## Wavefunctions

• group Representations of the Infinite Square Well

group Small Group Activity

120 min.

##### Representations of the Infinite Square Well
Quantum Fundamentals 2023 (3 years)

Warm-Up

• group Spin-1 Time Evolution

group Small Group Activity

120 min.

##### Spin-1 Time Evolution
Quantum Fundamentals 2023

Students do calculations for time evolution for spin-1.
• 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 Fourier Transform of a Derivative

group Small Group Activity

10 min.

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

Fourier Transforms and Wave Packets

• 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

• assignment Fourier Transform of Cosine and Sine

assignment Homework

##### Fourier Transform of Cosine and Sine
Periodic Systems 2022
1. Find the Fourier transforms of $f(x)=\cos kx$ and $g(x)=\sin kx$.
2. 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.
3. 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.
• assignment Approximating a Delta Function with Isoceles Triangles

assignment Homework

##### Approximating a Delta Function with Isoceles Triangles
Static Fields 2023 (6 years)

Remember that the delta function is defined so that $\delta(x-a)= \begin{cases} 0, &x\ne a\\ \infty, & x=a \end{cases}$

Also: $\int_{-\infty}^{\infty} \delta(x-a)\, dx =1$.

1. Find a set of functions that approximate the delta function $\delta(x-a)$ with a sequence of isosceles triangles $\delta_{\epsilon}(x-a)$, centered at $a$, that get narrower and taller as the parameter $\epsilon$ approaches zero.
2. Using the test function $f(x)=3x^2$, find the value of $\int_{-\infty}^{\infty} f(x)\delta_{\epsilon}(x-a)\, dx$ Then, show that the integral approaches $f(a)$ in the limit that $\epsilon \rightarrow 0$.

• 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 Fourier Transform of a Plane Wave

group Small Group Activity

5 min.

##### Fourier Transform of a Plane Wave
Periodic Systems 2022

Fourier Transforms and Wave Packets

• keyboard Sinusoidal basis set

keyboard Computational Activity

120 min.

##### Sinusoidal basis set
Computational Physics Lab II 2023 (2 years)

Students compute inner products to expand a wave function in a sinusoidal basis set. This activity introduces the inner product for wave functions, and the idea of approximating a wave function using a finite set of basis functions.
• Quantum Fundamentals 2023 (3 years)

Consider the following wave functions (over all space - not the infinite square well!):

$\psi_a(x) = A e^{-x^2/3}$

$\psi_b(x) = B \frac{1}{x^2+2}$

$\psi_c(x) = C \;\mbox{sech}\left(\frac{x}{5}\right)$ (“sech” is the hyperbolic secant function.)

In each case:

1. normalize the wave function,
2. plot the wave function using Mathematica or other computer plotting tool (be sure to include the code you used and label your plots/axes appropriately),
3. find the probability that the particle is measured to be in the range $0<x<1$.