## Activity: Normalization of the Gaussian for Wavefunctions

Periodic Systems 2022
Students find a wavefunction that corresponds to a Gaussian probability density.
• This activity is used in the following sequences

The Gaussian $${\cal P}(x)=\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(x-x_0)^2}{2\sigma^2}}$$ is normalized so that the area under the curve is equal to one. If this Gaussian represents the probability density for a free quantum mechanical particle, what is a possible wavefunction?

• group Gaussian Parameters

group Small Group Activity

10 min.

##### Gaussian Parameters
Periodic Systems 2022

Fourier Transforms and Wave Packets

Students use an applet to explore the role of the parameters $N$, $x_o$, and $\sigma$ in the shape of a Gaussian $$f(x)=Ne^{-\frac{(x-x_0)^2}{2\sigma^2}}$$
• group Going from Spin States to Wavefunctions

group Small Group Activity

60 min.

##### Going from Spin States to Wavefunctions
Quantum Fundamentals 2023 (2 years)

Arms Sequence for Complex Numbers and Quantum States

Completeness Relations

Students review using the Arms representation to represent states for discrete quantum systems and connecting the Arms representation to histogram and matrix representation. The student then extend the Arms representation to begin exploring the continuous position basis.
• 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

• keyboard Mean position

keyboard Computational Activity

120 min.

##### Mean position
Computational Physics Lab II 2023 (2 years)

Students compute probabilities and averages given a probability density in one dimension. This activity serves as a soft introduction to the particle in a box, introducing all the concepts that are needed.
• face Compare \& Contrast Kets \& Wavefunctions

face Lecture

30 min.

##### Compare & Contrast Kets & Wavefunctions

Completeness Relations

In this lecture, the instructor guides a discussion about translating between bra-ket notation and wavefunction notation for quantum systems.
• assignment Differential Form of Gauss's Law

assignment Homework

##### Differential Form of Gauss's Law
Static Fields 2023 (6 years)

For an infinitesimally thin cylindrical shell of radius $b$ with uniform surface charge density $\sigma$, the electric field is zero for $s<b$ and $\vec{E}= \frac{\sigma b}{\epsilon_0 s}\, \hat s$ for $s > b$. Use the differential form of Gauss' Law to find the charge density everywhere in space.

• group Superposition States for a Particle on a Ring

group Small Group Activity

30 min.

##### Superposition States for a Particle on a Ring

Quantum Ring Sequence

Students calculate probabilities for a particle on a ring whose wavefunction is not easily separated into eigenstates by inspection. To find the energy, angular momentum, and position probabilities, students perform integrations with the wavefunction or decompose the wavefunction into a superposition of eigenfunctions.
• 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.
• keyboard Electric field for a waffle cone of charge

keyboard Computational Activity

120 min.

##### Electric field for a waffle cone of charge
Computational Physics Lab II 2022

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.
• group Grey space capsule

group Small Group Activity

30 min.

##### Grey space capsule
Contemporary Challenges 2021 (4 years)

In this small group activity, students work out the steady state temperature of an object absorbing and emitting blackbody radiation.

Learning Outcomes