Additional Problems For Set 1
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\begin{document} \emph{Problem }
Estimate the Bohr radius of the H-atom using the uncertainty principle between momentum and position. Use the radial variables.
\emph{Solution}
The Energy for the hydrogen atom is described by:
$E=\frac{P^{2}}{2m}-\frac{e^{2}}{r}$
$\Delta r$ Is defined as the average radius of localization for the electron.
The Uncertainty Principle can be generalized as:
$\Delta r\cdot\Delta p\backsimeq\hbar$
The momentum corresponding to$\Delta r$can be generalized as:
$p\backsimeq\Delta p\backsimeq\frac{\hbar}{\Delta r}$
Leaving a kinetic energy of:
$KE=\frac{P^{2}}{2m}\backsimeq\frac{\hbar^{2}}{2m(\Delta r)^{2}}$
The potential is defined as:
$V\backsimeq-\frac{e^{2}}{\Delta r}$
Giving the total energy:
$E\sim\frac{\hbar^{2}}{2m(\Delta r)^{2}}-\frac{e^{2}}{\Delta r}$
The minimum for $\triangle r$ can be found by differentiating the total energy with respect to $\triangle r$.
$\frac{\partial E}{\triangle r}=0\backsim\frac{-\hbar^{2}}{m(\bigtriangleup r)^{3}}+\frac{e^{2}}{(\triangle r)^{2}}$
Solve for $\triangle r$:
$\frac{\hbar^{2}}{m(\bigtriangleup r)^{3}}\thicksim\frac{e^{2}}{(\Delta r)^{2}}$
$\triangle r\thicksim\frac{\hbar^{2}}{me^{2}}$
This corresponds to the bohr radius given by
$r_{bohr}=k\frac{\hbar^{2}}{m_{e}e^{2}}$
$k=4\pi\varepsilon_{0}$ \end{document}