# Fermat’s Principle, a lecture in Optics. 3

Geometric Optics: When the size of objects that a wave of light interacts with are large compared to the wavelength of light λ, λ can be neglected for practical purposes and the light waves behave like rays of light. Rays of light are geometric line segments from one point of incidence of light to another. Study of optics under the limit of negligible wavelength — λ → 0, is called Geometric Optics.

Geometric Optics can be studied using Fermat’s Principle, much like motion of objects in the realm of classical mechanics are studied using Newton’s laws of motion. To know the basic grounding of Fermat’s Principle follow the links to read two articles which expound the subject matter of Fermat’s Principle, art1 — detailed, historical and long, art2 — conceptual but short. More…

# Primary Aberrations, a lecture in optics. 1

In our Lecture-I we discussed the phenomena of aberrations that arise because of a discrepancy of a first order theory and the 3rd order theory as depicted by the Maclaurin series; where we saw that first order theory represents the so called paraxial optical systems. Please have a look of the linked article to get a basic view of the ground on which we are discussing this topic. At-least going half-way through the lecture and stopping short of the derivation will do well.

We discussed that there are two kind of aberrations. Monochromatic and Chromatic. As the name suggests the monochromatic aberrations are a result of the discrepancy when we considered our incoming ray to be having a single wavelength of light. The chromatic rays on the other hand can have multiple colors or wavelength of light. The monochromatic aberrations are also called as Seidel or Primary aberrations and we will shed more light on them today. The chromatic aberrations were dealt in greater detail — eg the derivations pertained to the chromatic aberrations. We did so because the chromatic aberrations are simple to understand. More…

# Inherent ability = difficulty * accomplishment. 2

All of Physics is this “Inherent ability = difficulty * accomplishment”. Thats just intuitive but can easily be seen to correspond mathematically with the Principle of least action.

First the edifice: whats the problem? The problem is given you move in straight line when every direction is same around you, which direction will you chose? While you are waiting for a good answer from astrologers intelligent people already give a good hint. Think you have some inherent ability which is fixed.

fixed: which changes only if estimated wrong.

That inherent ability is actually action. Accomplishments are adjusted for difficulties, you waded through a swamp 5 meters you would have accomplished in sand 8 meters with that given inherent ability called action. Because action is abstract we have been sticking to time and path-length, but they are not as fundamental, they are merely specifics. More…

# Aberrations; a lecture in Optics. 4

Optical systems are studied under two assumptions

object point does not lie far away from the axis of the optical system.

rays taking part in image formation make a small angle with the axis of the optical system.

The domain of optics where above two assumptions are valid is called as Paraxial optics. Paraxial systems are highly idealized and in reality do not perfectly represent the situation. The consequential errors in image reconstruction are known as aberrations.
The paraxial assumption can be represented by truncating at the first term of the polynomial expansion of the sin function by the Maclaurin series. More…

# A c++ code for calculating pi value. Reply

Finally I am successful in calculating pi value — less than 0.3% error, by using random number generation. Although my computer needs some fixation on its compiler or path definition etc, there are very good online compilers which helps in testing and running c++ codes: try the given link.

OUTPUT
Computing the value of pi using std::rand()
Enter number of trials: 10000
Enter number of random (x,y) points per trial: 10
pi = 3.14376 +- 0.00519107
average – exact = 0.00216735
CPU time = 0.004027 secs

Here is the code I found by searching a good deal on the web. Yes I did tinker around but only because my own compiler (Turbo C++ on windows 10, 64 bits) was throwing some exceptions on the included headers.

#include
#include
#include
#include
//#include
using namespace std;

double pi_estimate(const unsigned long points) More…

# Schrodinger Equation; Concepts and Problems in Quantum Mechanics. Reply

A long and technical discourse on Quantum Wave Function.

A 64 slide presentation styled discourse on the Quantum Wave Function. It consists of detailed solution of 5 important and interesting problems, apart from a threadbare discussion of the concepts.