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Old May 21st 2019, 05:16 PM   #1
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Probability graphs and reading them

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
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Old May 21st 2019, 07:18 PM   #2
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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 $\displaystyle \int _{20} ^{30}\psi ^* (x) \psi (x) ~dx \to \dfrac{4b^2 \cdot 10}{140b^2} = \dfrac{40}{140} = 0.286$

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