hughes Hallett, Gleason, et al.
- MHG 19.1:20
- MHG 20.2:16
- MHG 20.3:10,12,20
- MHG 20.4:22
Essay questions
- Which operation, curl or divergence is easier to understand?
- Which is more useful?
- Do you prefer to gauge curl from a plot or from a calculation? What about divergence?
Enrichment
- Emphasize the importance of divergence and curl in applications.
- Ask students how to determine which vector fields are conservative! (A single closed path with nonzero circulation suffices to show that a vector field is not conservative. The best geometric way we know to show that a vector field is conservative is to try to draw the level curves for which the given vector field would be the gradient.)
- Discuss the fact that \(\boldsymbol{\hat{r}}\over r\) and \(\boldsymbol{\hat{\phi}}\over r\) are both curl-free and divergence-free; this is counterintuitive, but crucial for electromagnetism. (These are, respectively, the electric/magnetic field of a charged/current-carrying wire along the \(z\)-axis.)
- Discuss the behavior of \(\boldsymbol{\hat{r}}\over r^n\) and \(\boldsymbol{\hat{\phi}}\over r^n\), emphasizing that both the divergence and curl vanish when \(n=1\).
- Relate these examples to the magnetic field of a wire (\(\boldsymbol{\vec{B}}={\boldsymbol{\hat{\phi}}\over r}\)) and the electric field of a point charge (\(\boldsymbol{\vec{E}}={\boldsymbol{\hat{r}}\over r^2}\); this is the spherical \(r\)).
- Show students how to compute divergence and curl of these vector fields in cylindrical coordinates.
- Trying to estimate divergence and curl from a single plot of a vector field confronts students with the need to zoom in. Technology can be useful here.
- Point students to our paper on Electromagnetic Conic Sections, which appeared in Am. J. Phys. 70, 1129--1135 (2002), and which is also available on the Bridge Project website.
- Most physical applications of the divergence are 3-dimensional, rather than 2-dimensional. Each vector field in this activity could be regarded as a horizontal 3-dimensional vector field by assuming that there is no \(z\)-dependence, in which case the flux can be computed through a 3-dimensional box whose cross-section is the loop, and whose horizontal top and bottom do not contribute.
These lecture notes covering week 8 of https://paradigms.oregonstate.edu/courses/ph441 include a small group activity in which students derive the Carnot efficiency.
These notes from the fifth week of https://paradigms.oregonstate.edu/courses/ph441 cover the grand canonical ensemble. They include several small group activities.
This is the first activity relating the surfaces to the corresponding contour diagrams, thus emphasizing the use of multiple representations.
Students work in small groups to interpret level curves representing different concentrations of lead.
This small group activity introduces students to constrained optimization problems. Students work in small groups to optimize a simple function on a given region. The whole class wrap-up discussion emphasizes the importance of the boundary.
This mini-lecture demonstrates the relationship between \(df\) on the tangent plane to its “components“ in coordinate directions, leading to the multivariable chain rule.
In class, you measured the isolength stretchability and the isoforce stretchability of your systems in the PDM. We found that for some systems these were very different, while for others they were identical.
Show with algebra (NOT experiment) that the ratio of isolength stretchability to isoforce stretchability is the same for both the left-hand side of the system and the right-hand side of the system. i.e.: \begin{align} \frac{\left(\frac{\partial {x_L}}{\partial {F_L}}\right)_{x_R}}{\left(\frac{\partial {x_L}}{\partial {F_L}}\right)_{F_R}} &= \frac{\left(\frac{\partial {x_R}}{\partial {F_R}}\right)_{x_L}}{\left(\frac{\partial {x_R}}{\partial {F_R}}\right)_{F_L}} \label{eq:ratios} \end{align}
Hint
You will need to make use of the cyclic chain rule: \begin{align} \left(\frac{\partial {A}}{\partial {B}}\right)_{C} = -\left(\frac{\partial {A}}{\partial {C}}\right)_{B}\left(\frac{\partial {C}}{\partial {B}}\right)_{A} \end{align}Hint
You will also need the ordinary chain rule: \begin{align} \left(\frac{\partial {A}}{\partial {B}}\right)_{D} = \left(\frac{\partial {A}}{\partial {C}}\right)_{D}\left(\frac{\partial {C}}{\partial {B}}\right)_{D} \end{align}
Make sure that you have memorized the following identities and can use them in simple algebra problems: \begin{align} e^{u+v}&=e^u \, e^v\\ \ln{uv}&=\ln{u}+\ln{v}\\ u^v&=e^{v\ln{u}} \end{align}
The gravitational field due to a spherical shell of matter (or equivalently, the electric field due to a spherical shell of charge) is given by: \begin{equation} \vec g = \begin{cases} 0&\textrm{for } r<a\\ -G \,\frac{M}{b^3-a^3}\, \left( r-\frac{a^3}{r^2}\right)\, \hat r & \textrm{for } a<r<b\\ -G\,\frac{M}{r^2}\, \hat r & \textrm{for } r>b \\ \end{cases} \end{equation}
This problem explores the consequences of the divergence theorem for this shell.
- Using the given description of the gravitational field, find the divergence of the gravitational field everywhere in space. You will need to divide this question up into three parts: \(r<a\), \(a<r<b\), and \(r>b\).
- Briefly discuss the physical meaning of the divergence in this particular example.
- For this gravitational field, verify the divergence theorem on a sphere, concentric with the shell, with radius \(Q\), where \(a<Q<b\). ("Verify" the divergence theorem means calculate the integrals from both sides of the divergence theorem and show that they give the same answer.)
- Briefly discuss how this example would change if you were discussing the electric field of a uniformly charged spherical shell.
Problem
Find the course syllabus and schedule on Canvas. Read through them carefully and bring your questions to the second day of class.
(Messy algebra) Convince yourself that the expressions for kinetic energy in original and center of mass coordinates are equivalent. The same for angular momentum.
Consider a system of two particles of mass \(m_1\) and \(m_2\).
- Show that the total kinetic energy of the system is the same as that of two “fictitious” particles: one of mass \(M=m_1+m_2\) moving with the velocity of the center of mass and one of mass \(\mu\) (the reduced mass) moving with the velocity of the relative position.
- Show that the total angular momentum of the system can similarly be decomposed into the angular momenta of these two fictitious particles.
The goal of this problem is to show that once we have maximized the entropy and found the microstate probabilities in terms of a Lagrange multiplier \(\beta\), we can prove that \(\beta=\frac1{kT}\) based on the statistical definitions of energy and entropy and the thermodynamic definition of temperature embodied in the thermodynamic identity.
The internal energy and entropy are each defined as a weighted average over microstates: \begin{align} U &= \sum_i E_i P_i & S &= -k_B\sum_i P_i \ln P_i \end{align}: We saw in clase that the probability of each microstate can be given in terms of a Lagrange multiplier \(\beta\) as \begin{align} P_i &= \frac{e^{-\beta E_i}}{Z} & Z &= \sum_i e^{-\beta E_i} \end{align} Put these probabilities into the above weighted averages in order to relate \(U\) and \(S\) to \(\beta\). Then make use of the thermodynamic identity \begin{align} dU = TdS - pdV \end{align} to show that \(\beta = \frac1{kT}\).
- Find the Fourier transforms of \(f(x)=\cos kx\) and \(g(x)=\sin kx\).
- Find the Fourier transform of \(g(x)\) using the formula for the Fourier transform of a derivative and your result for the Fourier transform of \(f(x)\). Compare with your previous answer.
- In quantum mechanics, the Fourier transform is the set of coefficients in the expansion of a quantum state in terms of plane waves, i.e. the function \(\tilde{f}(k)\) is a continuous histogram of how much each functions \(e^{ikx}\) contributes to the quantum state. What does the Fourier transform of the function \(\cos kx\) tell you about which plane waves make up this quantum state? Write a sentence or two about how this makes sense.
The goal of this problem is to show that once we have maximized the entropy and found the microstate probabilities in terms of a Lagrange multiplier \(\beta\), we can prove that \(\beta=\frac1{kT}\) based on the statistical definitions of energy and entropy and the thermodynamic definition of temperature embodied in the thermodynamic identity.
The internal energy and entropy are each defined as a weighted average over microstates: \begin{align} U &= \sum_i E_i P_i & S &= -k_B\sum_i P_i \ln P_i \end{align} We saw in clase that the probability of each microstate can be given in terms of a Lagrange multiplier \(\beta\) as \begin{align} P_i &= \frac{e^{-\beta E_i}}{Z} & Z &= \sum_i e^{-\beta E_i} \end{align} Put these probabilities into the above weighted averages in order to relate \(U\) and \(S\) to \(\beta\). Then make use of the thermodynamic identity \begin{align} dU = TdS - pdV \end{align} to show that \(\beta = \frac1{kT}\).
Problem
Suppose \(g(U) = CU^{3N/2}\), where \(C\) is a constant and \(N\) is the number of particles.
Show that \(U=\frac32 N k_BT\).
Show that \(\left(\frac{\partial^2S}{\partial U^2}\right)_N\) is negative. This form of \(g(U)\) actually applies to a monatomic ideal gas.
Consider the magnetic field \[ \vec{B}(s,\phi,z)= \begin{cases} 0&0\le s<a\\ \alpha \frac{1}{s}(s^4-a^4)\, \hat{\phi}&a<s<b\\ 0&s>b \end{cases} \]
- (2pts) Use step and/or delta functions to write this magnetic field as a single expression valid everywhere in space.
- (4pts) Find a formula for the current density that creates this magnetic field.
- (2pts) Interpret your formula for the current density, i.e. explain briefly in words where the current is.
Consider the electric field \begin{equation} \vec E(r,\theta,\phi) = \begin{cases} 0&\textrm{for } r<a\\ \frac{1}{4\pi\epsilon_0} \,\frac{Q}{b^3-a^3}\, \left( r-\frac{a^3}{r^2}\right)\, \hat r & \textrm{for } a<r<b\\ 0 & \textrm{for } r>b \\ \end{cases} \end{equation}
- (4pts) Use step and/or delta functions to write this electric field as a single expression valid everywhere in space.
- (4pts) Find a formula for the charge density that creates this electric field.
- (2pts) Interpret your formula for the charge density, i.e. explain briefly in words where the charge is.
Show that \begin{align} f(\mu+\delta) &= 1 - f(\mu-\delta) \end{align} This means that the probability that an orbital above the Fermi level is occupied is equal to the probability an orbital the same distance below the Fermi level being empty. These unoccupied orbitals are called holes.
Show that a Fermi electron gas in the ground state exerts a pressure \begin{align} p = \frac{\left(3\pi^2\right)^{\frac23}}{5} \frac{\hbar^2}{m}\left(\frac{N}{V}\right)^{\frac53} \end{align} In a uniform decrease of the volume of a cube every orbital has its energy raised: The energy of each orbital is proportional to \(\frac1{L^2}\) or to \(\frac1{V^{\frac23}}\).
Find an expression for the entropy of a Fermi electron gas in the region \(kT\ll \varepsilon_F\). Notice that \(S\rightarrow 0\) as \(T\rightarrow 0\).
In this entire problem, keep results to first order in the van der Waals correction terms \(a\) and $b.
Show that the entropy of the van der Waals gas is \begin{align} S &= Nk\left\{\ln\left(\frac{n_Q(V-Nb)}{N}\right)+\frac52\right\} \end{align}
Show that the energy is \begin{align} U &= \frac32 NkT - \frac{N^2a}{V} \end{align}
Show that the enthalpy \(H\equiv U+pV\) is \begin{align} H(T,V) &= \frac52NkT + \frac{N^2bkT}{V} - 2\frac{N^2a}{V} \\ H(T,p) &= \frac52NkT + Nbp - \frac{2Nap}{kT} \end{align}
Problem
Submit your video on Canvas. After you submit your video, complete the reflection questions for the course on Gradescope.
- Simplify the following expressions:
\(\ln{x}+\ln{y}\)
\(\ln{a}-\ln{b}\)
\(2\ln{f}+3\ln{f}\)
\(e^{m}e^{k}\)
\(\frac{e^{c}}{e^{d}}\)
- Expand the following expressions:
\(e^{(3h-j)}\)
\(e^{2(c+w)}\)
\(\ln{h/g}\)
\(\ln(kT)\)
\(\ln{\sqrt{\frac{q}{r}}}\)
Find the rectangular coordinates of the point where the angle \(\frac{5\pi}{3}\) meets the unit circle. If this were a point in the complex plane, what would be the rectangular and exponential forms of the complex number? (See figure.)
Students use completeness relations to write a matrix element of a spin component in a different basis.
Transitioning from the PDM back to thermodynamic systemsHeating
In the partial derivative machine, the change in internal energy corresponds to the work done on the left string and the right string: \begin{align} dU &= F_L dx_L + F_R dx_R \end{align} The ”thing we changed” was \(dx_L\) or \(dx_R\). From that we could determine the change in internal energy.
When we transfer energy to something by heating, it's hard to measure the “thing we changed,” which was entropy. It is, however, possible in some cases to measure the amount of energy transfered by heating, and from that we can work backwards to find out how much the entropy changed.
The amount of energy transfered into a system by heating is generally written as \(Q\). (There is a historical misconception built deeply into our language that heat is a property of a material, i.e. a state property. This is caloric theory. You don't need to know any of this history, but you do have to be careful when using the word “heat”.)
An infinitesimal amount of energy transfered by heating is called \({\mathit{\unicode{273}}} Q\). The symbol \({\mathit{\unicode{273}}} \) indicates an inexact differential, which you can think of as a “small chunk” that is not the change of something. \({\mathit{\unicode{273}}} Q\) is not a small change in the amount of energy transfered by heating, but rather is a small amount of energy transfered by heating.
When playing with the partial derivative machine, we can say the work done on the left string, \(F_Ldx_L\), is analogous to heat entering a thermodynamic system.
Latent heat
A phase transition is when a material changes state of matter, as in melting or boiling. At most phase transitions (technically, abrupt phase transitions as you will learnin the Capstone), the temperature remains constant while the material is changing from one state to the other. So you know that as long as you have ice and water coexisting in equilibrium at one atmosphere of pressure, the temperature must be \(0^\circ\)C. Similarly, as long as water is boiling at one atmosphere of pressure, the temperature must be \(100^\circ\)C. In both of these cases, you can transfer energy to the system (as we will) by heating without changing the temperature! This relates to why I keep awkwardly saying “transfer energy to a system by heating” rather than just “heating a system” which means the same thing. We have deeply ingrained the idea that “heating” is synonymous with “raising the temperature,” which does not align with the physics meaning.
So now let me define the latent heat. The latent heat is the amount of energy that must be transfered to a material by heating in order to change it from one phase to another. The latent heat of fusion is the amount of energy required to melt a solid, and the latent heat of vaporization is the amount of energy required to turn a liquid into a gas. We will be measuring both of these for water.
A question you may ask is whether the latent heat is extensive or intensive. Technically the latent heat is extensive, since if you have more material then more energy is required to melt/boil it. However, when you hear latent heat quoted, it is almost always the specific latent heat, which is the energy transfer by heating required per unit of mass. It can be confusing that people use the same words to refer to both quantities. Fortunately, dimensional checking can always give you a way to verify which is being referred to. If \(L\) is an energy per mass, then it must be the specific latent heat, while if it is an energy, then it must be the latent heat.
Heat capacity and specific heat
The heat capacity is the amount of energy transfer required per temperature to raise the temperature of a system. If we hold the pressure fixed (as in our experiment) we can write this as: \begin{align} {\mathit{\unicode{273}}} Q &= C_p dT \end{align} where \(C_p\) is the heat capacity at fixed pressure. You might think to rewrite this expression as a derivative, but we can't do that since the energy transfered by heating is not a state function.
Note that the heat capacity, like the latent heat, is an extensive quantity. The specific heat is the the heat capacity per unit mass, which is an intensive quantity that we can consider a property of a material independently of the quantity of that material.
I'll just mention as an aside that the term “heat capacity” is another one of those unfortunate phrases that reflect the inaccurate idea that heat is a property of a system.
Entropy
Finally, we can get to entropy. Entropy is the “thing that changes” when you transfer energy by heating. I'll just give this away: \begin{align} {\mathit{\unicode{273}}} Q &= TdS \end{align} where this equation is only true if you make the change quasistatically (see another lecture). This allows us to find the change in entropy if we know how much energy was transfered by heating, and the temperature in the process. \begin{align} \Delta S &= \int \frac1T {\mathit{\unicode{273}}} Q \end{align} where again, we need to know the temperature as we add heat.
- Work as area under curve in a \(pV\) plot
- Heat transfer as area under a curve in a \(TS\) plot
- Reminder that internal energy is a state function
- Reminder of First Law
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).