Probability graphs and reading them

CBM

May 2019
4
0
Given a probability over x graph, my questions were what is b in nm^-1 and what is the probability that you could find the particle between 20nm and 30nm.

In attempts to this problem I tried to square the ψ(x) to get a probability density. I then assumed that under x the graph cancelled out what was about leaving me with 4b^2*10=1 and rearranging for b but this was incorrect and the actual answer was 0.0845 and I have no idea how they got to that answer.

For the second part I realize that I have to intergrate the probability density in order to find the probability with an intergral of an upper limit of 30nm and lower limit of 20nm but I am also unsure of how exactly to intergrate the probability density
 

Attachments

topsquark

Forum Staff
Apr 2008
2,936
611
On the dance floor, baby!
The probability of the particle's position being between 20 and 30 nm is
\(\displaystyle P = \int _{20~nm}^{30~nm} \psi ^* (x) \psi (x)~dx\) as you say.

Since everything is nice and real all we need to do is square the wavefunction. This gives
\(\displaystyle \psi ^* (x) \psi (x) = \begin{cases} 0 & 0 \leq x < 10 \\ 9b^2 & 10 \leq x < 20 \\ 4b^2 & 20 \leq x < 30 \\ b^2 & 30 \leq x < 40 \\ 0 & 40 \leq x \end{cases}\)

So we have that \(\displaystyle \int _{20} ^{30} \psi ^* (x) \psi (x)~dx = 4b^2 \cdot 10\)

You didn't ask this so I'm assuming you know how to grok it: The total probability of the particle being anywhere is 1. So we know that \(\displaystyle 140b^2 = 1\). Thus we normalize our \(\displaystyle \psi (x)\) we get \(\displaystyle \int _{20} ^{30}\psi ^* (x) \psi (x) ~dx \to \dfrac{4b^2 \cdot 10}{140b^2} = \dfrac{40}{140} = 0.286\)

-Dan