Physics

(Work under progress)

1. Atomic beginnings

1.1 Hydrogen

The hydrogen atom is the simplest bound quantum system — it contains a single electron of charge  \(\color{blue}{-e}\)  bound to a proton of charge  \(\color{blue}{+e}\)  by the Coulomb potential:

As shown in eq. 1.1, the energy scales as \(\tfrac{1}{n^2}\).

import math

def run_calc_slider(gamma):
    n = 300
    t = [i * 0.1 for i in range(n)]
    return t

The time-independent Schrödinger equation in spherical coordinates is:
\(\displaystyle \left[ -\tfrac{\hbar^2}{2m_e}\nabla^2 + V(r) \right] \psi(r,\theta,\phi) = E\psi(r,\theta,\phi)\)

Separating variables, the wavefunction factorizes as:
\(\displaystyle \psi_{n\ell m}(r,\theta,\phi) = R_{n\ell}(r)\, Y_\ell^m(\theta,\phi)\)

where \(\displaystyle R_{n\ell}\) are the radial wavefunctions and \(\displaystyle Y_\ell^m\) are the spherical harmonics.

Damped Oscillator (Test plot)

Hydrogen Atom Wave Functions