91Ó°ÊÓ

As the gas decays and gives off electrons, the detector uses a magnet to trap them in a magnetic bottle. A radio antenna then picks up very weak signals emitted by the electrons, which can be used to map the electrons' precise activity over several milliseconds.

Probability of defection in width centred on position is given by

$p\left(x\right)={\mathrm{ψ}}_n^2\left(x\right)dx$

Probability density$\mathrm{ψ}{}_n{}^2\left(x\right)$ for the trapped electron is

${\mathrm{ψ}}_n^2\left(x\right)=A^{2}si{n}^2\left(\frac{n\mathrm{π}}{L}x\right),   [forn=1,2,3,...]$

Centred of the well, $x=\frac{L}{2}$

$\begin{array}{l}=\frac{200pm}{2}\\ =100pm\end{array}$

Probability of detection at x ,

$$\begin{gathered} p\left(x\right)={\mathrm{ψ}}_n^2\left(x\right)dx \\ ={\left[\sqrt{\frac{2}{L}\sin \left(\frac{n\mathrm{π}}{L}x\right)}\right]}^{2}dx \\ =\frac{2}{L}{\sin }^2\left(\frac{n\mathrm{π}}{L}x\right)dx \end{gathered}$$

Here,

n=3

Length, $L=200pm$

Width of the probe, dx=2.00 pm

Probability

$$\begin{gathered} \mathrm{p}\left(\mathrm{x}=\frac{\mathrm{L}}{2}\right)=\frac{2}{\mathrm{L}}{\sin }^2\left(\frac{\mathrm{²ÔÏ€}}{\mathrm{L}}\mathrm{x}\right)\mathrm{dx} \\ =\frac{2}{\mathrm{L}}{\sin }^2\left[\left(\frac{3\mathrm{Ï€}}{\mathrm{L}}\right)\left(\frac{\mathrm{L}}{2}\right)\right]\mathrm{dx} \\ =\frac{2}{\mathrm{L}}{\sin }^2\left(\frac{3\mathrm{Ï€}}{2}\right)\mathrm{dx} \\ =\frac{2}{\mathrm{L}}\mathrm{dx} \\ =\frac{2}{200\mathrm{pm}}\left(2.00\mathrm{pm}\right) \\ =0.020 \end{gathered}$$

Hence, the probability of detection is 0.020.

Number of independent insertion,

$N=1000$

Number of times the electron to be detected,$n=Np$

$$\begin{gathered} =\left(1000\right)\left(0.020\right) \\ =20 \end{gathered}$$

Hence,20 times should we expect the electron to materialize on the end of

the probe.