Wave function for Hydrogen and Probability

[Twigs]

Hey, if anyone can throw in some thoughts I am a little lost. Not sure If I need to integrate, or what. Thanks for any help.

The wave function for a hydrogen atom in the 2 s state is:(attachment)


I need to Calculate the probability that an electron in the 2 s state will be found at a distance less than 3.00 a from the nucleus.

 

[dextercioby]

Of course you need to inegrate.The Probability is the integral of the probability density...

\mathcal{P}=\iiint_{\mathcal{D}} \left| \Psi \left(\vec{r},t\right) \right|^{2} \ dV

is the probability of finding the electron (infinite mass nucleus) in the domain \mathcal{D}\subseteq \mathbb{R}^{3}

Daniel.

 

[thepatientmental]

To calculate the probability, you will need to use the probability density function, which is given by the square of the wave function. So, for the 2 s state of a hydrogen atom, the probability density function would be:

P(r) = Ψ(r)^2

Where r is the distance from the nucleus and Ψ(r) is the wave function. Since you are looking for the probability at a specific distance (less than 3.00 a), you will need to integrate the probability density function from 0 to 3.00 a.

∫P(r)dr = ∫Ψ(r)^2dr

You can use any integration method to solve this integral. Once you have the value, it will give you the probability of finding the electron at a distance less than 3.00 a from the nucleus. Remember, the probability density function gives the probability per unit volume, so you will need to multiply the result by the volume of a sphere with radius 3.00 a to get the actual probability.

Hope this helps!