Find the eigenvalues and normalized eigenvectors of the Pauli
matrices \(\sigma_x\), \(\sigma_y\), and \(\sigma_z\) (see the Spins Reference Sheet posted on the course website).
Einstein condensation temperature Starting
from the density of free particle orbitals per unit energy range
\begin{align}
\mathcal{D}(\varepsilon) =
\frac{V}{4\pi^2}\left(\frac{2M}{\hbar^2}\right)^{\frac32}\varepsilon^{\frac12}
\end{align} show that the lowest temperature at which the total number
of atoms in excited states is equal to the total number of atoms is
\begin{align}
T_E &=
\frac1{k_B}
\frac{\hbar^2}{2M}
\left(
\frac{N}{V}
\frac{4\pi^2}{\int_0^\infty\frac{\sqrt{\xi}}{e^\xi-1}d\xi}
\right)^{\frac23}
T_E &=
\end{align} The infinite sum may be numerically evaluated to be 2.612.
Note that the number derived by integrating over the density of
states, since the density of states includes all the states
except the ground state.
Note: This problem is solved in the text itself. I intend to
discuss Bose-Einstein condensation in class, but will not derive this
result.
Students work in small groups to use Coulomb's Law
\[\vec{E}(\vec{r}) =\frac{1}{4\pi\epsilon_0}\int\frac{\rho(\vec{r}^{\,\prime})\left(\vec{r}-\vec{r}^{\,\prime}\right)}{\vert \vec{r}-\vec{r}^{\,\prime}\vert^3} \, d\tau^{\prime}\]
to find an integral expression for the electric field, \(\vec{E}(\vec{r})\), everywhere in space, due to a ring of charge.
In an optional extension, students find a series expansion for \(\vec{E}(\vec{r})\) 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.
Students integrate numerically to find the electric field due to a cone of surface charge, and then visualize the result. This integral can be done in either spherical or cylindrical coordinates, giving students a chance to reason about which coordinate system would be more convenient.
electric fieldcone Found in: Computational Physics Lab II course(s)Found in: Computational integrating charge distributions sequence(s)
Consider the finite line with a uniform charge density from class.
Write an integral expression for the electric field at any point in space due
to the finite line. In addition to your usual physics sense-making, you must
include a clearly labeled figure and discuss what happens to the direction of
the unit vectors as you integrate.Consider the finite line with a uniform
charge density from class.
Perform the integral to find the \(z\)-component of the electric field. In
addition to your usual physics sense-making, you must compare your result to
the gradient of the electric potential we found in class. (If you want to
challenge yourself, do the \(s\)-component as well!)
Students need to understand that the surface represents the electric potential in the center of a parallel plate capacitor. Try doing the activity Electric Potential of Two Charged Plates before this activity.
Students should know that
objects with like charge repel and opposite charge attract,
object tend to move toward lower energy configurations
The potential energy of a charged particle is related to its charge: \(U=qV\)
The force on a charged particle is related to its charge: \(\vec{F}=q\vec{E}\)
Students examine a plastic "surface" graph of the electric potential due to two charged plates (near the center of the plates) and explore the properties of the electric potential.
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.
Write the equation for the electrostatic potential due to a point charge.
Instructor's Guide
Prerequisite Knowledge
Students will usually have seen the electrostatic potential due to a point charge in their introductory course, but may have trouble recalling it.
Whole-Class Conversations
As students try to remember the formula, many will conflate potential, potential energy, force, and electric field. Their answers may have some aspects of each of these. We use this question to get the iconic equation into the students' working memory in preparation for subsequent activities. This question also be used to help student disambiguate these different physical quantities.
Correct answers you're likely to see
\[V=\frac{kq}{r}\]
\[V=\frac{1}{4\pi\epsilon_0}\frac{q}{r}\]
You may want to discuss which constants to use in which contexts, e.g. \(k\) is short and easy to write, but may be conflated with other uses of \(k\) in a give problem whereas \(\frac{1}{4\pi\epsilon_0}\) assumes you are working in a particular system of units.
Incorrect answers you're likely to see
Two charges instead of one
\[\cancel{V=\frac{kq_{1}q_{2}}{r}}\]
Distance squared in the denominator
\[\cancel{V=\frac{kq}{r^2}}\]
Possible follow-up questions to help with the disambiguation:
Relationship between potential and potential energy \(U = qV\)
Which function is the derivative of the other: \(1/r\) or \(1/r^2\)?
Which physical quantity (potential or electric field, potential energy or force) is the derivative of the other?
What is the electrostatic potential conceptually?
Which function falls off faster: \(1/r\) or \(1/r^2\)?
What are the dimensions of potential? Units?
Where is the zero of potential?
Wrap-up
This could be a good time to refer to the (correct) expression for the potential as an iconic equation, which will need to be further interpreted (”unpacked”) in particular physical situations. This is where the course is going next.
This SWBQ can also serve to help students learn about recall as a cognitive activity. While parts of the equations that students write may be incorrect, many other parts will be correct. Let the way in which you manage the class discussion model for the students how a professional goes about quickly disambiguating several different choices. And TELL the students that this is what you are doing. Deliberately invoke their metacognition.
Many students may not know that the electrostatic potential that we are talking about in this activity is the same quantity as what a voltmeter reads, in principle, but not in practice. You may need to talk about how a voltmeter actually works, rather than idealizing it. It helps to have a voltmeter with leads as a prop. Students often want to know about the “ground” lead. We often tie a long string to it (to symbolize making a really long wire) and send the TA out of the room with the string, “headed off to infinity” while discussing the importance of setting the zero of potential. The extra minute or two of humerous byplay gives the importance of the zero of potential a chance to sink in.
We use this small whiteboard question as a transition between The Distance Formula (Star Trek) activity, where students are learning about how to describe (algebraically) the geometric distance between two points, and the Electrostatic Potential Due to a Pair of Charges (with Series) activity, where students are using these results and the superposition principle to find the electrostatic potential due to two point charges.
This activity is the initial activity in the sequence Visualizing Scalar Fields addressing the representations of scalar fields in the context of electrostatics.
Found in: Static Fields, None course(s)Found in: Warm-Up, E&M Ring Cycle Sequence sequence(s)
Students work in small groups to use the superposition principle
\[V(\vec{r}) =\frac{1}{4\pi\epsilon_0}\int\frac{\rho(\vec{r}^{\,\prime})}{\vert \vec{r}-\vec{r}^{\,\prime}\vert} \, d\tau^{\prime}\]
to find an integral expression for the electrostatic potential, \(V(\vec{r})\), everywhere in space, due to a ring of charge.
In an optional extension, students find a series expansion for \(V(\vec{r})\) 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.
Consider a system consisting of four point charges arranged on the corners of a square in 3D Cartesian space of coordinates \((x,y,z)\).
Write a python function that returns the potential at any point in space caused by four equal point charges forming a square. Make the sides of the square parallel to the \(x\) and \(y\) axes and on the \(z=0\) plane.
To do this you will need the expression for a the potential due to a single point charge \(V= \frac{k_Cq}{r}\) where \(r\) is the distance from the point charge. You will also need to use the fact that the total potential is the sum of the potentials due to each individual point charge.
Once you have written the above function, use it to plot the electrostatic potential versus position along the three cartesian axes.
Label your axes.
Work out the first non-zero term in a power series approximation for the potential at large \(x\), small \(x\), etc. Plot these approximations along with your computed potential, and verify that they agree in the range that you expect. Useful 1\(^{st}\) order Taylor expansions are:
\begin{eqnarray}
\sqrt{1+\epsilon} &\sim& 1+\frac{\epsilon}{2} \\
\frac{1}{1+\epsilon} &=& 1-\epsilon
\end{eqnarray}
where \(\epsilon\) is a small quantity.
Extra fun
Create one or more different visualizations of the electrostatic potential. For example a 2D representation in the \(z=0\) plane.
More extra fun
Create a plot of the potential along a straight line that is not one of the axes. Hint: start from a line on the \(z=0\) plane, then try a random straight line. You can use your browser for help.
Even more extra fun
Move the charges around (e.g., off the \(z=0\) plane) and see what happens to your graphs
Dipole fun
Repeat the above (especially the limiting cases!) for four point charges in which half are positive and half negative, with the positive charges neighbors.
Quadrupole fun
Repeat the above (especially the limiting cases!) for four point charges in which half are positive and half negative, with the positive charges diagonal from one another. It will help in this case to place the charges on the axes (rotating the square by 45 degrees), since otherwise the potential on each axis will be zero.
electrostatic potentialpython Found in: Computational Physics Lab II course(s)Found in: Computational integrating charge distributions sequence(s)
Students solve numerically for the potential due to a spherical shell of charge. Although this potential is straightforward to compute using Gauss's Law, it serves as a nice example for numerically integrating in spherical coordinates because the correct answer is easy to recognize.
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.
Consider a system of
fixed volume in thermal contact with a resevoir. Show that the mean
square fluctuations in the energy of the system is \begin{equation}
\left<\left(\varepsilon-\langle\varepsilon\rangle\right)^2\right>
= k_BT^2\left(\frac{\partial U}{\partial T}\right)_{V}
\end{equation} Here \(U\) is the conventional symbol for
\(\langle\varepsilon\rangle\). Hint: Use the partition function
\(Z\) to relate \(\left(\frac{\partial U}{\partial T}\right)_V\) to
the mean square fluctuation. Also, multiply out the term
\((\cdots)^2\).
For electrons with an energy \(\varepsilon\gg mc^2\), where
\(m\) is the mass of the electron, the energy is given by
\(\varepsilon\approx pc\) where \(p\) is the momentum. For electrons
in a cube of volume \(V=L^3\) the momentum takes the same values as
for a non-relativistic particle in a box.
Show that in this extreme relativistic limit the Fermi energy of a
gas of \(N\) electrons is given by \begin{align}
\varepsilon_F &= \hbar\pi c\left(\frac{3n}{\pi}\right)^{\frac13}
\end{align} where \(n\equiv \frac{N}{V}\) is the number density.
Show that the total energy of the ground state of the gas is
\begin{align}
U_0 &= \frac34 N\varepsilon_F
\end{align}
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 \(\beta = \frac1{kT}\).
Found in: Thermal and Statistical Physics course(s)
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 \(\beta = \frac1{kT}\).
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.
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 nota 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.
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.
Suppose that a
system of \(N\) atoms of type \(A\) is placed in diffusive contact
with a system of \(N\) atoms of type \(B\) at the same temperature and
volume.
Show that after diffusive equilibrium is reached the total entropy
is increased by \(2Nk\ln 2\). The entropy increase \(2Nk\ln 2\) is
known as the entropy of mixing.
If the atoms are identical (\(A=B\)), show that there is no increase
in entropy when diffusive contact is established. The difference has
been called the Gibbs paradox.
Since the Helmholtz free energy is lower for the mixed \(AB\) than
for the separated \(A\) and \(B\), it should be possible to extract
work from the mixing process. Construct a process that could extract
work as the two gasses are mixed at fixed temperature. You will
probably need to use walls that are permeable to one gas but not the
other.
Note
This course has not yet covered work, but it was covered in
Energy and Entropy, so you may need to stretch your memory to finish
part (c).
Students are prompted to consider the scalar superposition of the electric potential due to multiple point charges. First a single point charge is discussed, then four positive charges, then an electric quadrupole. Students draw the equipotential curves in the plane of the charges, while also considering the 3D nature of equipotentials.
Students practice identifying whether events on spacetime diagrams are simultaneous, colocated, or neither for different observers. Then students decide which of two events occurs first in two different reference frames.
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.
Students calculate the expectation value of energy and angular momentum as a function of time for an initial state for a particle on a ring. This state is a linear combination of energy/angular momentum eigenstates written in bra-ket notation.
We would like to get to know you a bit and learn about your expectations for this course.
Please take a few minutes to share your thoughts on the following questions.
What have you heard or learned about quantum mechanics? Where did you learn it?
What do you expect to get out of this class? (This could include how you hope your understanding, ways of thinking, confidence, or feelings about quantum mechanics will develop over the term.)
How do you feel about taking this class? Nervous, excited, bored, etc.? What makes you feel that way?
Are there aspects of quantum mechanics you are particularly curious about?
Consider a system which has an internal energy \(U\) defined by:
\begin{align}
U &= \gamma V^\alpha S^\beta
\end{align}
where \(\alpha\), \(\beta\) and \(\gamma\) are constants. The internal
energy is an extensive quantity. What constraint does this place on
the values \(\alpha\) and \(\beta\) may have?
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.
(Use the equation for orbit shape.) Gain experience with unusual force laws.
In science fiction movies, characters often talk about a spaceship “spiralling in” right before it hits the planet. But all orbits in a \(1/r^2\) force are conic sections, not spirals. This spiralling in happens because the spaceship hits atmosphere and the drag from the atmosphere changes the shape of the orbit. But, in an alternate universe, we might have other force laws.
In class, we discussed how to calculate the shape of the orbit for an inverse square potential. More generally, the equation for the orbit of a mass \(\mu\) under the influence of a central force \(f(r)\) is given by:
\begin{align}
\frac{d^2 u}{d\phi^2} + u &=-\frac{\mu}{\ell^2}\frac{1}{u^2}f\left(\frac{1}{u}\right)\\
\Rightarrow f\left(\frac{1}{u}\right)&=-\frac{\ell^2}{\mu}u^2
\left(\frac{d^2 u}{d\phi^2} + u\right)
\end{align}
where \(u=r^{-1}\).
Find the force law for a mass \(\mu\), under the influence of a central-force field, that moves in a logarithmic spiral orbit given by \(r = ke^{\alpha \phi}\), where \(k\) and \(\alpha\) are constants.
Students are asked to "find the derivative" of a plastic surface that represents a function of two variables. This ambiguous question is designed to help them generalize their concept of functions of one variable to functions of two variables. The definition of the gradient as the slope and direction of the "steepest derivative" is introduced geometrically.
The diagonal of the rectangle on the left below shows (a blown-up picture of)
an infinitesimal displacement from the point (\(x\), \(y\)) to the nearby point
(\(x+dx\), \(y+dy\)).
Label the rectangle with the lengths of the sides.
Express the sides of the rectangle indicated by arrows as vectors.
Use the unit vectors \(\boldsymbol{\hat{x}}\) and \(\boldsymbol{\hat{y}}\).
The diagonal of this rectangle is the vector differential \(d\vec{r}\). Express
\(d\vec{r}\) in terms of \(\boldsymbol{\hat{x}}\) and \(\boldsymbol{\hat{y}}\).
Find the length \(ds=|d\vec{r}|\) of the diagonal.
The diagonal of the “rectangle” on the right above shows (a blown-up
picture of) the same infinitesimal displacement, now expressed in
polar coordinates, from the point (\(r\), \(\phi\)) to the nearby point
(\(r+dr\), \(\phi+d\phi\)).
Label the rectangle with the lengths of the sides.
Careful!
Express the sides of the rectangle indicated by arrows as vectors.
Use the natural orthonormal basis defined by the picture, that is, let
\(\hat{r}\) be the unit vector which points in the direction of increasing \(\vec{r}\),
and let \(\hat{\phi}\) be the unit vector which points in the direction of increasing
\(\phi\). Do not attempt to express these vectors in terms of \(\boldsymbol{\hat{x}}\) and
\(\boldsymbol{\hat{y}}\)! You do not need to worry about the fact that some sides of the
rectangle aren't straight; the rectangle is so small that this error is
negligible.
The diagonal of this rectangle is again the vector differential \(d\boldsymbol{\vec{r}}\).
Express \(d\boldsymbol{\vec{r}}\) in terms of \(\hat{r}\) and \(\hat{\phi}\)
Find the length \(ds=|d\vec{r}|\) of the diagonal.
Essentials
Main ideas
Introduces \(d\boldsymbol{\vec{r}}\), the key to vector calculus, as a geometric object.
Don't skip this activity if you use nonrectangular basis
vectors!*
Prerequisites
Familiarity with \(\boldsymbol{\hat{r}}\) and \(\boldsymbol{\hat{\phi}}\). The Acceleration activity is a good introduction to those vectors.
Warmup
Draw a picture on the board showing \(d\boldsymbol{\vec{r}}\) as the infinitesimal change in the
position vector \(\boldsymbol{\vec{r}}\) between two infinitesimally close points.
Props
whiteboards and pens
Big arrows, perhaps made of straws, which can represent an orthonormal basis,
and which can be moved around a curve on the board.
Wrapup
Emphasize that \(d\boldsymbol{\vec{r}}\) is the same geometric object regardless of how it is
expressed.
Discuss the geometry of \(ds\) as the magnitude of \(d\boldsymbol{\vec{r}}\), that is, \(ds=|d\boldsymbol{\vec{r}}|\).
This is a good place to introduce the idea of “what sort of a beast is it”.
The vector differential \(d\boldsymbol{\vec{r}}\) is an infinitesimal differential having both
direction and (infinitesimal) length. When writing an expression for \(d\boldsymbol{\vec{r}}\),
students should make sure that each term has these same properties.
Details
In the Classroom
Most groups will miss the factor of \(r\) in the \(\boldsymbol{\hat{\phi}}\) component of \(d\boldsymbol{\vec{r}}\).
Watch for this as you walk around the classroom. A good thing to point out is
that \(d\phi\) is not a length.
Some groups will then remember the formula for arclength and be able to figure
out the rest on their own. Other groups will need to be reminded about the
relationship between arclength and radius on a circle. A good way to do this
is to ask them for the formula for the circumference of a circle, then half a
circle, a quarter, etc. Make sure to give the angles in radians! Eventually,
they get the point.
Some students may wonder whether the top of the (Cartesian) rectangle is \(\pm
dx\,\boldsymbol{\hat{x}}\). This question is ill-posed, since the sign of \(dx\) itself depends
on which way you're going; you can't change your mind in the middle of a
problem. The safest way to resolve such problems is to anchor all vectors to
the same point, as shown in the figures.
For the polar rectangle, many students will realize that that there are
second-order differences between the two arcs, but few will realize that there
are also second-order differences in the radial sides, due to changes in
\(\boldsymbol{\hat{r}}\).
Plan to spend some extra time addressing the nature of \(d\vec{r}\). Basis vectors, arc length, dot product and magnitude; there's a great deal to take in and it's easy to lose sight of the forest for the trees. People will benefit from a deeper understanding at this stage.
Subsidiary ideas
This is a good place to emphasize the relationship between the dot product and
the Pythagorean Theorem.
Homework
Have students determine \(d\boldsymbol{\vec{r}}\) in 3 dimensions in rectangular, cylindrical and
spherical coordinates.
(Spherical coordinates are
tricky; most students miss the factor of \(\sin\theta\) in the \(\boldsymbol{\vec{\phi}}\)
component.)
Find \(d\boldsymbol{\vec{r}}\) along the diagonal of a square.
Enrichment
Emphasize that \(d\boldsymbol{\vec{r}}\) is the concept which unifies most of vector calculus.
It may be helpful to some students to be asked to orient the arrows (see
Props) themselves at various points in the plane.
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.
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 small group activity, students multiply a general 3x3 matrix with standard basis row/column vectors to pick out individual matrix elements. Students generate the expressions for the matrix elements in bra/ket notation.
