Dual nature of light and matter

    Isaac Newton was able to assert the particle theory of light by describing phenomena like refraction, reflection and dispersion. However, this particle theory of light could not explain everything that people saw light do. Christiaan Huygens thought that light was actually a wave and came up with the concept of wavefronts and his idea was able to model diffraction. Later on, Thomas Young's famous two-slit experiment (demonstrated interference and diffraction) and James Clerk Maxwell's idea showing it's the electric and magnetic fields that wave in light, supported wave theory of light.

    The wave theory seemed an appropriate description of light until scientists began looking at the photoelectric effect in the early 1900s. To observe this effect, two electrodes connected to a battery were placed in a glass ball (with vacuum). As light waves incident on these electrodes, the electrons were supposed to absorb energy. As the electrons absorbed enough energy, they were expected to eject from the atoms and travel as current between the electrodes. The results of this experiment showed four problems with the wave theory of light:

1. No electrons were ejected if the frequency of light was below a certain cutoff frequency, no matter how bright the light was. According to wave theory, electrons would just wait until they absorbed enough energy to leave the metal.
2. The maximum kinetic energy of electrons was independent of the incident light intensity with frequency above cutoff frequency. As per wave theory, greater intensity would mean more energy thereby increasing kinetic energy of electrons.
3. The maximum kinetic energy increased with the frequency of incident light. As per wave theory kinetic energy should depend on incident intensity and not incident frequency.
4. The electrons were ejected from the electrodes instantaneously even when the incident intensity was very low. Wave theory suggested electrons should have waited before being ejected until they absorb enough energy from incident light intensity.

    Wave theory could not explain the above results from the photoelectric effect. In 1905, Albert Einstein came up with the idea of light having a dual nature, both particle and wave. He proposed light being a wave but with finite extent unlike the wave theory. He called this localized wave as a quanta and also later suggested that it's energy depended on the frequency of light. In 1923, Louis de Broglie postulated that matter, similar to photons, has both wave and particle properties. Using Einstein's idea about momentum and energy for photons, he suggested similar properties for electrons, protons, neutrons, atoms, etc. With light, it's the electric and magnetic fields that are waving. With particles, what is waving is not clear. Wave function for a particle is a probability distribution function (likelihood of finding a particle in a particular location) in quantum mechanics.

    As time progressed, things got more interesting as single electrons were sent through Young's double-slit setup. More number of electrons were collected at some locations (in places like a bright fringe in case of light) and almost no electrons were collected at other locations (in places like a dark fringe in case of light). This result was equivalent to the interference pattern produced by light. Electrons also exhibited diffraction when passed through crystals and analyzing the angle and spacing of the atoms that did the scattering led to the first experimental confirmation of the wavelength of the electrons. Also, these results exactly matched de Broglie's hypothesis. Concentrated electron counts in some areas and almost zero elsewhere could only be explained by the wave theory.

    In a similar fashion, single photons were passed through Young's double-slit. It was observed that a single photon exists at both slits at the same time which showed there might be more to the wave theory. With that, perhaps photons can communicate with each other even though they are separated by vacuum. This property is referred to as entanglement. Entangled photons are two or more photons with a special link. This phenomenon is so mysterious that Albert Einstein called it a spooky action. Recently, many experiments have shown that entangled photons can exchange information instantly even if the distance between them is in miles. Although bizarre, photon entanglement opens possibilities for applications in the realm of super-secure communication and computing.

(These are notes taken from the book "Optics For Dummies" by Galen Duree, Jr., PhD.)

Depth of Field

    It is the range of object distances over which the image is "sufficiently well" focused, i.e. the range over which the blur(b) is less than the pixel size of a camera sensor.

(Taken from a YouTube video of Shree K. Nayar from Columbia University)

Unsupervised Learning in Bioinformatics

    There are a few methods in machine learning that can be applied in the field of bioinformatics like studying a genome or classification of different proteins/sub-genes in a gene. Such methods known to me are classified under unsupervised learning approaches and are listed as follows:

1) Hierarchical Clustering (Dendrograms).

2) Matrix Factorization.

Eigenfaces: An application of Principal Component Analysis (precisely Dimensionality Reduction)

(The above image represents 16 eigenfaces for about a 1000 different images of human faces. Image Source: same link as for "towardsdatascience" page mentioned below.)

    I just came across the concept of having human faces as eigen vectors and this is simply awesome! This concept can be used to store compressed version of individual human faces(this saves storage space) and those individual faces can still be reconstructed to have a good approximation of their original versions. All this can be done using the basic concept of machine learning called Principal Component Analysis! Please checkout this crisp explanation from "towardsdatascience" page to know more about this topic.

Link: https://towardsdatascience.com/eigenfaces-recovering-humans-from-ghosts-17606c328184

Interpretability v/s Flexibility for different statistical learning methods

 

(Source: An Intoduction to Statistical Learning with Applications in R written by James G., Witten D., Hastie T. and Tibshirani R. and published by Springer Text in Statistics. This is a part of my machine learning specialization course from CU Boulder.)

    I learned from this topic that we can choose a method from multiple statistical learning topics based on our goal. This book shares an example saying that if we are interested in predicting the accurate values in the stock market and not much concerned about inference (how each predictor variable is associated with the output), we can use high-flexibility approaches like deep learning. However, if we are concerned about the inference, we can use high-interpretability models like Lasso or Least-squares. The book also states that sometimes the high-interpretability models give more accurate results.

Birefringence

    Aside from dispersion, materials usually have the same index of refraction for all orientations of the electric field in the light. However, some materials, usually nearly pure crystals, have two indices of refraction for the same wavelength of light. The difference is due only to the orientation of the light's electric field. Therefore, light with its electric field oscillating in the vertical direction experiences a different index of refraction from the index of refraction for an electric field oscillating in the horizontal direction. This is called birefringence. (Source: Optics for Dummies by Duree)

Variables in a Linear Regression model (machine learning)

There are two types of variables in a dataset in general:
1) Real valued (ex.: 1,2,3,4)
2) Categorical (ex.: Blood Group)
    Categorical variables have two sub-types:
    a) Ordinal (ex.: Age group 25-55, or Grades A-D)
    b) Non-ordinal (ex.: Male/Female, Plant/Animal)

    It is comparatively easy to build linear regression model around real-values and ordinal-categorical variables that can relate to regression. However, nonordinal-categorical variables pose a problem for a "regression" model. This is  because the regression equations,

y = a0 + a1X1 + a2X2 +....

cannot rely on their arbitrary nature to predict output values. Nonetheless, these nonordinal categorical variables can always be transformed into real-valued or binary forms to make use of them in a regression problem.