Probability Electron at Point: H Atom

atomicpedals
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The problem I'm working on asks: At what radius does the probability of finding an electron at a point in the hydrogen atom fall to 50% of its maximum?

I've already worked through the problem to find the probability at the maximum, which was to take the first derivative of r^2 e^(-2r/a) and then set it equal to zero and solve for r (results in r = a). For the 50% of maximum do I simply set that derivative equal to 0.5? Sent from my iPad using Physics Forums
 
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ah, no that wouldn't get the right answer. the question is about probability. you have the wavefunction, so how do you calculate the probability of finding the electron at a certain place?
 
Integral of the probability density; from 0 to 2pi in this case (I think).
 
you don't need to do any integration. what's important is the probability density. if the probability of finding an electron in one place is 1/2 the probability of finding it in another specific place, then what can you say about the ratio of the probability density at these two points?
 
Hi, I had an exam and I completely messed up a problem. Especially one part which was necessary for the rest of the problem. Basically, I have a wormhole metric: $$(ds)^2 = -(dt)^2 + (dr)^2 + (r^2 + b^2)( (d\theta)^2 + sin^2 \theta (d\phi)^2 )$$ Where ##b=1## with an orbit only in the equatorial plane. We also know from the question that the orbit must satisfy this relationship: $$\varepsilon = \frac{1}{2} (\frac{dr}{d\tau})^2 + V_{eff}(r)$$ Ultimately, I was tasked to find the initial...
The value of H equals ## 10^{3}## in natural units, According to : https://en.wikipedia.org/wiki/Natural_units, ## t \sim 10^{-21} sec = 10^{21} Hz ##, and since ## \text{GeV} \sim 10^{24} \text{Hz } ##, ## GeV \sim 10^{24} \times 10^{-21} = 10^3 ## in natural units. So is this conversion correct? Also in the above formula, can I convert H to that natural units , since it’s a constant, while keeping k in Hz ?
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