How to determine the region that would contain a quantum particle

(a) A hydrogen atom is in its ground state. If space is divided into identical infinitesimal cubes, in which cube is the electron most likely to be found? If instead space is divided into 31 concentric shells of infinitesimal thickness, like the layers of an onion, centered on the proton, what is the radius of the shell in which the electron is most likely to be found?

I believe it is safe to say $\psi(x,y,z)$ is

$$\sqrt{\frac{8}{abc}}\sin\biggl(\frac{n\pi x}{a}\biggr)\sin\biggl(\frac{n\pi y}{b}\biggr)\sin\biggl(\frac{n\pi z}{c}\biggr)$$

(obtained from 3-d schrodinger eq.) For a single cube, I would say the highest probability is at position $(\pi/2, \pi/2, \pi/2)$ because $\sin$ is max at these positions. However, the part that confuses me is the infinitesimal cubes. Could I simply state that the cube that would contain the hydrogen is the cube with center $(\pi/2,\pi/2,\pi/2)$?

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Oh. thanks for letting me know! This is my first question and I just signed up today. –  user26658 Jul 4 '13 at 20:47
No worries ;-) I think your edit takes care of the issue, so I'll reopen this for you. –  David Z Jul 4 '13 at 22:25
You may have been misled by the wording of the question. The physical system you are being asked to think about is the ground state of the hydrogen atom; not the wave function you've written. The "infinitesimal cubes" in the text are volume elements in which you are expected to evaluate the probability of finding the electron (they must have some size or $P = \int_V \mathrm{d}V\, \psi^*\psi$ goes to zero). Your solution to the 3D-infinite square well looks fine up to the constant which I have not verified, but does not apply to this question. –  dmckee Jul 4 '13 at 22:54

The wavefunction squared (strictly speaking $\psi^*\psi$, but this is equal to $\psi^2$ if the wavefunction is real) gives you the probability density of finding the electron. The probability of finding the electron in some infinitesimal volume $dV$ is given by the probability density times the volume:

$$P = \psi^2 dV$$

So if you divide space up into lots of identical cubes of volume $dV$ the cube with the highest probability of finding the electron is the one where $\psi^2$ is a maximum. The ground state of the hydrogen atom is:

$$\psi_{100}(r) = A e^{-r/a_0}$$

where $A$ is a normalising constant and $a_0$ is the Bohr radius $\hbar^2/me^2$, or about 0.0529 nm. Note we generally express the wavefunction in polar co-ordinates since these make it simple. So you just need to work out where $\psi^2(r)$ is a maximum.

The second part of the question is a bit more subtle, because it's asking what spherical shell is most likely to contain the electron. The point of the question is that the volume of the spherical shell of radius $r$ and thickness $dr$ is $dV = 4 \pi r^2 dr$, so the probability of finding the electron in this shell is given by:

$$P = \psi^2 dV = A e^{-2r/a_0} 4 \pi r^2 dr$$

You need to find the value of $r$ at which this function is a maximum.

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Thank you! I figured it out yesterday and I wanted to respond, but unfortunately I have not figured out how to get mathjax (I found the how to, but finally I have time this weekend to get it up and running), so I did not want to give a sloppy reply. –  user26658 Jul 6 '13 at 16:00