## Student handout: Gaussian Parameters

Periodic Systems 2022
Students use an applet to explore the role of the parameters $N$, $x_o$, and $\sigma$ in the shape of a Gaussian $$f(x)=Ne^{-\frac{(x-x_0)^2}{2\sigma^2}}$$
What students learn
• Increasing $N$ increases the height of the Gaussian,
• increasing $x_0$ shifts the Gaussian to the right,
• increasing $\sigma$ makes the Gaussian wider.

A Gaussian is a function of the form $$f(x)=Ne^{-\frac{(x-x_0)^2}{2\sigma^2}}$$ Use the applet at Gaussians to explore the role of the parameters $N$, $x_o$, and $\sigma$ in the shape of a Gaussian. Make sure that not only do you know the role of each parameter, but also that you can EXPLAIN this behaviour based on the algebraic expression for the Gaussian function.

• assignment_ind Normalization of the Gaussian for Wavefunctions

assignment_ind Small White Board Question

5 min.

##### Normalization of the Gaussian for Wavefunctions
Periodic Systems 2022

Fourier Transforms and Wave Packets

Students find a wavefunction that corresponds to a Gaussian probability density.
• group Fourier Transform of a Gaussian

group Small Group Activity

10 min.

##### Fourier Transform of a Gaussian
Periodic Systems 2022

Fourier Transforms and Wave Packets

• group Changes in Internal Energy (Remote)

group Small Group Activity

30 min.

##### Changes in Internal Energy (Remote)

Warm-Up

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.
• group Electric Field of Two Charged Plates

group Small Group Activity

30 min.

##### Electric Field of Two Charged Plates
• 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
1. objects with like charge repel and opposite charge attract,
2. object tend to move toward lower energy configurations
3. The potential energy of a charged particle is related to its charge: $U=qV$
4. The force on a charged particle is related to its charge: $\vec{F}=q\vec{E}$
• group Thermal radiation at twice the temperature

group Small Group Activity

10 min.

##### Thermal radiation at twice the temperature
Contemporary Challenges 2021 (4 years)

This small group activity has students reasoning about how the Planck distribution shifts when the temperature is doubled. This leads to a qualitative argument for the Stefan-Boltzmann law.
• assignment One-dimensional gas

assignment Homework

##### One-dimensional gas
Ideal gas Entropy Tempurature Thermal and Statistical Physics 2020 Consider an ideal gas of $N$ particles, each of mass $M$, confined to a one-dimensional line of length $L$. The particles have spin zero (so you can ignore spin) and do not interact with one another. Find the entropy at temperature $T$. You may assume that the temperature is high enough that $k_B T$ is much greater than the ground state energy of one particle.
• group Covariation in Thermal Systems

group Small Group Activity

30 min.

##### Covariation in Thermal Systems

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.
• assignment Coffees and Bagels and Net Worth

assignment Homework

##### Coffees and Bagels and Net Worth
Energy and Entropy 2021 (2 years)

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.

1. 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 $P_B$?

2. Write down the total differential of your savings, in terms of $B$, $C$, $P_B$ and $P_C$.

3. 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.

• face Statistical Analysis of Stern-Gerlach Experiments

face Lecture

30 min.

##### Statistical Analysis of Stern-Gerlach Experiments
• group Fourier Transform of a Shifted Function

group Small Group Activity

5 min.

##### Fourier Transform of a Shifted Function
Periodic Systems 2022

Fourier Transforms and Wave Packets

Learning Outcomes