# Chemistry - Most probable point for finding an electron in the 1s orbital of a Hydrogen atom

## Solution 1:

It makes a difference if you look in a point or in a volume element.

For the radial distribution we sum up the probabilities within (very thin) shells at different distances from the core. If the radius get's bigger the volume in this shell gets bigger. Remember that the surface of a sphere goes by $4 \cdot \pi \cdot r^2$ so the volume of the shell is essentially for all practical purposes $4 \cdot \pi \cdot r^2 \cdot \Delta r$. At the core the radius is zero which means the volume is zero and thus the probability is zero too.

At the same time the probability of finding the electron in a point at a certain distance goes down, but the volume goes up. Because of the scaling of the volume and the "decay" of the probability to find it in a point further away it turns out that there's a maximum of finding the electron at a certain distance away from the core (but not a single point at that distance).

## Solution 2:

I provide here a few supplementary comments; the major part of your question has been answered by DSVA.

For the electron's position in a 1s orbital, the probability density $\rho(r)$ is maximal at the origin, whereas the radial probability density $p(r)$ is maximal at the Bohr radius.

*probability*versus*probability density*: The probability density integrated over a volume yields a probability. We cannot talk about probabilities at a point for a continuous system because a point has measure zero---that is, if we make a measurement of a random variable for a continuous system, we will never get exactly the value represented by the point. Instead, we need to consider a higher-dimensional object, which has non-negligible measure. A natural choice is a shell centered around the origin with radius $R$ and thickness $\Delta R$, assumed small. We can find the probability that the electron is located within the shell as $$P(R < r < R+\Delta R) = \int_\text{shell}\mathrm{d}\mathbf{r}\,\rho(r) = \int_R^{R+\Delta R}\mathrm{d}r\,4\pi r^2\rho(r)\approx 4\pi R^2 \rho(R)\,\Delta R.$$ Motivated by this calculation, we define the radial probability density $p(r) := 4\pi r^2\rho(r)$, which is a more important measure of where the electron actually is, because of the radial symmetry in the system---we are more interested in the distance between the electron and the origin than in the position vector of the electron.*probability density*versus*radial probability density*: The probability density takes as its fundamental unit a point---imagine a cube of infinitesimal volume, as your professor uses. The radial probability density takes as its fundamental unit a sphere---imagine a shell of infinitesimal width, as before. We will restate the contents of the first bullet point in this context: The probability of finding the electron in a given infinitesimal cube is maximized for a cube centered about the origin. The probability of finding the electron in a given infinitesimal shell is maximized for a shell at the Bohr radius. In the latter case, although the probability density is lowered for large $r$, the increased surface area means that there is more space we can hope to find the electron in, and the competition of these two factors result in a maximum for the radial probability density at the Bohr radius.