Shantanu Kallakuri

Shantanu Kallakuri

Materials Science & Engg. | Applied Materials | Cornell University | BITS Pilani

Physika

This page is meant to be a comprehensive consolidation of various scientific topics I have learnt and been learning, presented in a simple, cohesive manner. It will cover areas of mechanics, quantum mechanics, optics, photonics, electromagnetics, field theory, relativity, gravity, as well as details from other topics depending on the relevance. We will try to blur disciplinary boundaries here, integrate and correlate learnings as much as possible while embracing an open approach starting from basic fundamentals.

It is physically impossible to cover every topic in the realm of science of course, but the goal is to serve as a personal repository as well as to introduce various topics sequentially, so that one can venture into more depth once the fundamentals are understood. I will include references where applicable and write code, simulations, and plots for easy elucidation and my own learning. Thank you for your patience, let’s begin! :)

1. Atomic beginnings

1.1 The Bohr model

\[ E_n = -\dfrac{13.6\ \mathrm{eV}}{n^2}\quad (\text{hydrogen-like stub}) \]
\(n_{\max}\), value = 5
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1.2 Hydrogen

The hydrogen atom: electron \(\color{blue}{-e}\), proton \(\color{blue}{+e}\), Coulomb potential.

\(\color{#000080}E_n = -\tfrac{13.6\text{ eV}}{n^2}\) ..... (eq 1.1)
\(\color{#000080}V(r) = -\tfrac{e^2}{4\pi\epsilon_0 r}\) ..... (eq 1.2)
\(\color{#000080}E_n = -\tfrac{13.6\text{ eV}}{n^2}\) ..... (eq 1.3)

Energy scales as \(1/n^2\) (eq. 1.1).

Code
  import math
  OSCILLATOR_X_RANGE = (0.0, 60.0)
  OSCILLATOR_Y_RANGE = (-1.25, 1.25)
  OSCILLATOR_X_AXIS_TITLE = "Time (dimensionless), t"
  OSCILLATOR_Y_AXIS_TITLE = "Displacement (dimensionless), x"
  def run_calc_slider(gamma, t_max=30.0):
    n = 300
    dt = t_max / max(n - 1, 1)
    t = [i * dt for i in range(n)]
    if gamma < 1.0:
        wd = math.sqrt(1.0 - gamma**2)
        y = [math.exp(-gamma * ti) * math.cos(wd * ti) for ti in t]
    elif abs(gamma - 1.0) < 1e-9:
        y = [math.exp(-ti) * (1.0 + ti) for ti in t]
    else:
        s = math.sqrt(gamma**2 - 1.0)
        y = [math.exp(-gamma * ti) * math.cosh(s * ti) for ti in t]
    ep = [math.exp(-gamma * ti) for ti in t]
    en = [-math.exp(-gamma * ti) for ti in t]
    return t, y, ep, en, OSCILLATOR_X_RANGE, OSCILLATOR_Y_RANGE, OSCILLATOR_X_AXIS_TITLE, OSCILLATOR_Y_AXIS_TITLE
\(\color{#000080}\psi_{n\ell m} = R_{n\ell}(r)\,Y_\ell^m(\theta,\phi)\) ..... (eq 1.4)
\(\color{#000080}\left[ -\tfrac{\hbar^2}{2m_e}\nabla^2 + V(r) \right] \psi = E\psi\) ..... (eq 1.5)

2. Bands and Orbitals

2.1 Damped oscillator

\[ x(t)=e^{-\gamma t}(A\cos\omega_d t+B\sin\omega_d t),\quad \omega_d=\sqrt{1-\gamma^2}\ (\gamma<1) \]
\(t_{\mathrm{end}}\), value = 30.0
γ, value = 0.10
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2.2 Hydrogen demos

\[ \text{Radial:}\quad R_{n\ell}(r),\quad r^2|R_{n\ell}(r)|^2 \]
\(n\), value = 1
ℓ, value = 0
\[ \text{Orbital slice }(xz)\text{:}\quad |\psi_{n\ell}(x,z)|^2 \]
\(n\), value = 1
ℓ, value = 0
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