this degeneracy will be doubled to 8 when we take into account the
intrinsic spin of the electron, sec. 13.4.6, p. 934). The degeneracy
of the differentzstates follows from symmetry, as in our original example of degeneracy on p. 920, and is therefore exact. The de- generacy with respect to different values offor the samenis not
at all obvious, and is in fact not exact when effects such as rela-
tivity are taken into account. We refer to this as an “accidental”
degeneracy. The very high level of degeneracy in the hydrogen atom
means that when you observe it the hydrogen spectrum in your lab
course, there is a great deal of structure that is effectively hidden
from you. Historically, physicists were fooled by the apparent sim-
plicity of the spectrum, and more than 70 years passed between the
measurement of the spectrum and the time when the degeneracies
were fully recognized and understood.
Figure h on page 928 shows the lowest-energy states of the hy-
drogen atom. The left-hand column of graphs displays the wave-
functions in thex−yplane, and the right-hand column shows the
probability distribution in a three-dimensional representation.Discussion Questions
A The quantum numbernis defined as the number of radii at which
the wavefunction is zero, includingr=∞. Relate this to the features of
figure h.
B Based on the definition ofn, why can’t there be any such thing as
ann= 0 state?
C Relate the features of the wavefunction plots in figure h to the
corresponding features of the probability distribution pictures.
D How can you tell from the wavefunction plots in figure h which ones
have which angular momenta?
E Criticize the following incorrect statement: “The`= 8 wavefunction
in figure f has a shorter wavelength in the center because in the center
the electron is in a higher energy level.”
F Discuss the implications of the fact that the probability cloud in of the
n= 2,`= 1 state is split into two parts.13.4.5 Energies of states in hydrogen
History
The experimental technique for measuring the energy levels of
an atom accurately is spectroscopy: the study of the spectrum of
light emitted (or absorbed) by the atom. Only photons with certain
energies can be emitted or absorbed by a hydrogen atom, for ex-
ample, since the amount of energy gained or lost by the atom must
equal the difference in energy between the atom’s initial and final
states. Spectroscopy had become a highly developed art several
decades before Einstein even proposed the photon, and the SwissSection 13.4 The atom 927