91Ó°ÊÓ

An electron wave function is given for the interval of $\left|x\right|≤1nm$,

$\mathrm{ψ}\left(x\right)=cx$and for the interval of $\left|x\right|â‰\yen 1nm$is, $\mathrm{ψ}\left(x\right)=\frac{c}{x}$.

Any wave function should satisfy the equation,known as the normalization condition which is the probability of finding a particle at position $x$.

Substituting the values of $\mathrm{ψ}\left(x\right)$in the given interval, we get,

$$\begin{gathered} {∫}_{-∞}^{-1}\frac{c^2}{x^2}dx+{∫}_{-1}^0{c}^2{x}^{2}dx+{∫}_0^1{c}^2{x}^{2}dx+{∫}_1^{∞}\frac{c^2}{x^2}dx=1 \\ 2{∫}_1^{∞}\frac{c^2}{x^2}dx+2{∫}_0^1{c}^2{x}^{2}dx=1 \\ 2c^2{\left[-\frac{1}{x}\right]}_1^{∞}+2c^2{\left[\frac{x^3}{3}\right]}_0^1=1 \\ \frac{2}{3}{c}^2+2c^2=1 \\ \frac{8}{3}{c}^2=1 \\ c=\sqrt{\frac{3}{8}} \end{gathered}$$

Therefore, the value of normalization constant is$c=\sqrt{\frac{3}{8}}$

Plotting the values of wave function $\mathrm{ψ}\left(x\right)$along y-axis and the position $x$along x-axis we get the graph as:

Within the given limit, we can write the probability density, substituting the value of normalization constant as,

${\left|\mathrm{ψ}\left(x\right)\right|}^2=\frac{3}{8}{x}^2$when $\left|x\right|≤1nm$and

${\left|\mathrm{ψ}\left(x\right)\right|}^2=\frac{3}{8x^2}$when $\left|x\right|â‰\yen 1$

Therefore, the ${\left|\mathrm{ψ}\left(x\right)\right|}^2$versus $x$graph should be as following:

The probability of electron density within the given limit can be written as,

$$\begin{gathered} P\left(-1nm≤x≤1nm\right)={∫}_{-1}^1{\left|\mathrm{ψ}\left(x\right)\right|}^{2}dx \\ P\left(-1nm≤x≤1nm\right)={∫}_{-1}^1{c}^2{x}^{2}dx \\ P\left(-1nm≤x≤1nm\right)=c^2{\left[\frac{x^3}{3}\right]}_{-1}^1 \\ P\left(-1nm≤x≤1nm\right)=\frac{2}{3}{c}^2 \\ P\left(-1nm≤x≤1nm\right)=\frac{2}{3}×\frac{3}{8} \\ P\left(-1nm≤x≤1nm\right)=\frac{1}{4} \end{gathered}$$

The number of electrons detected is$\left({10}^6×\frac{1}{4}\right)=2.5×{10}^5.$