x. If we try to draw such a wave, we fail.LxandLzwould also be
an incompatible set.Complete and compatible sets of quantum numbers
Let’s summarize. Just as we expect everyone to have a first and
last name, we expect there to be a complete and compatible set of
quantum numbers for any given quantum-mechanical system. Com-
pleteness means that we have enough quantum numbers to uniquely
describe every possible state of the system, although we may need
to describe a state as a superposition, as with the state`x= 0 in
figure d on p. 923. Compatibility means that when we specify a set
of quantum numbers, we aren’t making a set of demands that can’t
be met. These ideas are revisited in a slightly fancier mathematical
way on p. 987.
13.4.4 The hydrogen atomf/A cross-section of a hydrogen wavefunction.Deriving all the wavefunctions of the states of the hydrogen atom
from first principles would be mathematically too complex for this
book. (The ground state is not too hard, and we analyze it on
p. 931.). But it’s not hard to understand the logic behind the wave-
functions in visual terms. Consider the wavefunction from the be-
ginning of the section, which is reproduced in figure f. Although the
graph looks three-dimensional, it is really only a representation of
the part of the wavefunction lying within a two-dimensional plane.
The third (up-down) dimension of the plot represents the value of
the wavefunction at a given point, not the third dimension of space.
The plane chosen for the graph is the one perpendicular to the an-
gular momentum vector.
Section 13.4 The atom 925