Quantum Mechanics for Engineers 

© Leon van Dommelen 

N.8 Why the s states have the least energy
The probability of being found near the nucleus, i.e. the origin, is
determined by the magnitude of the relevant hydrogen wave function
near the origin. Now the power series expansion of
in terms of the distance from the origin starts with
power , (D.8). For small enough , a p,
(i.e. ), state involving a factor will be much smaller
than an s, (), state without such a factor. Similarly a
d, (), state involving a factor will be much less
still than a p state with just single factor , etcetera. So
states of higher angular momentum quantum number stay increasingly
strongly out of the immediate vicinity of the nucleus. This reflects
in increased energy since the nuclear attraction is much greater close
the nucleus than elsewhere in the presence of shielding.