Assignment 1: Mathematical Preliminaries

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import mmf_setup;mmf_setup.nbinit()
import logging;logging.getLogger('matplotlib').setLevel(logging.CRITICAL)
%matplotlib inline
import numpy as np, matplotlib.pyplot as plt

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Assignment 1: Mathematical Preliminaries#

Due Fri 29 Aug 2025 at the start of class

Important

These problems are all very well known. A quick search on the internet, using LLMs, or Wolfram Alpha will give you solutions. Unfortunately, by using these resources, you will learn very little.

You must be able to solve these problems without appealing to any outside resources.

For example: if you want to use the Euler–Maclaurin formula, then learn it well enough that you can derive it from scratch, and compute the Bernoulli numbers. Otherwise, you will not really be learning anything other than how to apply a formula someone else has developed.

This will be a critical skill as you move forward in your research and start tackling problems that are not common, and which have not been used to train LLMs, or coded into a system like Wolfram Alpha.

1. Series#

Determine if the following series converge. If they converge, find an expression for the sum, or compute the sum to 12 digits of accuracy:

\[\begin{align*} \text{(a)} &\quad \sum_{n=2}^{\infty}\frac{1}{\ln n} & \text{(d)} &\quad \sum_{n=1}^{\infty}\frac{1}{\sqrt{n(n+1)}} \\ \text{(b)} &\quad \sum_{n=1}^{\infty}\frac{n!}{10^n} & \text{(e)} &\quad \sum_{n=0}^{\infty}\frac{1}{2n^2+1},\\ \text{(c)} &\quad \sum_{n=1}^{\infty}\frac{1}{n2^n} & \text{(f)} &\quad \sum_{n=1}^{\infty}\frac{(-1)^{n}}{n}. \end{align*}\]

2. Euler-Mascheroni Constant#

Show that the following limit exists, and compute the Euler-Mascheroni constant \(\gamma\) to 6 digits (or 12 if you want more of a challenge):

\[\begin{gather*} \gamma = \lim_{N \rightarrow \infty}\left( \sum_{n=1}^{N}\frac{1}{n} - \ln N \right) \end{gather*}\]

3. Sine and Cosine#

By analogy with Euler’s identity \(e^{\I x} = \cos{x} + \I \sin{x}\), show using series that

\[\begin{gather*} \exp(\mat{I}x) = \mat{1}\cos{x} + \mat{I}\sin{x}, \qquad \mat{I}^2 = -\mat{1}, \end{gather*}\]

for any matrix that squares to the negative identity. In particular

\[\begin{gather*} \exp\begin{pmatrix} 0 & \theta\\ -\theta & 0 \end{pmatrix} = \begin{pmatrix} \cos \theta & \sin\theta\\ -\sin \theta & \cos \theta \end{pmatrix}. \end{gather*}\]
# Check that I got the right signs etc.
import numpy as np
from scipy.linalg import expm
theta = 0.1
M = np.array([[0, theta], [-theta, 0]])
assert np.allclose(
    np.array([[np.cos(theta), np.sin(theta)],
              [-np.sin(theta), np.cos(theta)]]),
    expm(M))

4. Bose-Einstein Distribution (10 points)#

Consider a thermal ensemble of 1D harmonic oscillator with quantized energies

\[\begin{gather*} E_n = \underbrace{\hbar \omega}_{\epsilon_0} (n + \tfrac{1}{2}). \end{gather*}\]

At temperature \(T\), each state is occupied with Maxwell-Boltzmann weight

\[\begin{gather*} f_T(E) \propto e^{-\beta E}, \qquad \beta = \frac{1}{k_B T}. \end{gather*}\]

Thus, the average energy in the system is

\[\begin{gather*} \braket{E} = \frac{\sum_{n} E_nf_T(E_n)}{\sum_{n} f_T(E_n)} \end{gather*}\]

Show that, by appropriately shifting the energy, the zero-point energy \(\hbar\omega/2\) drops out of the calculation, giving the expression in Arfken’s Exercise 1.3.14:

\[\begin{gather*} \braket{E} = \frac{\sum_{n=1}^{\infty} n \epsilon_0 e^{-\beta \epsilon_0 n}} {\sum_{n=0}^{\infty} e^{-\beta \epsilon_0 n}}. \end{gather*}\]

By identifying the numerator and denominator as binomal expansions, show that

\[\begin{gather*} \braket{E} = \frac{\epsilon_0}{e^{\beta \epsilon_0} - 1}. \end{gather*}\]

Show that in the limit of high temperature \(k_B T \gg \epsilon_0 = \hbar\omega\) we obtain the classical result \(\braket{E} \rightarrow k_B T = 1/\beta\).

5. Vector Cross Products (5 points)#

Prove that

\[\begin{gather*} (\vect{A}\times\vect{B})\cdot(\vect{A}\times\vect{B}) = (AB)^2 - (\vect{A}\cdot\vect{B})^2. \end{gather*}\]
# Check that I got the right signs...
import numpy as np
rng = np.random.default_rng(seed=2)
A = rng.random(size=3)
B = rng.random(size=3)
assert np.allclose(
    np.dot(np.cross(A, B), np.cross(A, B)),
    (np.linalg.norm(A)*np.linalg.norm(B))**2 - np.dot(A, B)**2)
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