In a previous message the question came up of whether atomic states have
quantized energy. People commonly assume the answer is "yes" but I would
like to argue that "yes and no" is a better answer. Specifically:
* Yes, if you measure the energy, you will force the atom into a state of
quantized energy.
* OTOH there are other perfectly reasonable states other than these
quantized-energy states.
The rest of this note explains this point in a little more detail.
1) Consider Feynman's classic discussion of a spin-1/2 atom going through
modified Stern-Gerlach machines.
* In a Z-axis analyzer, the atom takes the +Z path or the -Z path, never both.
* If the atom takes the +Z path in one analyzer, it will take the +Z path
in all subsequent analyzers.
We can summarize by saying that the filters in the set {+Z, -Z} are
idempotent, mutually orthogonal, and form a complete set.
From this, it would be tempting to conclude that the atom always "is" in
the +Z or -Z state, but that would be a mistake, as discussed in item (4)
below.
2) Imagine that (in addition to whatever inhomogeneous fields there are
inside the S-G machines), there is an overall field in the Z direction so
that the +Z and -Z states have different energies (say 1 and 0, respectively).
3) Consider using an electromagnetic process to flip the atom from the -Z
state to the +Z state. In the language of photons, we say that the atom
absorbs exactly one photon. Equivalently, in the language of NMR, we say
that we have applied a pi pulse. It corresponds to grabbing the atom and
turning it 180 degrees.
4) Now instead of a pi pulse, let's apply a pi/2 pulse. This corresponds to
grabbing the atom and turning it 90 degrees. This puts the atom into a
state that is a superposition of the +Z and -Z states. This state is not an
eigenstate of the parity operator. It is not an eigenstate of the energy
operator. But it is still a perfectly fine state.
It would be too crude to say that this state is "halfway between" the +Z
and -Z state. If you try to measure it using the energy operator, you will
get the E=1 and E=0 eigenvalues with 50/50 probability, and you will never
observe the halfway-between energy.
By the same token it would be too crude to say that in an ensemble of such
atoms, half "are" in the +Z state and half "are" in the -Z state.
The truth is that all such atoms are in the same state. This is a perfectly
fine state, and there are observations we can make to confirm the atoms are
in this state. Energy measurements are not the world's only measurements.
Perhaps it can be summarized this way: The quantized energy states are
favored states of the combination of atom plus energy-measuring apparatus.
They are not favored states of the atom by itself.
Similar remarks apply to systems other than atoms, including modes of the
electromagnetic field and other harmonic oscillators.