Answer
See the detailed answer below.
Work Step by Step
We have here 3 stages,
- 5000 electrons are symmetrically distributed on both sides around $x=0$ on the range of $-1\leq x\leq 1$
- 7500 electrons are symmetrically distributed on both sides around $x=0$ on the range of $-2\leq x\leq 2$
- 10000 electrons are symmetrically distributed on both sides around $x=0$ on the range of $-3\leq x\leq 3$
We know that the area under the graph of $|\psi(x)|^2$ must be 1, and we know that the region from -1 cm to 1 cm contains 50% of the electrons. This means that the area under the curve for this stage must be half.
$$A_1=bh$$
where $b$ is the width in the $x$-direction and $h$ is the height in the $y$-direction.
$$h_1=\dfrac{A_1}{b}=\dfrac{0.5}{1-(-1)}=\bf 0.25\;\rm cm^{-1}$$
So the first stage area is from -1 to 1 in the $x$-direction and from 0 to 0.5 in the $y$-direction.
The second stage contains 75% of the electrons, where 5000 electrons are in the center and the rest (2500 electrons) are uniformly distributed in two regions from 1 cm to 2 cm, and from -1 cm to -2 cm.
$$h_2=\dfrac{A_2-A_1}{b}=\dfrac{0.75-0.5}{2}=\bf 0.125\;\rm cm^{-1}$$
And by the same approach,
$$h_3=\dfrac{A-A_2}{b}=\dfrac{1-0.75}{2}=\bf 0.125\;\rm cm^{-1}$$
See the graph below.
