Notes & musings
- Bohr Radius from Variational Principle: This is an elegant way to compute the Bohr radius without looking at the Schrodinger equation. It also confirms the virial theorem for a central inverse-square force. This approach was presented to me first in Dr. Yuri Gartstein's PHYS 4301 Quantum Mechanics 1 course.
- Classical Particle in a Cone: the motion of a particle subject to gravity g inside of a cone of opening angle α, solved using the Lagrangian formulation. The equations of motion and numerical solution are shown below.
- Coriolis and Centrifugal Forces in the Plane: the motion in the reference frame of a mass given some initial velocity on a frictionless, flat, surface rotating at ω = 1 rad/s, solved numerically. The Coriolis and centrifugal forces are "fictitious" since they arise from applying Newton's laws to a non-inertial frame; in the inertial frame, the mass will travel in a straight line. The equations of motion and numerical solution for three different initial velocities are shown below.
Since ω points in the z-direction, the above can be reduced to
- Cycloid Motion: the motion of a charged particle subject to perpendicular uniform electric and magnetic fields E = Ez and B = Bx, solved numerically. Amazingly, the particle's average trajectory lies perpendicular to both the electric and magnetic fields.
- Electromagnetic Field and Dual Tensors: I have seen that several electrodynamics texts leave the derivation of the field and dual tensors as an exercise to the reader. I have Lorentz transformed an antisymmetric tensor twice below and have matched the set of transformation rules to previous results from special relativity to arrive at the field and dual tensors.
From our previous knowledge of the Lorentz transformations we know that fields transform as
We see there are six transformations and feel inspired to find an object with six distinct elements—a rank-2 antisymmetric tensor—that matches these results. Such a tensor is of the form
Let's Lorentz transform each of the six distinct elements of the above twice
where the Lorentz transformation matrix Λ is
where γ is the Lorentz factor and β ≡ v/c:
So the set of transformations is
There are two ways to correspond equations (1)-(6) with (7)-(12) respectively, giving rise to the field and dual tensors.
- Electromagnetism in Tensor Notation: using the above-found field and dual tensors, Maxwell's four equations in differential form
become two tensor equations:
where
is the current density 4-vector.
I will "run the indexes" below, showing that equations (5) and (6) capture the same information as equations (1)-(4). This is often left as an "exercise to the reader" in electrodynamics texts. Running indexes on equation (5):
That is equation (1).
Putting the above three components together gives
which is equation (4).
Now running indexes on equation (6):
which is equation (3).
Putting the above three components together gives
which is equation (2).
In summary, equations (1) & (4) are expressed in (5); equations (2) & (3) are expressed in (6).
- Electromagnetism in the Lorenz Gauge: we reduced Maxwell's four equations to two tensor equations in the above section. We can do better, reducing the two tensor equations to one equation. Recall from electrostatics that
Also recall Faraday's law:
and that the magnetic field is the curl of the magnetic vector potential:
Substituting (3) into (2),
Putting together (1) and (4),
So we see that the scalar electric potential and the magnetic vector potential entirely describe time-dependent electric and magnetic fields. We can construct a 4-vector:
I will verify that the field tensor (derived above) can be written as
For reference, also recall that the position four-vector is
Running the indexes,
So the electric and magnetic fields can indeed be described by (7). Now, substituting (7) into the inhomogeneous Maxwell Equation
gives
But in the Lorenz gauge,
So
So (8) becomes
Griffiths calls the above the "most elegant formulation of Maxwell's equations." I think it is beautiful.
- Guided Electromagnetic Waves: a loose end from undergrad electrodynamics that I tied up in grad school.
- Numerical Solutions to the Stellar Structure Equations: these solutions are not normalized but can be solved for the boundary conditions of any main-sequence star.
- The Planck quantities: the Planck quantities provide intuition on the various regimes of physics (classical, quantum, relativistic mechanics). This was the starting point of Dr. Xiaoyan Shi's PHYS 4352 Concepts of Modern Physics course.
- Symmetric Top in Gravity: the motion of a symmetric top (I1 = I2 ≠ I3) of mass M (center of mass at height h) with tip fixed in gravity g solved using Euler angles and the Lagrangian formulation, assuming there is no acceleration in the ψ direction. The Lagrangian, equations of motion, and numerical solution are shown below.
- Thoughts on the Maxwell equations: A semi-analytical and semi-reflective document of thoughts on the Maxwell equations.
- Vector calculus theorems: quick, intuitive derivations of the divergence theorem and Stokes' theorem
