An experiment finds electrons to be uniformly distributed over the interval 0 cm x 2 cm, with no electrons falling out-side this interval.

a. Draw a graph of 0 c1x2 0 2 for these electrons.

b. What is the probability that an electron will land within the interval 0.79 to 0.81 cm?

c. If 106 electrons are detected, how many will be detected in the interval 0.79 to 0.81 cm?

d. What is the probability density at x = 0.80 cm?

Here,

${\left|\psi \left(x\right)\right|}^2$is the probability density

According to the problem, the electrons are uniformly distributed over an interval of $0cm\le x\le 2cm$. so the probability ${\left|\psi \left(x\right)\right|}^2$is constant over the interval $0cm\le x\le 2cm$and thus ${\left|\psi \left(x\right)\right|}^2$is a square wave function

The condition for normalization is

${\left|\psi \left(x\right)\right|}^2=1$

Determine the probability of an electron to be found over the interval $0cm\le x\le 2cm$

${\left|\psi \left(x\right)\right|}^2\delta x=1$

Substitute 2cm for $\delta x$

${\left|\psi \left(x\right)\right|}^2\delta x=1$

${\left|\psi \left(x\right)\right|}^2=0.5c{m}^{-1}$

Thus the probability of an electron to be found over the interval 0.79 cm to 0.81cm is 1%

$N\left(in\delta xatx\right)=Nxprob\left(in\delta xatx\right)$

Calculate the number of electrons detected in the interval 0.79cm to 0.81 cm

$N\left(in\delta xatx=0.80cm\right)=NxProb\left(in\delta xatx=0.80cm\right)$

Thus the number of electrons detected in the interval 0.79cm to 0.81cm is 10⁴

prob$$\begin{gathered} \left(x=0.80cm\right)={\left|\psi =0.80cm\right|}^2 \\ Prob\left(x=0.80cm\right)=0.5c{m}^{-1} \end{gathered}$$

Thus the probability density at x-0.80 cm is 0.5 cm^-1