Quantum Vocabulary

Vocabulary of physics, the fountain of research ideas.

0. Fine Structure Constant

\(\alpha = \frac{k_\mathrm{e} e^2}{\hbar c} = \frac{1}{(4 \pi \varepsilon_0)} \frac{e^2}{\hbar c} = \frac{e^2 c \mu_0}{2 h}\)

In electrostatic cgs units, \(\alpha = \frac{e^2}{\hbar c}\).

In natural units, \(\alpha = \frac{e^2}{4 \pi}\) .

1. Hydrogen Atom

Potential \(V(r) = -\frac{Z e^2}{4\pi \epsilon_0 r}\).

Energy levels: \(E_{n} = -\left(\frac{Z^2 \mu e^4}{32 \pi^2\epsilon_0^2\hbar^2}\right)\frac{1}{n^2} = -\left(\frac{Z^2\hbar^2}{2\mu a_{\mu}^2}\right)\frac{1}{n^2} = \frac{\mu c^2Z^2\alpha^2}{2n^2}.\)

Ground state of hydrogen atom \(\psi_{100}(r)=\frac{1}{\sqrt{\pi}}\frac{1}{a^{3/2}} e^{-Z r/a}\).