These
quantum numbers describe electronic states of different energy and
geometry. One radical difference between the new quantum theory
and the Bohr theory is that we have to abandon forever any hope
of knowing the exact path of an electron around the nucleus. Classical
mechanics would have led us to believe that if we were skillful
enough, we could know the position of the electron at every instant
in time, and plot its precise trajectory. The Bohr theory replaced
a billiard-ball electron by a stationary wave around a circular
orbit, but still let us give a precise numerical value for the radius
of the orbit. Wave mechanics take even this away from us. Instead
of an electron's position, all that we know is the probability
that an electron will be at any selected point in space. Solving
the mathematical wave equations for an electron in a particular
quantum state (n,l,m)
around an atom yields a wave function, (x,y,z),
which varies from one point in space (x,y,z)
to another. The wave function, ,
has no direct physical meaning, but the square of the wave
function, ,
is proportional to the probability of finding the electron at point
(x,y,z) rather than somewhere else. The end
result, when plotted, is a fuzzy cloud of electron probability around
the nucleus for each (n,l,m)
quantum state. The density of the probability cloud at each region
in space represents the probability that the electron will be found
there, and not elsewhere.

These
electron probability clouds are sketched on the next
page for I = 0, 1, and 2, or for s,
p, and d states.