Month: September 2019

Notes Update 9/20/19; Fun SR Facts

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I’ve posted an update to the notes with some reorganization of the section on quantum symmetry. Specifically, I included a subsection describing the momentum state eigenbasis and the Fourier relation between position and momentum bases.

I came across a fun article the other day that highlights something that is rather surprising about how length contraction would appear in the real world. Usually when I’ve described length contraction, we imagine it from the side in a rather two-dimensional sort of way. We draw a picture of a 2D car with the car’s image squeezed along the direction of its motion. This is fine for getting the idea across, but it doesn’t actually reflect what we’d really see. You see, objects are three dimensional and light from different parts of the object would reach us at different times. It turns out that as a result of this, length contraction would actually appear to be something like a rotated version of the object.

For more details and some very helpful graphics that explain what’s going on, check out this article from Physics World:

https://physicsworld.com/a/the-invisibility-of-length%E2%80%AFcontraction/

 

Homework Problems and Notes Update: 9/10/19

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I noticed some egregious typos in my notes, so I’ve updated the file.

This semester, the first homework assignment is going to be based on our discussion of rotations in 2D. There were a few claims I made without proof (exercise numbers are from my notes.):

1) Prove that the rotated vector components really are supposed to be what I claimed. (Ex. 2.1.)

2) Prove that the length of the vector is unchanged by rotations. (Ex. 2.2.)

3) 2D rotations commute. Note that this is not true for higher dimensions. (Ex. 2.3.)

We didn’t discuss this in class, but there are an infinite number of complex representations of the 2D rotation group. You can work these out by mapping the 2D vectors to complex numbers. For example, you can let the x-basis vector map to 1 and the y-basis vector map to i. After doing that, any other 2D vector maps to a complex number. A fun exercise is to see that our vector rotations correspond to multiplication by a phase exp(i theta). (Ex. 2.4.)