Probability Electron at Point: H Atom

[atomicpedals]

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

 

[BruceW]

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?

 

[atomicpedals]

Integral of the probability density; from 0 to 2pi in this case (I think).

 

[BruceW]

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?

 

[paranormal]

App

Based on the information provided, it seems like you are on the right track in solving this problem. To find the radius at which the probability of finding an electron at a point in the hydrogen atom falls to 50% of its maximum, you can set the first derivative of the probability function equal to 0.5 and solve for r. This will give you the radius at which the probability is 50% of its maximum value. Keep in mind that this is just an approximation, as the probability function is continuous and the exact value may lie between two points. Additionally, it is important to consider the units of your answer, as r is typically measured in units of the Bohr radius (a). I hope this helps!