Finding the probability of an electron in the forbidden region

[reminiscent]

Homework Statement

For the hydrogen atom in the first excited state, find the probability of finding the electron in a classically forbidden region.

Homework Equations

E₁ = -Z²μe'⁴/2n²ħ²
ψ₂₀₀ =
∫ψ(r)²dr from the limit to ∞

The Attempt at a Solution

https://imgur.com/a/fWIMq

I feel like this answer is too weird... I would like my work to be checked on.

 

[CitizenSentry]

Bump.

 

[BvU]

Do I see an $e^{14}$ in your $E$ ?

 

[TSny]

I feel like this answer is too weird... I would like my work to be checked on.

Your work looks pretty good. However, you need to integrate over 3 dimensional space, not just over $r$. How would you express the volume element $dV$ in terms of spherical coordinates?

What is $Z$ for hydrogen?

For the $r$ part of the integration, let the variable of integration be $\sigma = r/a_0$. The lower limit will then become a simple number.

 

[TSny]

Do I see an $e^{14}$ in your $E$ ?

I think the OP is using $e'$ for the proton charge in order to distinguish it from the base of the natural log function.