An electron is described by the $1$D wave function:

$(x)=\{\begin{array}{ccc}0& \text{f}\text{o}\text{r}& x<mn>0</mn>\\ \sqrt[\phantom{2}]{12a}(e^{-ax}-e^{-2ax})& \text{f}\text{o}\text{r}& x>0\end{array}$

where $a$ is a real constant with dimensions $m^{-1}$

$($i$)$ What is the probability of finding the particle at some specific $x$ for Why?

$($ii$)$ What is the probability of finding the particle at some specific $x$ for $x>0?$ Why?

$($iii$)$ Around what value of $x$ is the probability of finding an electron the highest? Why? What

role does the probability density play versus probability here? Calculate the relevant position

and give your answer in terms of $\frac{1}{a}.$ Hint: bear in mind that $e^{ax}{e}^{bx}=e^{(a+b)x}$ and treat the

problem like you would any other quadratic equation.

$($iv$)$ Calculate the expectation value $($:$x$:$)$ of measurements of position performed on an ensemble of

identically prepared particles.

$($v$)$ Explain why the results of parts $($c$)$ and $($d$)$ are different.

$($vi$)$ Is this wave function for the electron an eigenfunction of momentum? Give your reason.