Copasetic Flow

I'm still studying for quals. I got stuck for awhile on problem 2.11 of Griffith's Introduction to Quantum Mechanics first edition. The problem involves manipulating raising and lowering operators to perform a normalization. The problem I was having was doing too much of the math too early. I tried to expand all the operators and got stuck. I tried to apply integration by parts as the book suggests and applied it to the wrong term the first time, and carried the operation too far on a second try. The series of conservative steps that finally did the trick is shown in the video below.

On page 123 of “Gropus and Their Graphs” by Grossman and Magnus I struggled understanding Theorem 6 and the test it provides. The theorem read more clearly for me with the following substitution: “the set K that contains all elements of G such that...” becomes “the set K containing every element of the group G such that...” The following paragraph that contained the test was easier for me to understand with the following substitution: “If f maps all elements onto I...” becomes “If f maps all elements of G onto I...” Hope this helps! Any comments or questions?

This installment of “It’s Obvious. Not!” looks at: Book: “Statistical Mechanics” Edition: second Authors: R.K. Pathria Publisher: Elsevier Butterworth Heinemann Page: 12 The idea in Pathria is to start with a function for the number of microstates as a function of energy and then maximize it to study the implications of a system in equilibrium, (the maximum number of microstates). Pathria skips a few steps in the differentiation and simplification. There shown below to help me and others along. Have fun! Starting with Maximize with respect to . Keep in mind that is a function of : Use the chain rule of differentiation to expand the second partial derivative: The last derivative term simplifies to -1: So, to maximize we have: Now, consider the differentiation of a function . The chain rule gives: Applying this to our result above, we get:

This installment of “It’s Obvious. Not!” looks at: Periodical: “Physical Review” Volume: 21 Page: 483, 486 Title: "A Quantum Theory of the Scattering of X-Rays by Light Elements" Author: Arthur H. Compton Excerpt from page 486: In the above excerpt, Compton discusses how to calculate the momentum of an electron that caused x-ray or gamma scattering. The momentum added to the electron is the momentum of the incident photon minus the momentum of the scattered photon. Problem: The angle of scattering, (theta), appears to be mislabeled in the above figure vs. the usage of the angle in formula 1. In formula 1, Compton calculates the magnitude of the electron momentum as the vector difference of the momentum of the incident and scattered photons. To subtract two vectors, you place their tails together, the resulting vector that points from the head of the second to the head of the first is the difference vector as shown below and described in this Wikipedia article . To get t...

This installment of “It’s Obvious. Not!” looks at: Book: “Classical Dynamics of Particles and Systems” Edition: third Authors: Jerry B. Marion and Stephen T. Thornton Publisher: Harcourt Brace Jovanovich Page: 172-177 Chapter Five in the third edition is titled "The Calculus of Variations" The book does a great job of giving a very detailed bottom-up derivation of Euler's equation and a second form of Euler's equation. I had a much easier time with the material once I figured out that Euler's equation was actually the goal of the derivation and how Euler's equation is used. Since the top-down view made things simple for me, I decided to post it here for other top-down thinkers. Chapter five is simply building a set of tools to be used in chapter six with regard to Hamilton's Principle and Lagrangian mechanics. So, maybe the first question should be, 'Why are Hamilton's Principle and Langrangian mechanics important?' Newtonian Mechanics esse...

This installment of “It’s Obvious. Not!” looks at: Book: “Classical Dynamics of Particles and Systems” Edition: third Authors: Jerry B. Marion and Stephen T. Thornton Publisher: Harcourt Brace Jovanovich Page: 68 This post looks at Example 2.10 that investigates the motion of a charged particle in a magnetic field. The example is fairly straightforward with one exception. When determining the equations of motion, the authors propose a solution for the system of differential equations discussed below and reference example C.2 of Appendix C. It’s not immediately apparent how to use Example C.2 to arrive at the authors’ solution, so the steps are outlined in detail here. If you have questions, or suggestions, all comments are always welcome! The original system of coupled differential equations is: First, the authors’ differentiate both equations and then substitute the results into the other: at this point, the book suggests using the technique of ex...

This installment of “It’s Obvious. Not!” looks at: Book: “Classical Dynamics of Particles and Systems” Edition: third Authors: Jerry B. Marion and Stephen T. Thornton Publisher: Harcourt Brace Jovanovich Page: 52 This installation of the series provides a few clarifications into the example presented in the textbook and asks even more questions. I have a feeling that readers steeped in differential equations will immediately follow the reasoning of the example as it is written in the textbook. Please, if you have answers to the remaining questions below, or even ideas, please comment. Thanks! The first example of the chapter titled “Newtonian Mechanics” asks the reader to find the velocity of an object sliding down a ramp. The solution for the acceleration, (second derivative of the position x), has already been derived as: The process for deriving the velocity as a function of position illustrated by the author starts with the above equation for accelera...