Additional Problems For Set 1: Difference between revisions

Revision as of 16:47, 22 January 2011

emph{Problem }

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-\emph{Problem }
+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
-$r_{bohr}=k\frac{\hbar^{2}}{m_{e}e^{2}}$
-$k=4\pi\varepsilon_{0}$