a/1. Eight wavelengths fit
around this circle (`= 8). This is
a standing wave. 2. A traveling
wave with` = 8, depicted ac-
cording to the color conventions
defined in figure u, p. 914.What properties should we use to classify the states? The most
sensible approach is to used conserved quantities. Energy is one
conserved quantity, and we already know to expect each state to
have a specific energy. It turns out, however, that energy alone is
not sufficient. Different standing wave patterns of the atom can
have the same energy.
Momentum is also a conserved quantity, but it is not particularly
appropriate for classifying the states of the electron in a hydrogen
atom. The reason is that the force between the electron and the pro-
ton results in the continual exchange of momentum between them.
(Why wasn’t this a problem for energy as well? Kinetic energy and
momentum are related byK=p^2 / 2 m, so the much more massive
proton never has very much kinetic energy. We are making an ap-
proximation by assuming all the kinetic energy is in the electron,
but it is quite a good approximation.)
Angular momentum does help with classification. There is no
transfer of angular momentum between the proton and the electron,
since the force between them is a center-to-center force, producing
no torque.
Like energy, angular momentum is quantized in quantum physics.
As an example, consider a quantum wave-particle confined to a cir-
cle, like a wave in a circular moat surrounding a castle. A sine
wave in such a “quantum moat” cannot have any old wavelength,
because an integer number of wavelengths must fit around the cir-
cumference,C, of the moat. The larger this integer is, the shorter
the wavelength, and a shorter wavelength relates to greater momen-
tum and angular momentum. Since this integer is related to angular
momentum, we use the symbol`for it:
λ=C/`The angular momentum is
L=rp.Here,r=C/ 2 π, andp=h/λ=h`/C, soL=
C
2 π·
h`
C
=
h
2 π`
In the example of the quantum moat, angular momentum is quan-
tized in units ofh/ 2 π. This makesh/ 2 πa pretty important number,
so we define the abbreviation~=h/ 2 π. This symbol is read “h-
bar.”
In fact, this is a completely general fact in quantum physics, not
just a fact about the quantum moat:Section 13.4 The atom 919