This is a unit that introduces the Fourier transform and its properties and then applies the Fourier transform to free particle wave packets in non-relativistic quantum mechanics. The activities and homework are listed here. Appropriate text materials for mini-lectures can be found in the chapter Fourier Transforms and Wave Packets in the free online textbook
The Geometry of Mathematical Methods.
Warm-Up (Welcome Activity Reviewing Material from Undergraduate Physics)
This content is used in the Physics Department at OSU with incoming graduate students to remind them of undergraduate content before classes start and to help them to decide whether or not to take some Bridge Courses. This sequence is intended to run in two blocks of three hours each. The sessions should be run by someone with a deep knowledge of all of the relevant courses, the specific activities, and active engagement in general.
This session may be the first opportunity for the incoming graduate students to meet each other as well as some faculty and other graduate students. So start with a 1/2 hour dedicated to introductons.
Consider inviting some or all of the following people to participate:
At least one faculty member to run the session who has broad experience with the curriculum and the activities--typically the Paradigms Director.
Graduate students who have TAd for courses that incorporated these exact activities, as needed to provide one experienced person to sit with each group of three graduate students. The Head Graduate Advisor has often asked these graduate students for evaluative input regarding the members of their group. CAM thinks that they should be given a heads-up about what will be expected.
The Head Graduate Advisor (n.b. In the past the Grad Advisor has roamed the classroom, hovering over the groups as they work. CAM thinks this can appear intimidating/judgmental. Consider asking the grad advisor to SIT with groups, even if they move frequently from group to group.
Members of the Core Advising Committee
Faculty who will be teaching the Bridge Courses so that they are available to answer student questions, especially individual questions during breaks.
Graduate students who have take Bridge Courses in the past who are comfortable discussing their choices and experiences.
A valuable model for figuring out how we're going to save the Earth
Let's start by visualizing the energy flow associated with driving a gasoline-powered car. We will use a box and arrow diagram, where boxes represent where energy can accumulate, and arrows show energy flow.
The energy clearly starts in the form of gasoline in the tank. Where does it go?
Actually ask this of students.
Visualize the energy as an indestructable, incompressible liquid.
“Energy is conserved”
The heat can look like
Hot exhaust gas
The radiator (its job is to dissipate heat)
Friction heating in the drive train
The work contribute to
Rubber tires heated by deformation
Wind, which ultimately ends up as heating the atmosphere
The most important factors for a coarse-grain model of highway driving:
The 75:25 split between “heat” and “work”
The trail of wind behind a car
What might we have missed? Where else might energy have gone?
We ignored the kinetic energy of the car, and the energy dissipated as heat in the brakes. On the interstate this is appropriate, but for city driving the dominant “work” may be in accelerating the car to 30 mph, and with that energy then converted into heat by the brakes.
Let's apply the relationship of heat, entropy, and temperature to a contemporary challenge!
We'd like to maximize the efficiency of any process that is based on heat flow as an input.
Just a few examples of heat engines.
Energy flow diagram
Energy flow diagram
The efficiency of the machine is
\begin{align}
\text{efficiency} &= \frac{W}{Q_{\text{in}}}
\\
\textit{e.g.} &=\frac{500\text{ J}}{1000\text{ J}} = 50\%
\end{align}
For a car engine, \(T_H\approx 600\text{ K}\) and \(T_C\approx 300\text{ K}\).
Remember that \(\Delta S=\frac{Q}{T}\), and \(\Delta S_{\text{tot}} \ge 0\).
In this activity, students apply the Stefan-Boltzmann equation and the principle of energy balance in steady state to find the steady state temperature of a black object in near-Earth orbit.
Google “phet blackbody spectrum”' and open the simulation.
At what wavelength is the peak in spectral intensity
\(\lambda_{\text{peak}}\) for a black rock on the Earth's surface,
\(\lambda_{\text{peak}}\) for the black walls of a pizza oven,
\(\lambda_{\text{peak}}\) for a light bulb,
\(\lambda_{\text{peak}}\) for the sun.
Check that the peak wavelength decreases with temperature following a \(1/T\) relationship.
Use the numerical integration feature (the checkbox labelled “intensity” near the upper-right corner of the graph) to find the total intensity, in units of \(\text{W/m}^2\), emitted by
a black rock on the Earth's surface,
the black walls of a pizza oven,
the surface of a tungsten light bulb filament,
the surface of the sun.
Check that these intensities are proportional to \(T^4\). Note, the quick way to check involves ratios: Does \(\frac{I_1}{I_2} = \left(\frac{T_1}{T_2}\right)^4\)?
How cold should you make an object if you want zero thermal radiation emitted?
(Extra---if your group has time)
For an incandescent light bulb with a filament surface area of \(A\), estimate how efficiently it converts electrical energy into visible photons. Hint: you will need to estimate the following ratio:
\begin{align*}
\frac{\text{Electromagnetic radiation in visible wavelengths}}{
\text{Total electromagnetic radiation}
} = \frac{
A\int_{400\text{ nm}}^{700\text{ nm}} S_\lambda(\lambda, T)d\lambda
}{
A\int_{0}^{\infty} S_\lambda(\lambda, T)d\lambda
}
\end{align*}
Estimate the filament surface area \(A\) for a 60 W light bulb.
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.
Students consider the change in internal energy during three different processes involving a container of water vapor on a stove. Using the 1st Law of Thermodynamics, students reason about how the internal energy would change and then compare this prediction with data from NIST presented as a contour plot.
Students are asked to draw lines of constant \(u\) and \(v\) in a \(u,v\) coordinate system. Then, in the same coordinate system, students must draw lines of constant \(x\) and constant \(y\) when
Students consider how changing the volume of a system changes the internal energy of the system. Students use plastic graph models to explore these functions.
Transitioning from the PDM back to thermodynamic systems
Heating
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.
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.
Find the Fourier transform of the (simplified) Gaussian function
\begin{equation}
f(x)=e^{-x^2}
\end{equation}
You may want to use the value of the following integral
\begin{equation}
\int_{-\infty}^{\infty} e^{-x^2}\, dx = \sqrt{\pi}
\end{equation}
Students struggle with understanding that entropy can be created. It's an extensive quantity, and is the only one that isn't normally conserved, so that makes it pretty weird. We (professors) don't always realize how very weird this is, and students don't have the vocabulary to explain it to us, and are often afraid to try.
Students use an applet to explore the role of the parameters \(N\), \(x_o\), and \(\sigma\) in the shape of a Gaussian
\begin{equation}
f(x)=Ne^{-\frac{(x-x_0)^2}{2\sigma^2}}
\end{equation}
These lecture notes for the first week of https://paradigms.oregonstate.edu/courses/ph441 include a couple of small group activities in which students work with the Gibbs formulation of the entropy.
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.
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(c)
(d)
