# Mathematical and Computational Methods in Physics and Engineering (fall 2016)

## Daily outline for Mon, Nov 21, 2016 9:10 AM (1 year ago)

### Fourier Transform applications

assigned today:
• standard-fourier transform applications:

I can apply Fourier transform theory

1. Calculate the shape of a diffraction pattern for an interesting 2D mask.
2. An electron is in the lowest level of a SHO with a frequency of 10^15 Hz. Then the potential is shut off. Describe the particle's wave function after one nanosecond.
3. What is the maximum frequency response (and width of the resonance peak) for a damped system excited by a delta function with m=1, b=0.2, and k=10?

Due today:

# planning thoughts

• Quantum
• momentum and position representations
• Damped Driven
• both theory and fft?
• Diffraction

Do diffraction before x/p (and only do 1d diffraction?)

# in class

• Quiz on Nyquist (no change to the problem)
• had them plot two different repetitive functions with different periods
• asked them to consider which frequencies would be needed
• big idea: period goes up, frequency spacing goes down
• extended this to period going to infinity

$$f(t)=\frac{1}{\sqrt{2 \pi}}\int_{-\infty}^{\infty}F(\omega)e^{i\omega t}\,d\omega$$

$$F(\omega)=\frac{1}{\sqrt{2 \pi}}\int_{-\infty}^{\infty}f(t)e^{-i\omega t}\,dt$$

• Talked about how the first one is an interesting statement about how adding waves can produce anything and how it's a useless statement without the second one that tells you how to do it
• Had them calc the time derivatives of the first one to set up the damped driven stuff
• compared and contrasted to the Laplace approach
• talked quickly about the quantum stuff

edit