The probability density for an electron that has passed through an experimental apparatus is shown in the figure. If 4100 electrons pass through...

Saturday, January 21, 2012

The probability density for an electron that has passed through an experimental apparatus is shown in the figure. If 4100 electrons pass through...

Hello!

First we determine the probability $\displaystyle{p}$ of the event that one electron will land in a specified strip. Then the expected number of such electrons will be $\displaystyle{p}\cdot{N},$ where $\displaystyle{N}$ is the total number of electrons.

By the definition of a probability density $\displaystyle{P}{\left({x}\right)},$ the probability of being between $\displaystyle{a}$ and $\displaystyle{b}$ is equal to $\displaystyle{\int_{{a}}^{{b}}}{P}{\left({x}\right)}{\left.{d}{x}\right.}.$ Here $\displaystyle{a}=-{0.1}$ mm and $\displaystyle{b}={0.1}$ mm. Because the given density is an even function, $\displaystyle{p}={2}{\int_{{0}}^{{0.1}}}{P}{\left({x}\right)}{\left.{d}{x}\right.}.$

It is simple to write a formula for $\displaystyle{P}{\left({x}\right)}$ for $\displaystyle{x}$ between 0 and 3, it is $\displaystyle{P}{\left({x}\right)}=\frac{{1}}{{3}}-\frac{{x}}{{9}}$ (a straight line). Therefore

$\displaystyle{p}={2}{\int_{{0}}^{{0.1}}}{\left(\frac{{1}}{{3}}-\frac{{x}}{{9}}\right)}{\left.{d}{x}\right.}={2}{\left(\frac{{x}}{{3}}-\frac{{{x}}^{{2}}}{{18}}\right)}{{\mid}_{{{x}={0}}}^{{0.1}}}={2}\cdot{\left(\frac{{0.1}}{{3}}-\frac{{0.01}}{{18}}\right)}\approx{0.0656}.$

And the expected number is about $\displaystyle{4100}\cdot{0.0328}\approx{269}.$ This number is dimensionless.

Note that this number is the most probable, but the neighboring numbers are also very probable.

Ninenotes

Saturday, January 21, 2012

The probability density for an electron that has passed through an experimental apparatus is shown in the figure. If 4100 electrons pass through...

No comments:

Post a Comment

find square roots of -1+2i