Angular Momentum of the Hydrogen Atom via Scalar Model
A natural continuation of analyzing the model is to classically calculate the angular momentum of an electron in orbit of a proton. In order to calculate angular momentum, velocity is required. The velocity is:
\(
\begin{equation}\label{q1h} \tag(1)
v=\sqrt{\frac{e^2}{4 \pi \epsilon_0 m_e r}}
\end{equation}
\)
Using the value of \(r=\frac{r_e}{\alpha}\) v is:
\(
\begin{equation} \tag{2}
v=\sqrt{\frac{\alpha e^2}{4 \pi \epsilon_0 m_e r}}
\end{equation}
\)
Combining the radius, mass, and velocity the classically derived angular momentum at r is:
\(
\begin{equation}\label{eAngularH} \tag{3}
\mathcal{L}_T=r m_e v
\end{equation}
\)
Sub \(\alpha \hbar c = \frac{e^2}{4 \pi \epsilon_0} \) into equation \ref{eAngularH} one obtains:
\(
\begin{equation}\label{eAngularH2} \tag{4}
\mathcal{L}_T=\sqrt{r_e m_e \hbar c}
\end{equation}
\)
Recall from the model, the advance of the electron is an invariant distance \(2\pi r_e\). If one steps the radial value by integer multiples of \(r_e\) the following equation is generated:
\(
\begin{equation}\label{eAngularH3} \tag{5}
\mathcal{L}_T=\sqrt{n r_e m_e \hbar c}
\end{equation}
\)
Where n is an integer value. Next, convert the value of \(E_A(r)\) to reflect the \(r_e\) steps from equation \ref{eAngularH3}, again using \(\alpha \hbar c = \frac{e^2}{4 \pi \epsilon_0}\):
\(
\begin{equation}\label{eAngularH4}\tag{6}
\mathcal{L}_A=\frac{\alpha \hbar c r_e}{ \left(\frac{r_e}{\alpha}\right)^2} = \frac{\alpha^3 \hbar c r_e}{(n r_e)^2}
\end{equation}
\)
Lastly, equate the angular momentum \(\mathcal{L}{‘}\) again using \(\alpha \hbar c = \frac{e^2}{4 \pi \epsilon_0}\):
\(
\begin{equation}\label{qAngularPrime}\tag{7}
\mathcal{L}{‘}=
\left( \frac{\left( \frac{n\; r_e}{\alpha}\right)^2}{r_e} \right) m_e
\sqrt{\left(
\frac{\alpha \hbar c}{m_e \left( \frac{\left( \frac{n\;r_e}{\alpha}\right)^2}{r_e} \right) }
\right)}=n \hbar
\end{equation}
\)
Equation \ref{qAngularPrime} provides a straight forward understanding of the model’s prediction. The angular momentum of the system must step in discrete increments of energy. When \(\lambda = \lambda_{ebar} \) this discrete energy step is \(\hbar\). This correlation is a one to one relation between the gravitational induced advance of the electron and the discrete angular momentum levels allowed. This is a direct link from gravitational effects to a causal effect of quantum discrete energy levels in the angular momentum of an electron proton system.
The figure below shows the breakout of equation the main equation. The equation is derived from the predicted advance of the electron orbit. The advance causes an additional energy \(E_A(r)\) equated as traditional Coulomb energy \(E’_C(r’)\). The binding energies allowable and the angular momentum states allowable are discrete energy levels.
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