The Bohr Model of the Hydrogen Atom — Question 1

Question 1

Coulomb force and angular-momentum quantization jointly determine Bohr energies

An electron of mass m moves in a circular orbit around a fixed nucleus of charge $+Ze$. Use $mv^2/r=k_eZe^2/r^2$, $mvr=n\hbar$ and potential energy $U=-k_eZe^2/r$, with zero energy at infinite separation.

1. Derive the radius and speed as functions of n and Z.

2. Derive the total energy and its relation to kinetic and potential energies.

Question 1 - Solution

Coulomb force and angular-momentum quantization jointly determine Bohr energies

(a) Method and calculation

$$v=\frac{n\hbar}{mr},\quad\frac{n^2\hbar^2}{mr^3}=\frac{k_eZe^2}{r^2}\Rightarrow\boxed{r_n=\frac{n^2\hbar^2}{mk_eZe^2}}.$$$$\boxed{v_n=k_eZe^2/(n\hbar)}.$$

(b) Interpretation and further calculation

$$K=\frac{k_eZe^2}{2r},\quad U=-\frac{k_eZe^2}{r}=-2K\Rightarrow E=-K=U/2.$$$$\boxed{E_n=-\frac{m(k_eZe^2)^2}{2n^2\hbar^2}}.$$

Physical check

The negative total energy indicates binding relative to a separated electron and nucleus. The Bohr orbit is a historical model, not a spatial trajectory assigned by the full wave-function description.