17.7 The Intrinsic Angular Momentum of the Electron. “Spin” 755
nucleus, then moving toward the nucleus, passing through the nucleus and repeating
the motion on the other side of the nucleus. Sometimes it is best not to visualize things
classically.PROBLEMS
Section 17.6: The Time-Dependent Wave Functions
of the Hydrogen Atom
17.38Describe verbally the how an electron appears to move
around the nucleus in the 211 state and in the 21,− 1
state.
17.39a.The energy of the 211 state of the hydrogen atom
relative to the ground-state energy is equal to
10.20 eV 1. 6340 × 10 −^18 J. Find the frequency of
oscillation of the orbital using this energy. Remember
that the frequency of oscillation of a de Broglie wave
depends on the choice of the zero of energy. Only
differences in frequencies are meaningful.
b.Find the angular speed around thezaxis of the angular
nodes of the wave function in radians per second for
the 211 state of the hydrogen atom. Convert this to
revolutions per second.
c.Assume that the electron is orbiting around thezaxis
at a distance equal to 4awhereais the Bohr radius.
This distance is equal to the most probable distance for
the 211 state of the hydrogen atom. Assume that
Lzh ̄and calculate the angular speed using the
classical formula
Lzmrω^2
whereωis the angular speed in radians per second.
Compare this result with the angular speed of the
nodes in parts a and b.
d.Calculate the speed of the electron with the same
assumptions as in part c.17.7 The Intrinsic Angular Momentum
of the Electron. “Spin”
The complex orbitals withm0 correspond to revolution of the electron around the
nucleus. The angular momentum of this motion is called theorbital angular momen-
tum. Electrons are found experimentally to have an additional angular momentum that
is not included in the Schrödinger theory. To obtain adequate agreement with exper-
iment this feature must be added to the Schrödinger theory. We call it theintrinsic
angular momentumor thespin angular momentum. There is a version of quantum
mechanics that is compatible with special relativity, based on the Dirac equation.
The intrinsic angular momentum occurs naturally in this theory, which we do not
discuss.^5The Dirac equation is named for
P. A. M. Dirac, 1902–1984, a famous
British mathematician and physicist who
shared the 1933 Nobel Prize in physics
with Schrödinger.
EXAMPLE17.9
Calculate the expectation value of the square of the speed of the electron in a hydrogen atom
in the 1sstate, and from this calculate the root-mean-square speed. Compare this speed with
the speed of light.
(^5) F. Mandl,Quantum Mechanics, Butterworths Scientific Publications, London, 1957, p. 203ff.