Modern Physics notes Paul Fendley fendley@virginia.edu Lecture 6 ...

The fact that the wavelength and hence the momentum and energy of a particle in a box
are quantized is quite important. Let’s study the atom by making some more hand-waving
arguments. First think about the radial direction. The arguments above indicated that to
minimize the energy and hence be stable, the electron should remain within some radius a. This
doesn’t violate the uncertainty principle: we can’t predict precisely where it appears in this
region (although later we will be able to compute the probability amplitude). Thus this is like
confining a particle to a box of radius a, and so the radial momentum is quantized. The angular
component of the momentum is quantized for a similar reason. Think of an orbit at fixed radius
and momentum. Since we are now in quantum mechanics, this is a probability wave going around
the atom. Instead of demanding that the probability vanish at the edges, we demand that the
probability be periodic: if you go around once, the probability must be the same. Put another
way, one must have an integer number of wavelengths n around the orbit. Thus
where a is the radius of electron’s orbit.
nλ ∼ 2πa
Don’t take this argument too seriously, but the result is very important. The momentum
and energy of the electron in an atom are quantized. By similar reasoning, the angular
momentum of the electron is quantized as well. (The angular component of the ordinary
momentum is related to the angular momentum, but the two are not the same. How do they
differ?)
This now explains why we saw the distinct spectral lines in the mercury vapor. The vapor
in the lamp is fairly dilute, so the atoms are far enough apart so that they can be treated
individually. This means that the electrons in the atoms have distinct energy levels. To illustrate
this, physicists often draw the diagrams of the type you see in Fig. 2-9 of Feynman.
So how do discrete energy levels translate into what we see? When you heat a gas, or run
an electrical current through the gas, it puts energy into the system. This “excites” the atom.
This means that the energy you put in lifts the electrons out of the lowest energy levels and into
higher ones. But as we said before, the electron won’t stay there, if it can. It will fall down
into the levels with lower energy. So what happens to the now-excess energy. When falling
down levels, a photon is emitted! The energy of this photon is not arbitrary: when an electron
falls from energy E 1 to the lowest-energy state (the “ground” state) with energy E 0 , the photon
emitted must have energy E 1 − E 0 . But now quantum mechanics comes in again: we know that
this photon of fixed energy must have a fixed frequency. Thus this transition will emit a photon
of frequency
ν = E 1 − E 0
.
h
The discrete lines we saw in mercury arise from the various transitions in mercury.
Quantum mechanics therefore explains something completely obscure classically: the existence
of spectral lines. The existence of spectral lines with known frequencies was of vital
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