s-orbitals
The lowest energy state of the H atom is the 1s orbital, for which
| ψ1,0,0 = ( | 1 | )½ |
| |||||
| πa03 |
Clearly the 1s orbital is spherically symmetric, but ψ depends on distance from the nucleus, decaying exponentially from its value at r = 0 (what is that value – you can calculate it!); the most probable point at which the electron will be found is at the nucleus. All s orbitals are spherically symmetric, differing in the number of radial nodes. For instance the 2s orbital has a single node where
| 2 - | r | = 0 ie. r = 2a0 |
| a0 |
We see that as n increases, the average distance of the electron from the nucleus increases. This average value is the expectation value
<r> = ∫rψ2dτ
where dτ = r2dr sin θdθdφ. But the angular integration yields 1 because the spherical harmonics are normalised. Thus
<r>1s = &int0∞ r(R1, 0)2r2dr
and you should be able to show that this equals 3a0/2.
Similarly it can be shown that <r>2s = 6a0.
