Physics Help Forum Probability graphs and reading them

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 May 21st 2019, 05:16 PM #1 Junior Member   Join Date: May 2019 Posts: 4 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 Attached Thumbnails
 May 21st 2019, 07:18 PM #2 Forum Admin     Join Date: Apr 2008 Location: On the dance floor, baby! Posts: 2,681 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 studiot and CBM like this. __________________ Do not meddle in the affairs of dragons for you are crunchy and taste good with ketchup. See the forum rules here.