Radial distribution probability

[hellomister]

Homework Statement

When I'm trying to find the probability of finding an electron within a sphere of a certain radius, do i integrate the radial distribution probability function with respect to r from 0 to infinity? My book says the product of the radial distribution function times dr would give the probability, but I always thought you had to integrate it.

Homework Equations

n/a just looking for a simple answer to my question... i didnt show the homework problem cos i want to do it myself i just want this question answered.

The Attempt at a Solution

I've attempted the problem, i just integrated and I want to know if you should integrate.

 

[kuruman]

The probability of finding the particle in a shell of thickness dr having radius r is

$$dP=|\psi|^2\:4\pi r^2dr$$

The probability of finding the particle anywhere within a sphere of radius R is

$$P=\int^R_0 |\psi|^2\:4\pi r^2dr$$

Does this help?

 

[hellomister]

yes! helps a ton! Thank you.

 

[elibj123]

Notice that $$4\pi r^{2}$$ is the area of the surface of a sphere. This factor looks like that because you only have radial distribution, independent of azimuthal direction,
while if your wavefunction looks like $$\psi(r,\theta,\phi)$$ things will get a little more complicated, and will involve additional integrations.

 

[paranormal]

I can confirm that integrating the radial distribution probability function is the correct approach to finding the probability of finding an electron within a sphere of a certain radius. The product of the radial distribution function and dr is simply the differential element used in the integration process. This is a common approach in many areas of science, where we use integration to calculate probabilities or other quantities of interest. Therefore, if you have integrated the function and obtained a solution, you have correctly approached the problem. Keep up the good work!