Question

Consider the electron wave function

$\psi \left(x\right)=\left\{\begin{array}{l}cx\left|x\right|\le 1nm\\ \frac{c}{x}\left|x\right|\le 1nm\end{array}\right.$

where x is in nm. a. Determine the normalization constant c.

b. Draw a graph of c1x2 over the interval -5 nm … x … 5 nm. Provide numerical scales on both axes.

c. Draw a graph of 0 c1x2 0 2 over the interval -5 nm … x … 5 nm. Provide numerical scales.

d. If 106 electrons are detected, how many will be in the interval -1.0 nm … x … 1.0 nm?

Short answer

The wave function of the electron$\psi \left(x\right)=\left\{\begin{array}{l}cx\left|x\right|\le 1nm\\ \frac{c}{x}\left|x\right|\le 1nm\end{array}\right.$

Step-by-step solution

The probability independent

$$\begin{gathered} {\int }_{-\infty }^{+\infty }\left|\psi \left(x\right)\right|dx=1 \\ {\int }_{-\infty }^{+\infty }\left|\psi \left(x\right)\right|dx+{\int }_{-1}^1\left|\psi \left(x\right)\right|dx+{\int }_1^2\left|\psi \left(x\right)\right|dx=1 \\ {\int }_{-\infty }^{+\infty }{\left(\frac{c}{x}\right)}^{2}dx+{\int }_{-1}^1\left(cx\right)dx+{\int }_1^2{\left(\frac{c}{x}\right)}^{2}dx=1 \\ {c}^2\left[2\frac{2}{3}\right]=1 \end{gathered}$$