Quantum mechanics - probability of finding an electron

1. 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?

2. Homework Equations
$$|\psi(r)|^2$$
3. 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...

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siddharth
Homework Helper
Gold Member
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?

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....

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.