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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 3: Vector Calculus

Due Fri 12 Sep 2025 at the start of class

Hint: Use the Levi-Civita symbol and the fact that

\[\begin{gather*} [\vect{A}\times\vect{B}]_c = \varepsilon_{abc}A_aB_b \end{gather*}\]

(with summation implied over indices \(a\) and \(b\)), and the property that cyclic permutations are even, so that

\[\begin{gather*} \varepsilon_{abc} = \varepsilon_{bca} = \varepsilon_{cab} =\\ = - \varepsilon_{acb} = -\varepsilon_{bac} = -\varepsilon_{cba}. \end{gather*}\]

1. Vector Manipulations

Express \(\vect{\nabla}\cdot(\vect{a}\times\vect{b})\) in terms of the curls \(\vect{\nabla}\times\vect{a}\) and \(\vect{\nabla}\times\vect{b}\). (Check your result explicitly with some simple vector fields \(\vect{a}\) and \(\vect{b}\).)

Based on Arfken, 3.7.2. The Helmholtz decomposition breaks a vector field into a sum of a pure divergence (irrotational) and pure curl (incompressible). It is not unique and you should be able to guess a simple decomposition that works here.

2. Non-conservative Work

Find the work \(\int \vect{F}\cdot\d{\vect{r}}\) done by moving on a unit circle in the \(x\)-\(y\) plane, doing work against a force field given by

\[\begin{gather*} \vect{F} = \frac{-y\uvect{i}}{x^2+y^2} + \frac{x\uvect{j}}{x^2+y^2} \end{gather*}\]

1. The work \(W_1\) done moving counterclockwise from \(0\) to \(\pi\).

2. The work \(W_2\) done moving clockwise from \(0\) to \(-\pi\).

3. Explain why the work depends on the path by breaking \(\vect{F} = -\vect{\nabla} \phi + \vect{\nabla}\times \vect{A}\) using the Helmholtz decomposition. (Use an educated guess, don’t actually try to evaluate the integrals you might find in various references about the Helmholtz decomposition, but find an explicit example of potentials \(\phi\) and \(\vect{A}\) that give \(\vect{F}\).)

3. Maxwell’s Equations

Derive the integral form of Maxwell’s equations from the following differential form by applying the appropriate integral theorem and explain physically the meaning of each:

\[\begin{align*} \vect{\nabla}\cdot\vect{B} &= 0,\\ \vect{\nabla}\cdot\vect{E} &= \frac{\rho}{\epsilon_0},\\ \vect{\nabla}\times\vect{B} &= \epsilon_0\mu_0 \pdiff{\vect{E}}{t} + \mu_0 \vect{J},\\ \vect{\nabla}\times\vect{E} &= -\pdiff{\vect{B}}{t}. \end{align*}\]

Show that Maxwell’s equations have solutions that propagate at the speed of light \(c = 1/\sqrt{\epsilon_0\mu_0}\).

4. Net Current

See Arfkin 3.8.5. Hint: consider each component of \(\vect{J}\) separately and use Maxwell’s equations to show that \(J_i = \vect{\nabla}\cdot(x_i\vect{J})\). Then apply Gauss’s theorem.

A particular steady-state electric current distribution is localized in space. Choosing a bounding surface far enough out so that the current density \(\vect{J} = \vect{0}\) everywhere on the surface, show that the volume integral of the current is zero:

\[\begin{gather*} \int \vect{J}\d{\tau} = \vect{0}. \end{gather*}\]

5. Angular Momentum

See Arfkin 3.10.32. Please work through as many of the problems 3.10.28 through 3.10.34 as you need to be comfortable with the angular momentum operator: you will be expected to know this for quantum mechanics.

1. Show that (setting \(\hbar=1\))

   \[\begin{gather*} \vect{L} = -\I (\vect{r}\times\vect{\nabla}) = \I\left( \uvect{e}_\theta \frac{1}{\sin\theta}\pdiff{}{\phi} - \uvect{e}_{\phi}\pdiff{}{\theta} \right) \end{gather*}\]

   Note that \(\vect{L}\) only looks at angular variations.

2. From \(L^2 = \norm{\vect{L}}_2^2 = L_x^2 + L_y^2 + L_z^2\) show that

   \[\begin{align*} L^2 &= -\frac{1}{\sin\theta}\pdiff{}{\theta}\left( \sin\theta \pdiff{}{\theta}\right) - \frac{1}{\sin^2\theta}\pdiff[2]{}{\phi}\\ &=-r^2\nabla^2 + \pdiff{}{r}\left(r^2\pdiff{}{r}\right). \end{align*}\]

   Note: Although it is straightforward to do this using Cartesian coordinates, it is somewhat messy. Feel free to explore other techniques if you like.

Note

This last relation is critical for solving problems like the hydrogen atom in quantum mechanics. It allows us to replace the Laplacian in the Schrödinger equation with

\[\begin{gather*} -\nabla^2 = \frac{L^2}{r^2} - \frac{1}{r^2}\pdiff{}{r}\left(r^2 \pdiff{}{r}\right), \end{gather*}\]

allowing us to write solutions in terms of spherical harmonics, which are eigenfunctions of \(L^2\):

\[\begin{gather*} \psi(\vect{r}) = f(r)Y^{m}_{\ell}(\theta,\phi), \qquad L^2Y^{m}_{\ell}(\theta,\phi) = \ell(\ell+1)Y^{m}_{\ell}(\theta,\phi),\\ -\nabla^2\psi(\vect{r}) = Y^{m}_{\ell}(\theta,\phi)\left( \frac{\ell(\ell+1)}{r^2} - \frac{1}{r^2}\pdiff{}{r}\left(r^2 \pdiff{}{r}\right)\right)f(r). \end{gather*}\]

We can thus focus on the radial form of the solution, which picks up the additional “centrifugal” potential \(\hbar^2 \ell(\ell+1)/r^2\).

6. Rotation Matrices (from last week)

Hints: Consider the Pauli matrices. Also, note that rotation matrices can be expressed as \(\mat{R}_{\vect{\theta}} = e^{\mat{\vect{\theta}\times}}\).

Here \([\mat{A}, \mat{B}] = \mat{A}\mat{B} - \mat{B}\mat{A}\) is the commutator.

The effect of rotations on physical systems can be expressed in terms of Lie algebras and groups as we will discuss later in the course. For now, find examples of three 2D and 3D matrices \(\mat{T}_{x,y,z}\) that satisfy the following commutation relationships:

\[\begin{gather*} [\mat{T}_x, \mat{T}_y] = -\mat{T}_{z} \quad \text{and cyclic permutations:} \quad [\mat{T}_y, \mat{T}_z] = -\mat{T}_{x}, \quad [\mat{T}_z, \mat{T}_x] = -\mat{T}_{y}. \end{gather*}\]

Prove that the trace of any such matrices must be zero: \(\Tr\mat{T}_{i} = 0\). Show that if the matrices \(\mat{T}_{i}^\dagger = \pm\mat{T}_{i}\) are either hermitian or anti-hermitian, then they must be anti-hermitian: \(\mat{T}_{i}^\dagger = -\mat{T}_{i}\).

Can the matrices \(\mat{T}_{i}\) be real in 3D? What about 2D? Justify your answer.

Physicists typically include a factor of \(\I = \sqrt{-1}\) in the definition of the commutation relations, so you will often see

\[\begin{gather*} [\mat{M}_{i}, \mat{M}_{j}] = \I \epsilon_{ijk}\mat{M}_k,\qquad \mat{R}_{\theta} = e^{\I \vect{\theta}\cdot\vect{\mat{M}}}. \end{gather*}\]

Show how \(\mat{M}_{i}\) can be obtained from \(\mat{T}_{i}\).

Note

Armed with such matrices, one can form matrices that effect rotations on on spinors (2D) and vectors (3D) by exponentiating:

\[\begin{gather*} \mat{R}_{\vect{\theta}} = \exp\left(\sum_{i\in\{x,y,z\}} \theta_i \mat{T}_{i}\right) \equiv e^{\vect{\theta}\cdot\vect{\mat{T}}}. \end{gather*}\]

The linear combinations of \(\mat{T}_{i}\) form a representation of the Lie algebra \(\mathfrak{so}(3)\) and exponentiating these gives the rotation matrices, which form a representation of the corresponding Lie group \(SO(3)\). In the case of the 2D representation, this is related to the Lie group \(SU(2)\) (but the topology of these two groups differ as we will discuss later).