Quantum mechanics - probability of finding an electron

[endeavor]

Homework Statement

The wave function of an electron in the lowest (that is, ground) state of the hydrogen atom is
$$\psi(r) = (\frac{1}{\pi a_0^3})^{1/2} exp(-\frac{r}{a_0})$$
$$a_0 = 0.529 \times 10^{-10} m$$
(a) What is the probability of finding the electron inside a sphere of volume 1.0 pm³, centered at the nucleus (1pm = 10^-12m)?
(b) What is the probability of finding the electron in a volume of 1.0 pm³ at a distance of 52.9 pm from the nucleus, in a fixed but arbitrary direction?
(c) What is the probability of finding the electron in a spherical shell of 1.0 pm thickness, at a distance of 52.9 pm from the nucleus?

Homework Equations

$$|\psi(r)|^2$$

The Attempt at a Solution

(a) $$volume = 1.0 \times 10^{-36} m^3$$
using r = 0, the probability is 1.137 * 10^-16.
(b), (c) What equations should I use here?
$$R^2|\psi(r)|^2$$ ?
$$4\pi r^2 R^2|\psi(r)|^2$$ ?
but I don't have R...

 

[siddharth]

a) The probability of finding the electron with a real wave function $$\psi$$ in a small volume element $$d\tau$$ is $$|\psi|^2 d\tau$$. To obtain the probability of finding the electron inside the sphere, you integrate. Can you take it from here?

 

[endeavor]

I don't know how to integrate it, but my teacher said although the correct way is to integrate, we won't need to integrate...

 

[electrifice]

I would like some help on the same problem too... I'm not sure if I'm doing it correctly. The probability of finding the electron is given by ($$\Psi$$)$$^{2}$$dV... Though I know how to integrate I don't think its necesssary (we're not supposed to use integration). I am solving it by setting r=0 in the wave function, then squaring it, and multiplying it by dV, which I am taking to be 1.0 pm^3. I'm not sure if this is the correct way of doing it. Any help appreciated.