Approximating Probability for a Wave Function

[doggydan42]

Homework Statement

The wavefunction at t = 0 is given by
$$\Psi = N*e^{-\frac{r}{a_0}}$$
where $r = |\mathbf{x}|$. $a_0$ is a constant with units of length. The electron is in 3 dimensions.
Find the approximate probability that the electron is found inside a tiny sphere centered at the origin with $b_0 << a_0$ [Hint: the exact calculation is much harder than the approximate estimate]

Homework Equations

N was found in the previous part of the problem:
$$N = \frac{1}{\sqrt{\pi a_0^3}}$$

For probability:
$$P = \int_0^{b_0} |\Psi|^2 d^3x$$

The Attempt at a Solution

I first calculated $|\Psi|^2$.
$$|\Psi|^2 = N^2e^{-\frac{2r}{a_0}}$$

Using spherical coordinates to write the integral:
$$P = \int_0^{b_0} |\Psi|^2 d^3x = \int_0^{2\pi}\int_0^{\pi}\int_0^{b_0} N^2e^{-\frac{2r}{a_0}} r^2sin(\phi)drd\phi d\theta$$

Using $u = \frac{2r}{a_0}$ and simplifying the integral by plugging in N gives me:
$$P = \frac{1}{2}\int_0^{\frac{2b_0}{a_0}} e^{-u}u^2du$$

My issue is how to approximate it give $b_0 << a_0$.
All the answer choices are proportion to $\frac{b_0^2}{a_0^2}$ or $\frac{b_0^3}{a_0^3}$.

 

[vela]

If $b_0 \ll a_0$, then in the integral $u$ will be close to 0 so you can say $e^u \approx 1$.

 

[doggydan42]

If $b_0 \ll a_0$, then in the integral $u$ will be close to 0 so you can say $e^u \approx 1$.

Would I integrate first then approximate it, or use $e^{-u} = 1$ inside of the integal such that the integral just becomes the integral of $u^2$?