99: The de Broglie-Bohr Model for a Hydrogen Atom Held Together by a Gravitational Interaction

$\lambda = \frac{h}{mv}$

de Broglie's hypothesis that matter has wave-like properties.

$n \lambda = 2 \pi r$

The consequence of de Broglie's hypothesis; an integral number of wavelengths must fit within the circumference of the orbit. This introduces the quantum number, n, which can have values 1,2,3,...

$mv = \frac{nh}{2 \pi r}$

Substitution of the first equation into the second equation reveals that linear momentum is quantized.

$T = \frac{1}{2} m v^2 = \frac{n^2 h^2}{8 \pi^2 m_e r^2}$

If momentum is quantized, so is kinetic energy.

$E = T + V = \frac{n^2 h^2}{8 \pi^2 m_e r^2} - \frac{G m_p m_e}{r}$

Which means that total energy is quantized, where $$- \frac{G m_p m_e}{r}$$ is the gravitational potential energy interaction between a proton and an electron.

$\frac{d}{dr} \left( \frac{n^2h^2}{8 \pi^2 m_e r^2} - \frac{G m_p m_e}{r} \right) = 0 ~ \text{solve, r} \rightarrow \frac{h^2 n^2}{4 \pi^2 G m_e^2 m_p}$

Minimization of the energy with respect to orbit radius yields the optimum values of r. This expression is subtituted back in the energy expression below to find the allowed energies.

$E = \frac{n^2 h^2}{8 \pi^2 m_e r^2} - \frac{G m_p m_e}{r} \text{substitute, r} = \frac{h^2 n^2}{4 \pi^2 G m_e^2 m_p} \rightarrow E = \frac{2 \pi^2 G^2 m_e^3 m_p^2}{h^2 n^2}$

$\begin{matrix} \text{Fundamental constants:} & m_p = 1.67262 (10)^{-27} \text{kg} & m_e = 9.10939 (10)^{-31} \text{kg} \\ ~ & h = 6.62608 (10)^{-34} \text{joule sec} & G = 6.67259 (10)^{-11} \frac{m^3}{ \text{kg s}^2} \end{matrix}$

Energy:

$E(n) = - \frac{2 \pi^2 G^2 m_e^3 m_p^2}{h^2 n^2}$

$r(n) = \frac{h^2 n^2}{4 \pi^2 G m_e^2 m_p^2}$
$\begin{matrix} n = 1 .. 4 & \frac{E(n)}{J} = \begin{pmatrix} -4.233 \times 10^{-97} \\ -1.058 \times 10^{-97} \\ -4.704 \times 10^{-98} \\ -2.646 \times 10^{-98} \end{pmatrix} & \frac{r(n)}{m} = \begin{pmatrix} 1.201 \times 10^{29} \\ 4.803 \times 10^{29} \\ 1.081 \times 10^{30} \\ 1.921 \times 10^{30} \end{pmatrix} \end{matrix}$