Solutions to Exercises in A Short Introduction to Mathematical Concepts in Physics Jim Napolitano
tuf43817@temple.edu
November 1, 2023
2
Preface These are my own solutions to the end-of-chapter exercises in the textbook. I’ve tried hard to avoid my own mistakes, but you never know. Please get in touch with me if you see errors or otherwise have questions. Many of my solutions use Mathematica, for making plots, checking answers, or actually carrying out all or large parts of the exercise. Except for trivial uses to check results, all of the Mathematica notebooks I used are provided separately by the publisher. Jim Napolitano ⟨tuf43817@temple.edu⟩ November 1, 2023
Contents 1 Solutions
5
2 Solutions
13
3 Solutions
19
4 Solutions
37
5 Solutions
47
6 Solutions
53
7 Solutions
67
8 Solutions
73
9 Solutions
79
Contents
4
Chapter 1 Solutions (1) The answers are all straightforward: z12 = (1 + i)2 = 1 + 2i − 1 = 2i z1∗ = 1 − i z1 z2∗ = (1 + i)(3 + 4i) = 3 + 3i + 4i − 4 = −1 + 7i √ √ √ |z1 | = 1 + 1 = 2 and |z2 | = 9 + 16 = 5 |z1 z2 | = |(1 + i)(3 − 4i)| = |3 + 3i − 4i + 4| = |7 − i| = = |z1 ||z2 |
√
√ 49 + 1 = 5 2
(2) [P V ] = [Force]L−2 L3 = M LT −2 ]L−2 L3 = M L2 T −2 = [Energy]. From the ideal gas law, since N is dimensionless, kT must have the dimensions of energy. The SI unit of energy is Joule, so the units of k are Joule/K. (3) First find [k] = [Force]/[Distance] = M LT −2 L−1 = M T −2 . (a) Write ϵ = mx k y Az , so M L2 T −2 = M x M y T −2y Lz = M x+y T −2y Lz and y = 1, z = 2, and x = 0. That is, the energy scale is ϵ = kA2 . You’ll recall that the potential energy of a harmonic oscillator with amplitude A is kA2 /2 so the scale is the correct energy to within a factor of two. (b) Write ϵ = mx k y ℏz , so M L2 T −2 = M x M y T −2y L2z M z T −z = M x+y+z T −2y−z L2z and z = 1, 2y + z = 2y + 1 = 2 so y = 1/2, and x + y + z = x + 3/2 = 1 so x = −1/2. Therefore ϵ = ℏ(k/m)1/2 . In quantum mechanics you will learn that the energy spacing in the harmonic oscillator is indeed ℏω where ω = (k/m)1/2 . (4) The calculation is straightforward: τ = mx ℓy g z