Additional Problems For Set 1

\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

$r_{bohr}=k\frac{\hbar^{2}}{m_{e}e^{2}}$

$k=4\pi\varepsilon_{0}$