Lienard-Wiechert potentials via retarded time consideration

We can use the Lienard-Wiechert potentials formula to rediscover special relativity, by carefully taking into consideration the time of travel for the radiation from the moving charge to reach the chosen point.

Take a charge moving along the x-axis at the speed v. Consider a given point in spacetime with coordinates (x,y,z,t) : the electric potential felt at that position is due to the radiated influence originating from the charge at an earlier point (x′,0,0,t′).

The Lienard-Wiechert potential is then expressed as follows :

Hence we see the appearance of the Lorentz-transformed displacement in the third factor above. As for the Lorentz factor in the second part, it comes from the 4-vector nature of charge density-current density.

The vector potential x-component is expressed in a similar form as the scalar electric potential. So here is another 4-vector : the electromagnetic potential.

 

 

Rotation matrices for spin-half particles

These matrices will be useful for computing the probability amplitudes of an initial state vector, after some rotation, to project onto a given final state vector in the new reference frame.

The derivation idea is from Feynman, like most of the other formula in this blog. The setup uses the Stein-Gerlach apparatus with the magnetic field increasing in the positive z direction.

First, rotation about the z-axis : there is no effect on the magnitude of the amplitude, only its phase; but a 180° rotation should not bring the system back to the same physical state, hence the factor ½ in the exponent :

Next, to rotate by 180º about the y-axis :

Now it will be easy to get the matrix representing rotation about the x-axis :

Going back to rotation about the y-axis, we again use the idea of combining a few matrices :

So by purely geometrical thinking, we are able to derive all the rotation matrices about the x.y and z axes for spin ½ particles.

Path integral method for Harmonic Oscillators

We can learn so much Physics and Math by studying the derivation of the eigenfunctions of the quantum harmonic oscillators it’s well worth the effort. Anyone interested in going deeper in Physics must know the bolts and nuts of the harmonic oscillators equations thoroughly enough.

There are at least 3 ways to solve the Simple Harmonic Oscillator differential equation. Here we will go through the path integral method to understand how the eigenvalues for the energy and their related eigenfunctions are obtained.

To begin, the Schrödinger equation for the harmonic oscillator :

We introduce three basic quantities in path integral : the Lagrangian, the Action and the Kernel :

To derive the classical Action and the Kernel, we will need some integration tricks and manipulate x and its time derivative :

The derivation of F(T) takes us through the method of Fourier series. It’s done this way :

Once we have the Kernel (also called the “propagator”), we can work out the energy eigenfunctions. There are two ways to get them : the direct way, and the indirect way. We will go through them both.

First, the direct method :

Next, the indirect method :

We can then put the above result into the Gaussian Ψ(x) and obtain the eigenfunctions Φn, and also rediscover the formula for Hermite polynomials, that are so close-at-heart to the harmonic oscillators.

The whole exercise is mentally exhausting, but the satisfaction of getting the result brings great happiness at the end.

Classical and Quantum Statistics

There are 3 (or maybe 3 and a half?) distributions that we shall look at : Maxwell-Boltzmann distribution, Fermi-Dirac distribution and Bose-Einstein distribution.

Boltzmann distribution concerns distinguishable particles. Its half-cousin is the Maxwell-Boltzmann distribution, which applies to  dilute gases. They resemble one another so much that many people don’t even care to make a distinction between them.

Maxwell-Boltzmann statistics simply takes away the N! factor from w, which makes no difference to the partition function and the distribution function. Hence the two distributions are often taken to be the same.

Fermi-Dirac particles are loners – they cannot share the same quantum states with each other, i.e., they obey Pauli’s exclusion principle.

Bose-Einstein particles are social animals, they prefer staying together than being left alone.

 

Coupled pendulum

A coupled pendulum is a simple 2- state system. One way to make such a system is to attach a light spring between two pendulums. Starting with one of the pendulums swinging while the other one at rest, soon the oscillations of the one that moves first reduce in amplitude while the other one’s amplitude builds up. Then the cycle repeats itself and we get a continuous back-and-forth pouring of energy between the two pendulums.

The mathematics behind this interesting behaviour begins with a set of differential equations :

To solve these differential equations, we assume that their solutions are in the form as given below :

Plugging them back to the original differential equations and we now can solve for the unknown A and B :

The first solution simply have the two pendulums swinging together in unison : each keeps its own energy and nothing transfers between them, in other words nothing really fun to look at. The second solution is the one corresponding to the start-stop behaviour that sees alternately one pendulum with dying and the other pendulum with growing oscillations, definitely more gratifying when watched.

Hermite polynomials

After struggling for many years to understand harmonic oscillator, I’m finally making some headway.

To remember this happy moment, let’s also remember 4 nice formula related to Hermite polynomials.

“Any serious student of quantum mechanics must have a thorough understanding of the harmonic oscillator.”    –  J.J. Sakurai.

Euler angles

A rotation in 3-dimension space about any axis passing through the origin can be made equivalent to the combined effect of 3 successive rotations. Euler proved this theorem and the angles associated with the 3 rotations were named Euler angles, in particular, α, β and γ.

First a rotation about the z axis by an angle α results in new orientations for the original x and y axes, let’s call them x′ and y′. Then a rotation about the x′ axis by an angle β  turns y′ and z′ to two new directions : y″ and z″. Finally a rotation about the z″ axis by an angle γ brings the body to a final orientation (referenced by XYZ axes in red below) coinciding with that obtained by rotating about the initial prescribed axis.

This sequence of rotation can be represented as an algebraic operation of matrix multiplication :

Learning about Euler angles can help us form up the transformation matrix in passing from one representation to another when spin-half particles are filtered by 2 Stern-Gerlach apparatus oriented differently with respect to one another.

Complex Refractive Index

In working out the formula for the refractive index of a piece of material, we considered the oscillations of an electron, being impinged upon by the electric field carried by the incoming electromagnetic wave. In the end a complex number inadvertently creeped out. This is due to the inclusion of the drag force acting on the oscillator, which translated into absorption of the electromagnetic energy by the material as the electrons shake about.

The real part of the complex number gives the ordinary re-radiated wave (with a phase shift) ; the imaginary part will cause a drop in the amplitude of the wave as it penetrates deeper into the material.

We note the attenuation of the amplitude of the E field upon passing into the material. The coefficient of absorption β is deduced.

Another interesting observation we can make is that the index of refraction can be less than 1 ! But that doesn’t mean the wave speed is really higher than c. We need to make a distinction between phase speed and group speed, the speed of the carrier and the speed of the signal. The signal speed will not surpass c.

A few other useful quantities can be derived upon further considerations.

In a conductor the conduction electrons are unbounded, so we can ignore the natural frequency in the above formula. Also, by bringing in other electrical characteristics of the material, such as the conductivity and the characteristic time, the refractive index of metals is presented in a new form :

At low frequencies, we simplify the above expression by taking away 1 and the second order term in omega, then the skin-depth is deduced.

At high frequencies, we can simplify the denominator by ignoring the first order term in omega. Then we obtain the plasma frequency, which is related to the density oscillations of free electrons.

Negative temperature

Can temperature go lower than absolute zero kelvin ? It’s possible with statistical thermodynamics.

Consider a two-level system, for example magnetic dipoles in a magnetic field. Some of the dipoles may be aligned with the magnetic field while the rest are aligned opposite to the field. Those aligned with the field are at a lower energy level and those aligned opposite are at a higher energy level.

We can relate the number of dipoles of each type with energy and temperature this way :

So if there are more dipoles in the upper level than the lower level, we will have a negative thermodynamic temperature !

Electron radius, fine structure constant and HEC

Here’s a way to remember the formula for the Bohr radius, the classical electron radius and the fine structure constant : HEC (in French this acronym means Hautes Etudes Commerciales).

Bohr radius is the quantum mechanical radius for electron and it can be calculated by the ratio between Planck’s reduced constant squared and the product of the mass of an electron and its reduced charge squared.

The classical electron radius is the ratio of the square of the reduced electronic charge to the product of the mass of an electron and the square of the speed of light.

The ratio of Bohr radius to classical electron radius is about 137², and it’s the square of the reciprocal of the fine structure constant.